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A Practical Introduction
Stoclmstics Cfkfclfllfs; Practical fafrtdffclcn, Richazd Dllrrette ./1.
Chass Expfmsinn,Multiple Wcincr-ln Integrals and Victor Perez-Abreu
Probability and Stochascs Series
fmff
White Noise Dfsfrfsllfion Theonol,Htti-l-lsiung Kuo Topics n Cnntempnt.y Probabiliiy, J. Lattrie Snell
Applications, Chz-isti.artHoudre
Richard Durrtt C Boca Raton
C
CkC Press New York London
Tokm
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rtial diferential book (Chaptys 1-5). To this I have >ddd equations tq be the core pf other practically important topics: on dimensipnql diffuppp, semigrqups @nd geperators, Harri, chaipsi and peak convergence.. I have struggled Fith this materi>l for almst twepty years. I zpF Shipk that J Jmderstand most of it, ( ' sp to help you mmster it in less time, t have tmed to pxplain iti ms simply and clearly as I can. My students? motivations for learning this material have been diveyse: some have wanted to apply idemsfrom probability to analysis or diflbrential geometry, others have gone on to do zesearch on difusiop prcsses or stochmstic partial of thes ideas diferential equatins, some have been inte#ested in applications prpblemK ih operations reseyrch. My motiyation for writing to fnance, or to this book, like that for Pvobabilily Teorg and fzcpvled, wws to simplify my life as a teacher b# bringin: together in one plae useful material tht is scattered iu a variety of souicesk tfif you copy from e book that i plagiarism, but ' An old joke says that scholaiship.'' boks that is from From that viewpoint this is tez if you' copy contributors various subjects are (a)stochatic main book. the for Its cholarly and equations: and lkeda artd diferential Chung Williams integration (1900), Wataabe (1981), Revuz and Yor Karatzmq and Shreye (1991), Protter (1990), Stroock and Varadhan (1970); (1991),Rogers and Williams (1987), (b)partil equations: Folland Diedman Port diffrential (1976), (1964), (1975), and Stohe and Karlin and Zhao dimnsional diflksions: Chung (1978), (1995);(c) one and Yor semi-groups and Taylor (198.1)) Dynkin Revuz generators: (1965), (d) and Billlngsley weak Ethier Kurtz (1991); (e) convergence: (1986), (1968), If you bought a1l those books you would spend Stroock and Varadhan (1979). more than $1000 but for a fraction of that cost you can have this book, the intellectualequivalent of the ginzu knife. ,
'eading
.moye
,thi;
.
.
.
,
.
This book contains information obtained from authendc and highly regarded sources. Replinted material is quoted with pennission, and sources are indicated. A wide variety of refemnces are listed. Reasonable effortq have beert made to publislt reliabl data and information, but the autlzor atd the publisher cmnnot assume responsibility forthe validity of all materials or for the comsequence,sof theiruse. Neithertls booknoruypartmaybereproduced rtmnsmitted in any fonn orby anymeans.electronic or mechanicak including photocopying, microfllming, nd iecording, or by any information stomge or retrieval systenx witlmut prior permission in writing hp4p tlle publisher. CRC P=s, lnc.'s consent does not extend to copying for geneml distribution, for promodon: for creating new works, or for resale. Specitk pennission mstb obtained in wtiting from CRC Press for such copying. Direct all inquirie.sto C'RCPrss, lnc., 2000 Corpomte Blvd.. N.W., Boca katon,Florida 33431. g
@ 1996 by CRC Pmss, lnc. No clnim to original U.S. Government works Intemadonal Standard Book Number 0-8493-8071-5 Plinted in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acide,e paper
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vi
Preface
Shutting off the laugh-track and turning on the violins, the road from this book's frst publication in 1984 to its rebirth in 1996 hmsbeen a long and winding second half of the 80's I accumulated long list an embarrassingly one. In the edition. Some time at the beginning of the 90's I talked of typos from the flrst to the editor who brought my first three books in t o the world John Kimmel about preparing a second edition. llowever, after the work was done, te second edition was personally killed by 2ill Roberts, the President of Brooks/cole. At the end of 1992 I entered into contract with Wayne Yuhasz at CRC Press produce this book. In the first few months of 1993, June Meyerman typed to of book into TeX. In the Fall Semester of 1993 I taught a course from the most this material and began to organize it into the current form. By the summr of 1904 I.thought I was almost done. At this point 1 had the good (and'bad) fortune of having Nora Guertler, a student from Lyon, visit for two months. When she was through making an average of six corrections per pagej it was clear that the book was faz from flnished. During the 1994-95 academic year most of my tine was devoted to preparing the second edition of my qrst year graduate textbook Probability: Tct)r.q and Eamples. After that experience my brain cells could not bear to work on another book for another several months, but toward the end pf 1995 they decided is now or never.'' The delightftll Cornell tradition of a long winter break, which for me stretched from early December to late January, provided just enough time to finally fnish the book. I am grateful to my students who have read various versions of this book and also made numerous comments: Don Allers, Hassan Allouba, Robert Battig, Marty Hill, Min-jeong Kang, Susan Lee, Gang Ma, and Nikhil Shah. Earlier in the process, before 1 started writing, Heike Dengler, David Lando and I spent and Jacod and Shiryaev (1987), an enterprise a semester reading Protter (1090) ( which contributed greatly to my education. The ancient history of the revision process hms unfortunately been lostk At the time of the proposed second edition, 1 transferred a number of lists of typos to my copy of the book, but 1 have no record of the people who supplkd the lists. 1 remember getting a number of corrections from Mike Brennan and Ruth Williams, and it is impossible to forget the story of Robin Pemantle whq took Bvownian Motion, Martingales and Analysis as his only math book for a yearr long trek through the South Seas and later showed me his fully annotatvd copy. However, I must apologize to others whose contributions were recordd but and 1'11have something whose names were lost. Flame me at rtdllcornell.edu to say about you in the next edition. ,
,
About
the Author
'a
'
'tit
'
.
.
.
.
.E
. .
Rick Durrff
,
.
( ' . . . .
'. . . '
Ttick Durrett received his Ph.Dt in Opeitions research from Stanford University in 1976. He taught i ihe Mathematics Deprtment at UCLA for nine years before becoming a Professor of Matheniatics at Cornell University. He ws a Sloan Eellow' 1981-83, Guggenheim Fellpw 1988-89, and spoke at the ! International Congress of Math in Kyoto 1990. of > gradlpt tltbppk, Prqbqbility: Teorg and Dmrett is the The Eqentials of Jiropllip. 'zcmpled, and an tudergpdu>ty He hms one, ujjjj co-aythorj an' d en 19 students taj jj written ( almost 100 papers > 9 u. comylete their Ph.D.?s. under his dtxciippr lds recept rrsearc fjcuses n the spatial mpdl: it? vayiolp problerns in zlor. appllcations of stochuti '
E
Eauthor
EE
'
Calculus:
Stochastic 1.
Introduction
A Practical
5. Stochastic Diferential Equations 5.1 5.2 5.3 5.4 5.5 5.6
Brownin.n Motion 1.1 1.2 1.3 1.4
Deflnition and Construction 1 Markov Propexjy, Blumenthal's 0-1 Law 7 Stopping Times, Stropg Markov Property 18 First Formulms 26 ; '
E
,
2 Stochastic lntegraon 33 2.1 Integin ds i P !!Qitble Pr'eje Lc>l Mitikl Integrators: Continus 2.2 .
t.3
.
.
6. One Dimensional Difusions
;
.
.
.
.
'
.j
37
wa.tt
'
'
'
i
Reculzence and Transience 95 Occupation Times 100 Exit Times 105 Change of Time, Lvy's Theorem 111 Burkholder Davis Gundy lnequalities 116 Martingales Adapted to Brownian Filtrations
4. Partial Diferential Equations
B.
4.4 4.5 4.6 C. 4.7 4.8 4.9
Parabolic Equations The Heat Equation 126 The Inhomogeneous Equation 130 The Feynman-lfac Formula 137 Elliptic Equations The Dirichlet Problem 143 Poisson's Equation 151 Equation 156 The Shrdinger Applications to Brownian Motion Exit Distributions for the Ball 164 Occupation Times for the Ball 167 Laplace Transforzns, Azcsine Law 170
'
Constructiou 211 Feller's Test 214 Recurrence and Qansience 219 Green's Ftmctions 222 Boundary Behavior i29 Applications to Higher bimensions234
7. Difusions as Markov Processes 7.1 7.2 7.3 7.4 7.5
8.
3. Brownirm Motion, 11
A. 4.1 4.2 4.3
.. .
and
E
3.1 3.2 3.3 3.4 3.5 3.6
6.1 6.2 6.3 6.4 6.5 6.6
E ('
Covariahe Prcse 42 Bded 52 Myitinzle Ih t 4 tity 59 2.5 The Kit-Watab 2.6 Integration w.r.t. Local Maztingales 63 2.7 Change of Variables, lt's Formula 68 2.8 Integration w.r.t. Semimartingales 70 2.9 Associative Law 74 2'.10Ftmctions of Several Semimaztingales 76 Chapter Summax.y 79 2.11 Meyer-rfnaka Formula, Local Thne 82 2.12 Girsanov's Formula 90 Variane
2.4 Integration
Examples 177 It's Apmoach 183 Extension 190 Weak Solutions 196 Change of Memsure 202 Change of Time 207
119
Semigroups and Generators 245 Examples 250 Aansition Probabilities 255 Harris Chains 258 Convergce Theorems 268
s7eakConvergence 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8
ln Metric Spaces 271 Prokhorov's Theorems 276 The Space C 282 Skorohod's Existence Theorem for SDE 285 Donsker's Theorem 287 The Space D 293 Convergence to DiAsions 296 Examples 305
Solutions to Exercises 311 References 335 lndex 339
1
Motioh
Btownian
''''''' 1.1. Defnltlop
Constructlon
and
yyytj (yyjytyjjoj k( sjju tvaut, . . (. . . .wit . aftei . birtlz of a . lili w itl b ible redtiin: th deqittn of a to have fun with our new arrival. We begin by starting with point d-dimensional Brownian motion to th>t 9 a one a geneqal itikt tw sttie t , (1r >nd (tiifl il s ioal BlWi rfttitt titi: t 0 (t * 2) 1 re part of tmi . .
Vl
jt
iS g 0llOh
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the
ja u o jsu ziow att
def/ child, i mssf g
We Will
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.
:
tzt
.
'jaihful,
. .
g
.
. . .g
.
'
t)
'lt
.
tteqnitioh.
(1.1) Tzmnslatlon hwarlnnce. (.Df Bz, k 0Jis independent of Bz the same distributioh as a Brownia motio with Bz 0. =
of coordlnate (1.2) Independence tj V g jwyj jgyllj fnotions startig Brownian dimensional 0) are independent one
Uj
g
j
: .
kw. .
.
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('
E'
* dte Process Bt .
,
g.
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. . . .. ..
k
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k 0 then
.,
..
:j .
..
...
zotin
.
.
+ t)
-
e .A) s
Bst
..
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() With p rpbability
one,
,
Bz
.5401)
=
Btc),
.
.
.
Btn)
.
.
y
(j
j
E
.....:.
tt
..
j
$
-
'
be a
.
:
.
.
:
B (t,z-1)are
-
z::
/..a(2vt).7'/2exp(s
' . ..
0 and t
.
. -+
t
is
.
.
continku:
:, '
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k
. .. . at ktaiung ..
.
.
.
.
.
)
.
.
ta then Btc),
t hms the fpllowing profitisi
(a) If to < z < independent. (b). If s, t
.
.
x on dimnsiiat
5
hms
nd
-'.
.
.
. :
' .
.
'
.
the
inrement lkmsindependent increments. () says th>t (a) says tat Bs + t) W(s)haq a normal distribution witlt mean vectpr 0 >nd variance t. The reader slkould note that above we have sometimes (c) is self-explanatory. -
Capter
2
1 fz.owan
we will contoirme in what
For each 0
the defnition that will be useful myuy times is:
#r,4z,...xatXl
wzitten Bs and sometimes follows.
written
An immediate consequence
of
(1.3) Scnllng relatlon.
lf Bz
.8(s),
a practice
0 then for any t
=
.(l,z,s z 0).
zfc
ftlzBs,
>
0,
where zp
k 0).
s
To be precise, the two families of random variables have the same flnite dimepsional distributions, i.e., if sz < < sg then .
.
(sazt, .
.
d (t'/2s,,,
../,-.,)
..,
,tl/2.s,-) ..
.
'
.
:
'
'
'
ln view of (1.2)it suces to prove this foi one dimensional motion. To check this when n q.z: 1, we note that 1/2 times a pormal Fith me>n 0 and s ls a ppifnt withmean 0 and variance si. The resvlt torn $ 1 toltows vayiqpce from indpendent increments. '
y
defttick' E' .
. . .witl .
frofn Bz 0, Which .we Va ltle cl PTOCeSS Sa tisf y in g =i
..
zrowntan
.
; q i seckztt equiilent
(a#) Bv is Gaussinn multivariate normal),
:
.
.
,
y
'
.
.
.
.
.
:
.
:
. .
.
.'
'
,
.
.
.
(i.e.,all
-
-
t S
m,,,,...,a (.A1 x
J-field
.
-
.
x .A,,)
Thij follow: from a generalization of Kolmogorov's extension theorem. We will not bo ther with te details since at this point we are at the dead end referrd to above. If C (td: t td(t) is continuous) theq C / So, that is, C is not a measurable set. The easiest way of proving C Fo is to do --+
=
6 So if and only if there is a sequence of times t1, tz , J-feld T xR k so that e (0,((J e T?/1,2,...) (theinfnite product a1l words, = Bj. In events in Fo depend on only : @(t1),(f2), .4 countablymauy coordinates.
Exerclse x)
.4
1.1. and > B
...
.)
.
...)
.
Capter
4
1
Prownan Mbton
The above problem is easy to solve. Let Q2 (m2-n: m, n k 0) be the (tJ : Qa dyadlc rationals. R) and Fq is the c-fleld generated by the If fl flnite dimensional sets, then enumerating and applying the rationals 1, q,, Kolmgorov ' s ex t ens ion theorem shows that we can construct a probability v z) : tJ(0) 1 and (+)in (1.4)holds when the on (f1gFq) so that vrf x) we will show: ii e Qz. To extend the defnition from Q2 to g0, ::zq
=
-+
.
.
.
.
=
,
=
(1.5) Theorem. Let T < (x) and z E R. vo mssignj probability tJ : Qa R that are uniformly continuous on Q2n (0,T).
with
sl1+G (1.6) Theorem. Suppose that Epxj .X)# % ff lt alp then with probability one thete is a contant Cw) 7
0, Q ((,./) (p(aY(./2-n),X(2-n)) Gn S j and t/P(IYl > c) S E lFl# we have .f
.
... .
P.
=
vc o
#-
#(G
((J
)S
i
KK
'
=
,
-+
S J'f 2n2n9
PG') where enough
=
so
.
p)(1 + a > 0. To that
(1
pyj
-
-
(1.7) Lemma.
Let
(, jj
-
complete
=
S Xl
-
rl-/
2-'T).
-
(1.6) follows
Proof of (1.5) By (1.1)and (1.3),we can without loss of generality supyose Bz 0 and prove the result for T 1. In this case, part (b)of the deqnltion and the scaling xelation (1.3)imply =
PH'
.E'n(If
/,14)
=
fnlf-,I4
=
Ct
-
s)2
where C = .E'1/114< x. From the last observation we get the desirqd uniform continuity by using a result due to Kolmogorov. In this proof, we do not use the independent increments property of Brownian motion; the only thing we
N
xxGn
=
with 1
rl
-
s
we
have
2-(1-8)X
(1.7):
from
G)
S
)S
On #w'
Q2n (0,1)
(x7
=
-
euily
we can pick p small show
for q, r E
E
p(dz).%(.)
2n9J
Since
ffz-''
=
p). Since y < alp, the prf ow we will
=
=
Remark. It will take quite a bit of work to prove (1.5).Befoye takipg pn t/ tmsk$ we will attend to the lwst memsure theoretic detail: we tidy tlkings up y = (t2 : movingour proability measures to C, c) where (continuous (0,x) R) and is the J-field generated by the coozdinate maps t w(), To do this, we observe that the map # that takes a uniformly continuous point jzi fl to its and we set extension in C is meyurable, ;
and Construction
Denition
Section 1.1
PGn'
n=N
) S ff
S
nN
rl-/ for This shows pxg, .X'rl < I rl pxq, Xr) S C(Y)I rl-f for q, r e (0,1). .1:
-
-
gg-N
2-n
% :(a)
=
1
-
2-
and implies that we have
-
Proof of so that
(1.7)
Let q, r E
Q2f3 (0,1) with 0
ths
with exporent
+
-
.p(.rt)
s
Exerclse 1/2 + 1/
The scaling relation
So using
If1/aIS
izic
'
3
-,z
.
exptvzi/z) 1. Letting n x show, q all emsing d cotpletes hows /(.A r. ) t f6r i r n
:
. '
i 3Cv 2(l-0hjr .
shows
'
.
5c' a n 5C 3
n P
=m
n motion then is to introduce te measure theory necessary to state and prove the result. The flrst step in doing this is to explain what we mean by ttwhat happened before time s.'' The technical name for what we are about to deqne is a flltrat 1 h a f n y term for an increasing collection of flelds F , i e if s < t thz F a c F f. Since we want Ba e $a,' i.e., B, is meadurale with respc' t t Fa', tlte flrst thing that comes to mind is ,
,
F:
tztfr
=
:
r
,
.
it.is convenient y+
to
-+
=
y
o
=
fzh
Likewise, if F4)
0,
J(patJ(t))
=
,
F
.
.
o
.tJf
p,
=
J((s
))
+
=
)
fBs-vt
then
a)
fB,+t
=
.
,
.
.
B,+tn
)
.,
,
K s)
replace
(:.1) The Markov then for al1 z E Rd
lf s k 0 and F is bounded
property.
f'zty
where the right hand side
F: by
njxay-jo
=
.
C, for s k 0
k0
o
%1F,+)
=
and
py)
=
C measurable
En.Y :
For technical reasons,
-+
In words, we cut of the part of the path before time s and then shift the path so that time s becomes time 0. To prepare for the next result, we note that if R is C memsurable then F o 0, is a function of the future after time s. ' :C J(tJ(t)). ln th case To see this, consider the simple example F(tJ)
Bsvj
.
%: C
s,'' it is convenient to deqne the shift transformations
9
EvY evaluate j at
.p
f
=
. .
().
Exylnnation. In words, this says that the conditional expectation of y o % given &+is just the expected value ot F for a Brownian motiop starting at B, ttgiven the present state B, any other ipforpatn To explain why this implies kbout wht happened before time s , i irilevant for? predictihg w jy t ju p jjyjjs after time s,'' we begin by recalling (see(1.1)in Chytei 5 of Duirett (199$)) .E'(Zlf). Applying this with that if c: F and .E'(gl.f') e f then .E'(Zl7) c(l,) = = F+ aud .F we have 4 .
The felds F,+ are nicer because they nt>,Jl+
=
are right
nf>, (nu>t.T-:)
=
continuous.
nuyyaF:
=
That is,
,
Ft
=
peek at the future,'' i.e,, In the Fs+ allow us an ttinflnitesimal e Ss'f if 6 it is in Ss*+zfor any > 0. If Bt is a Brownian motion and f is any memsurable function with fuj > wh.en > 0 the random variable words
.
.
.
E
:
.
.E'.(y lat'i'-o es1.,:-a+)
'tz
Bt limsup t ta
-
#a)/J(t
s)
-
=
o
0s 11,)
If we recall that X = .E'(ZlF) is our best guess at Z giyen the information in F tin the sense of minimizing E LZ X)2 over X E .4F, see (1.4)in Chapter 4 of then we see that our best guess at F o % given B, is the same Durrett (1995)) ..F9- is irrelevant fpr best guess given Fa+, i.e., any other information in as our redicting o % p -
is memsurable with respect to Fs+ but not F(. Exercises 2.9 and 2.10 onsider what happens when w take fn) lg log(1/u).' HoWever, V-uand fn) will see in (2.6),there are no interesting examples of sets that are in F+! ms we , but not in S:. The two J-flelds are the same (u to null sets). To state the Markoy property we need some notation. Firstj recall that we have a family of measures .P., z e Rd, on C, C) so that under Pr, .8t(tJ) () is a Brownian motion with Bz z. In order to de:ne happens after time =
=
'tz
'
'
.
Proof
By the deqnition of conditional
expectation,
what we need to show is
=
=
Gwhat
(MP)
f.(F
o
#, ; W) = &(.Eb.F;
d)
for a1l
.z4
6 F,+
Capter
10
1 Brownan Afoton
Section 1.2
We will begin by proving the result for a special clmss of F's and a special clss of W's. Then we will use the theorem and the monotone clmss theorem to extend to the gerteral cmse. Suppose 'm
lf.I y,,ttmll lKmsn
'i'(-)
=
.,1.
.
.
=
,
extend
the
E=pB,+
clmss of
,
#); .A) for a11 f.d. sets .A in
.A's to al1 of Fo ?,, a+
,
(2.2) The g' theorem. Let be a collection of subsets fl and is closed under intersection (i.e.,if B e Jt then collection satisfy of that sets a (i) if B E f and D B then W B E
'utl
'tzt
FXs
.4
of fl that contains
nB e
.4).
Let
f be
.4,
-
.
(ii) If d,z
.
=
11
we use
,4
.
.
To
%; .A) =
Blumenthal's 0-1 rw
.,
-+
.
o
-
where 0 < :1 < < ri and the fm : Rd R are bndd d fiisurable. Lt (tJ and let 0 < < 1, let 0 < sl : g(s/) < sk S s + E Aj 1 i j i k where Aj E 'Itd for 1 K j K k. We will call these .A's the flnlte dlmensional or f.d. sets in F*a+! Prom the definztioz of Brownian motion it follows that if 0 < < dezi of sity then the given Buzj Bu is by joint . z .
(MP-1) E.Y
-
Markov Property
If
.4
and
e
then the
C
dn
1 .A, then
J-feld
generated
E
.
by
.
d
t7'(.4)
.4,
.
'l-lz
.
.
,
.!%lfu,
=
g1,
.
.
.
Bue
.
.
.
pz)
=
z
=
f=1 where yz
z.
=
(MP-2) E.Y
yij (:w-1,
puf-uf-,
See (2.1)in the Appendix
Proof
..
.
.
''
''
o
#,;
.4)
Eg,pBaw;,
= ''
.
of
.
'
.
Durrey,t ,
#);
-)
(1995).
for
all
.. .
.
..
G Lh
.
..('
.
.
.
.
.
.
ikoof Fix F and 1et be the collection of sets X fpr whith the desired equality i? tiu. A simple subtraction shows that (i)in (1,2)hold, While the zootne shows that (ii)holds. If wr let t be t pllction pf jt nite nvergence theorem (i.4) dimensional sets in Sso..s then we hve shw F,%s / c: d whih is the desired conclusion. (:I
From this it follows that
jt
Er
z
giBui) 'I7 =1
dy puz-unlpa, !/1).1(g1)
=
.
.
dpz puz-uz-z
.
Applying this result with the ui given by th obvlou chois for the gi We have .E'.(F
o
p,;.)
=
1,
k 1x,(l,,)
T1
Ec
.
.
dzlpaztz, z1)
..
.
Wd
,
s +
t1,
.
.
.
+ tn
and
(MP-3) ErY
o
#a;.A)
w
F1'ym(s+,m)
*y)
m=1
,
. dpp,+?,-,.(zk,
p) py,
&)
dpl ;hz-?&(p, pz).f1(p1)
k gi Bui Using the formula for E= 1-Ifal
.
.
.
) again
dyn pta-ta-zlp,:-z, we
have
0); d) for a1t W E Fi >
0,
all we need
.
.
dyn pta-ta-z
0 in
convergence
theorem shows
-+
ezh, ) Using (MP-2) result.
-+
(pn-1,ynjfn (pn)
is bounded and memsurable. Using the dominated 0 and zs that if z then ynlfrtyuj
to do is let
J1(p1) dy1 pta-tztgz, p2)J2(p2)
=
..4.
where w(g,)
EzpBs
Proof Since St C F,*+s for any (MP-2). It is easy to see that
-+
=
=
dzk p,k-,p-z(zk-l;zk)
.
ykz
sk , s +
.
J=1
=
,
(a.??j)
laa
.
.
Our next step is
(pz-1,pz).z(gz)
and
=
J
dplpfz-hlzh,
the bounded
convergence
p1)/(p1)
-+
ez,
0)
theoyem now gives the desired 1Zl
Section 1.2 Afarov
1 Brownian Motion
Capter
12
(MP-3) shows that (MP) holds for F Fllzm
Pnt
S f)
=
j
h(r, Brj
dr1
pt(0, p)J5(% > 1
-
t) dp
:
Rd
j
a
hs + :
Bn4 du
'tz,
R x Rd
-+
R are bounded
.'%(
(T(fz-,)
other
and measurable
-a
R and
-+
#7s-
turn our attention now to
(.('-' (s+
exp
applications tttriviality''
.%4
u,
dn)
j
f the Markov property below, that hms surprising consequences.
.
%1.'F'a+ ) = En.Y
o
C F:
that it follows (see,e.g., (1.1)in Chapter 5 of Durrett (1995)) o
pa1F,+)
=
Eckr
o
pa1J7)
1.4. =
0J.
From the last (2.6) Theorem.
equation
it is a short step to
If Z E C is bounded then for all ErzLFt,
t) g =
=
R is bounded
tr,Bv) dr
(-ftr,m4 j Ljo ) exp
ErY in Sectin
f
t. lf
Ek (Y
with
zizj
n, Bt e ff ) k an .
be a collection of subsets qf fl that (2.3) Monotone class theornm. Let tmder intersection (i.e.,if .A,B G thep .An B E #). contains f apd is closed Let 7f be a vector space of real valued functions on f/ satipfying
ect
Property Blumenthal's 0-1 faw
=
0J.
Proof when
By .the monotone
clmss
z
=
=
k 0 and
z
e Rd,
Erz6Fsot.
theorem,
(2.3),it
'I'IJm(l(tm))
m=1
s
suces
to prove the result
S'
14
Section 1.2
Chapter 1 Brownian Motion
lh this cmse Z = .XIF o 0aj where and the Jm are bounded and measurable. expectation (see,e.g., X e F: and F E C, so using a property of conditional and the Markov property (2.1)gives (1.3) in Chapter 4 of Durrtt (1995)) f.(Z1Fa+) and the proof is
=
XEnLY
p, 1.T-,+ )
o
e F:
XEB.Y
=
If we 1et Z E F1' then (2.6)implies Z ErzkF:) t are h e same up to null sets. At flrst jlance this The fun starts when we take s = 0 in (2.6)to get =
,
0-1 law. lf
e Ft
.
then for
(iv)
.&,
Another typical application
=
is trivial
=
= >Q(1xIJ7) 1.4 = E (14 1.&+)
>.
.&(.)
=
This shows that the indicator function and it follows that .P.(.) E (0,1)
1.A
is a.s.
equal
.s.
to the number #.(X) I::I
Proof
J%('r
inft
K t) k .Pc(#
metric about 0. Letting t
t
.&l'r
k 0 : Bt >
0)
=
> 0) then
1/2
since
.&('r
=
0)
=
1.
the normal distribution is sym-
0 we conclude =
0)
=
S t) k 1/2
limatr tt0
so it follows from
(2.7)that
#otr
=
0)
(2.9) Theorem.
lf Tc = inftt
>
0 : Bt
=
immediately starting frpm 0, it this forces: Bt is continuous,
0) then .45(Tp
=
0)
E'
.
.
.
.
01/2. The 1aw of the
=
=
=
21/2 when
x J)c a.s., so with probability of oider 1/2 at 0.
.
=
1.
for
a
the
complexity
psitive
of
amount
Of course, on top of your name it will write everybody else's name, ms well as al1 the work of Shakespeare, everal pornographic novel, and' a lot of nonsense. precie statement Thinking of the fuction g' as ttr ijnture we can mak follows: ms
(2.10)Theorem. Let g : (0,1) let E > 0 and 1et trt t 0. Then fi sup 0s8s1 Vrn
D
-+
ft)
ttlf you run Brwnian mtion ih two dimensihs of time, it will write your name.''
Rd be a
-+
Botnj
1.
Once Brownian motion must hit (0,x) must also hit (-x, 0) immediately. Since t
'
shows that c (1995))
limsuptlclttl/l/z are not Holder cpntipuou
=
=
when
Comlc Rellef. There is a wonderful way pf expyssing Brownian paths that t learned tromWilfridKendallz
.
,
=
.
Remark. Let 7G@)be the set of times at Fhich the path u) e C is Hlder jy, continuous of order y. (1.6)shows tht #t7v s (0,(R))) = 1 gor j < Exercise 1.2 shows that Pt7G = 9) = 1 for @ > 1/2. The last exerclse shows #t e 7z/al = 0 for each t, but B. Davis (1983) hms shown P(7f1/2 # 9) 1.
t
In words, the last result says that the germ fleld, Fo+ ls triv j a j sus yesplt is very useful in studying the local behavioz of Brownian paths. Until further notice we will restrict our attention to one dimensiortal Brownian motion. lf r
l
' .
=
.
(2.8) Theorem.
(t
one Brownian (/.(./:,)
=
.
(2.7)is
of
In the next exercise we will see that c cx) iterated logarithm (seeSection 7.9 of Durrett 1oglog(1/t))1/2. T(f)
.zz
:
.
.
xerclqej.40. showthat paths
the fact that (ii)(2.6),(iii) e proof using(i) apd if is trivial E (aYlf) EX gives
under
. .
.
=
e Rd,
.$t,
.,4
'
Exercise 2.8. If c < b then with probability one there is a local maximum of Bt in (c, ). Since with probability one this holds for all rational c < b, the local maxima of Brownian motion are dense.
.!5
,
(0' 11
Pr(X) C
the Markov property you can prove ' .
=
a11 z
0-1 Tzaw 15
Exercise 2.9. Let ft) be a function with ftj > 0 for a11 t > 0. Use (2.7)to conclude that limsuptc #(t)//(t) c a.s. where c E (0,x) is a constant.
J-flelds so the two conclusion exciting. i not
6 F:
(2.8)and (2.9)with ..
IZI
complete.
(2.7) Blumenthal's
Combining
Markov Property Blumenthal's
goj
=
ad F: is no longer importnt, so we Fill inke on iul irprovemet E in our c'-felds aud remove the superscripts. Let
=
#.4A)
E I then either
.,1.
Remark.
witl
'
=
=
.
%Fsm
E
0)
.E'a(.G(1s o p1lF1))
=
.E'.(1s
=
/(2v)-d/2
o
=
expt-lp
-
zl2/2).Py(#) dy
Taking z = 0 we see that if #(44.A) 0 then PvBj 0 for a.e. y with respect again shows and using #.(.) = 0 for all z. the formula Lebesgue to measure, and observe f%(.Ac) 1 f%(.A) that dc T 0, so the last handle E the case To al1 IZI for result implies 0 z. =
=
Sa are the null sets FJ are the completed c-lelds tor #.. Since we do not want the flltration to depend on the initial stat we tak the intersection of This technicality will be mentioned at on ppint in all the completed J-felds. otherwise section be ignored. but can the next and
EzEnz 1s)
=
=
=
.&(.c)
=
.
of Bt ms i 0. y psing a (2.7)concerns the behavior about result beh>yior the information get this ms s+ &., to --+
trickwe
qll use
(2.11) Theorem. lf Bt is a Brownian motion starting at 0 then so is the process defned by Xz 0 and Xt = t #(1/t) for t > 0. =.
Proof By (1.2)it suces to prove the result in op dipension. We begin by observingthat the strong law of large numbers implies X3 0 ms t 0, so X hmscontinuous paths and we only have to check tht X has tlkeright f.d.d.'s. to show that (i) if By the second deqnition of Brownian motion, it suces rimltirit norwl ditribution .X'(tn)) hms a < ta then (I(t1), 0 < tl < obvious) and is with mean 0 (ii)if s < t the -+
-+
...
...
(which
ECX'XL)
=
s.E'(#(1/s)#(1/t))
=
s
C1
The next result is a typical application closerelativ of the one for (2.8).
(2.13)' nn (#t =
of
(2.12).The
Let B j be a on tltmensional Brownian for some k nJ. Then /.4.) = 1 for all
eorem:
j
argument
here is
motion and let
a
.4
=
.
ln words, one dimensional Brownian motion is recurrent. It will return to 0 often,'' i.e., there is a sequence of times @r& 0. We 1 cxnso that Bt have to be careful with the interpretation of the phrmse in quotes since starting from 0, Bt Fill return to 0 inqnitely many time: by time e > 0. Ginfnitely
=
.h/zt
Proof We begin by noting that under #., with mean zlvi and variance 1, so if we use PrBt
0 zesult the fact that Brownian are =
(2.9)(orthe
Symmetry implies that the lmst result holds for z < 0, while property) covers the lmst case z = 0. Combining the fact that #.IT: < x) k 1/2 for all property shows that PzBt
=
0 for some t
k
n)
Erpnn
=
(Tn
), LS = t) are in Fs. (ii) LS < T), LS > T), and (S T) are in Fs
(3.5) Theorem.
lkol Herp L is an extra point we add to C to covr the cmse in which th convztio that all Some authors like to adbpt the ath ts hifted away. f ttions hake J() = 0 to tak car of the second cmse. H6weker, We will j.t sually explicitly restrict our attention to (S < XJ so that the second a jj ojthe defnition will not come into play.
S
(
=
let A : Fs
and
time
21
time
(andin .T).
below
are:
thenFs
are stopping
times
n (T < )
(.4n ( s t)) n (T s ) e h.
=
are stopping
C F1'.
times then Sr
=
nnF(Ta).
Proof (3.5)implies F(Tn) D Fw for all n. To prove the other indusion, let n.T1Tn). Since n (T',z < ) e h and Tn t T, it follows that n (T < .t) eC F so .A Sv E) l e .4
.
The second quantity Ss, ttthe information known at time S,'' i a mor ubtle. We culd have deflned
y, so by analogy
=
g(sw(,+r),t
nroc
tittte
k ())
.
The last result is obvious from the
and Exerciss 3.2 and verbal deinition.
=
ne>()cBths-vzj,
(.
something
that
.
-+
t k 0) wor
with.
The next result goes in the opposite frz is increasing and f c'(fr,). n -+
=
to prove
=
The deflnition we will now give is less transparent but easier to Fs
allow us
3.7
Kxercise 3.8. Bs C Ss, i.e., the value of Bs is memsurable with respect to the information known at time S! To prov this lt Sh ((2nS) + 1)/2n be the (x) and stopping times deflned in Exercise 3.2. Show BSn) E Fsv. then 1et n use (3.6).
we could set
Fs
'
:
.An LS K t) E St for all t
k 0)
Jt that lies in In words, this makes the remspnable demand that the part of af time available information with measurable respect to the (S K ) should be and but in the choice made cmse of < between K t as t a @. Again we have useful that and the know is two it to makes disezkence stopping times, this no equivalnt. deinitious are
Exerclse 3.5. When Ft is right continuous, if we replace (5'K t) by (S < ). Eor practice with the definition
of
Fs do
the deflnition
of
Ss is
unchanged
direction from
=
(3.6).Here frj 1
meas
:7Nerc iS@ 3 9 Let S < (x) and Trj be stopping times and suppoqe th>t Tri 1 (x) as n 1 x. Show that Sshn 1 Fs ms n 1 x. .
.
We are now ready to state the strong Markov property, the Markov property holds at stopping times. (3.7) Strong Mazrkov property. Let (s,tJ) measurable. If S is a stopping time then for E=Ys
o
pslFs)
=
Ens4Ys
-+
F(s,
) be bounded
a1l z E Rd
on
which says that
(5'
pply the Markov property and E Fs then put the pieces back together. If we let Zn 'Ia @)and =
=
,1.
=
f.(Fs
o
0s
J1
n (S
< x))
=
V E=Zu
n
=1
,z)
#a ; X n (S
:.=:
o
-+
,
'''7'''l't )t,.))),
,
'
t lte b oun ded convergence theorem implles that (3.7)holds when J hks the given in (+). form ' '. j ttt. To complet, the pwof now we use the monotone clasp theorem, (2.3).Let k the colletion of bouded functions for which (2.7)holds. cleprly7-/ is i jl(ll'i . t))q'. and inciemse t a if ys e a vector space that satisnes are nonnegative (ii) t'.tll! chet i collection of sets of bounded y', then '. j;qrjjj!yl'tjl let be the now, yy (i) e '.y!.).t... the foim (u : (a(t/) e Gjj where Gj is an open set. If G is open the function ))!))i l (:111... j :,!pI. decjeasing limit of the continuous fupctionsfktz) ( )) ) (1 k disttz, G))+ )j)'tt'k('. . .j!!)()*411*:1 pt disttz,c) lx : .H. This here is the distance IE/iIII from then G, if .A t z so ; e '. ). . shows aud the desired conclusion holds from .j. (i) cl ollows (2.3). s
,E'.)t#,)t) ' -
-
''
:
E-(E
',tittjtt) ''itlt -
E -' ;jE .E ;. . ;. .'E.t)'tt) .
tnl)
os
o
'1Eil)
.
-zf
,y),;
.
.
.
''./
. EE :.E -
'
.
i
.tt.
'ro
.4
''l:''pp)j
Now if .A e Ss, A n (S (. n (5' K ts)) that the property follows the Markov from it o =
o
ysa
Elitl))
=
(.4(3 (5' K tn-1J)E above -
.F(in)?
E E
(:
EE ' E.i
. .
.
=
-1t:::;;,,'
'''''i7')t '''?ttt
suin is
-
,
.
-
' -- -
E-
-
-
.'
' .Ej.lEj ...).Etjt)t. .r(
EEi(.l.;.,jjlt't..jqj.',Eg. ..
=
E.E'.(.Eb(f-)zk;.
n fs
n=1
=
ta))
=
.G(.Eb(s)'Fs;.n
fs
< x))
.)..)yj y.y. yjjg
=
-+
;
'
.-
.
.
i.- .
.
n
Fa(=)
hst
E 'E
qE. . .
...
.
:
.. E;.Itg gy ' ' jj )! ( . . . .. .. '.;'...l(lli)ILfj.. tttl (. ' '. . '. . ' .!.. . lt'ty ..
,
'E E . !'
-
. .
! : E :
'
. .
-
.
' -. E
.
.
w(z,s)
=
EZYS
=
hsj
dplptz
:'
(z,pl).f(p1)
'Er'..'i-'...(ytj
.. . j..E. . .(
.
.
dps pta-za-,
(pn-1,pn)J'(pn)
>
o
as()tz
c)
>
.c
.
..
.
.
..y
E ...
..
.- .
E.
.
tz,(j
-
'
.
: . .. .
- : . -.-
E
EEq E EEE E 'E -E
!. . . .t'E(/,.k)( -
E: .
E . . . . . .. . . y. . y . .
E. .- . . - E. E. E . - . E . . .q
. .. - -- . . .
''k-'j
-
.
.
.
'?.l
=
.. . - .
q. E: ...
. .
.
.
..
..
,,.
E E. . E;)))t)y)'. E: . . E: E
=
-
.
. : ..
'itt').. .
.
E
g , tE)'tlE'.i.klE,
:.
E-
. .
. -
t, t :.
.
-
. : : E. E.E Eq.... . . );E),.
.
.
,g(td)
-.
.
E... .... -. . E bijji,, .: . . . : i E . :: . .: ; y.. (. . . E i. . .y . EE ,l,EE . .. -.. . q.. E -
i .. .
.
. .
..
l'll''i
-. . . -. . E E 'E
,
-
.
- .
. . . : y - E. . y. . . y .y . .
.
E::. ..
y.-i),),.
-
..)iyt.)(j
.
to op, cjsx,)
;::jI:;,.
.
. .. . q) Ey
q
that
=
' ji. . . . . Taking the expected value of the lmst equation we see that . .' jt) y .-( .: . ))' ( j.-. .''jiilittty.t o p y., '(..' () tl'ttttj op,> () sr some rational t) 0 ; (.. ;y ... ; )jiki f'tj'jj($-. . .q. .. y.....y . ' . ( r . . . E.. .. i '.'.'').l:jy!:
[email protected]('.j;.(j; .J(tJ) rq.rE, .. jh';;j(jjjgy';lll'r, with probability > it follows that From this ne, if a point u e (t : ',iE,t..t,,. r'i'l)r'i tz) ! js rational isolated the left thre is that < t on so a 1!)/)((1 st(w) c) (, n (i.e., @ . . (C;'.'.;il.)r.;@i:)'. .....E)y.. q. = j) then it is a decremsing limit of points in J@). This shows . that tlte .1...j y ( . y (. ..b3jl . . z(w) . ': pl i'). closed set z(w)has no isolated points and hence must be uncountable. For the tit).t jmst and step ; ! ( ) Hewitt Stromberg ) 72. ( pyg see .'.7771''1 . (1969), rr'. ( ry t ' ;ll!:t')pjr. !.i.. If we let lz()I denote the tebesguememsure of then Fubini's theorem . '.-'.. ri..j!:.k jy.(.... y.y.j(.jy.. .iyjpjjt'j. ..yEl/tt)lt))y, . . ) . )( j....y..y y.yy y ; Ltn . jji... j jy..j.... .. .-.....';;Lj,j$-q;3.;qL-1L.L 4(::)1 4(::), i!r .-..)... 41:) 4(:4 dr''..'lk 44:4 ..;!!iEiI, rqEilr''-'ljl p(.-.. rj,jlll (2)) 1(:21) 11 (zliIIII 11 1(:). -.-qyq!)'tri((t$r'. (;'..i;. i. .r . ) . ' . .(.y.j.y(.j..y.r.q.,,.,y.,. (.. . .ttljjijjy/yty )j ljtjj gy,jyjgyjgj ; j . ..yy j.yj jty'jjyy. jj.y. j.y..jyyy.j g y jjr y. j.yjy...y.yjy. . ........yj.(yyj.j'. j j(.jy qjgj..(.jy. ..gy .j.yy..j yrj. '...(.((1y)(j).)t'tjjy'jltjlphtr (r. .(y.yj.y.y......jjy...y..jj . ....y.yy.y(y yg. j.. jy yygj .y ! jjj ,Et
From this
q
E'
.
...','i!kjjll
and induction it follows
.
.
E f't))y E
y)E.
E
'tt
is continuous.
.
.
p
E
-
j jy . . ' yjjj
.-+
..
E-
.
.
..
.
=
E.. q EE)t))l)).j. . q! . . . . .
'
and continuous.
.
.
.
are real valued,
.
E'.
-
bounded < ts and A, where 0 < l < fn If f is botmded and continuous then the dominated convergence theorem implie that dylhz, ylfyj z .
. ..
=
=
q
-
.q.yy
.
.
.
=
.
- -
'
m=1
.
.
.
=
E. s'''jjjjl'. . . . . . ..'.iljtljtyj )qyj(1 j(' ..' .'.
1-fJm(a'(m))
..
.p.(a
-
::
=
. ..
-.
t'
[email protected] =
y ..
.
.
',''':)it) sxnmple3.1. zeros o Brownlan motlon. considerone dimensional Brow. q)jtij nian motion, let Rt 0) and let T'n inf (tz> 0 : Bu infttz > t : Bu 0). trytjyilypj., and x) strong implies Bt) :i k;. < the Markov 1, 0 ; . .. ) so xow(2.1a) ' rttyty lir: )1.)! . ..propeity and .. ..... . imply . . .. .tt'ytlll. . . (2.9) '.!'f)j) l -
+ 1)/2n where (z) the To prove the result in general we 1et Sa ((2n5') largest integer % z. In Exercise 3.2 you showed that Sa is a stopping time. To (:x) we restrict be able to 1et n our attention to F's of the form
.
..
.
...
.
....jj .....,...jjyjy.
E
).E.y...-E.....t.q. ....
-
'
E-
'. E . E- .. . - - . . . .. E ''l lEi!i'j!')itt)@.:...(...'...E....('E.y..(.'...';.',E. . .. . E..iEjlEji.) .), .
. .
--
. .
-
q,
...
E
ig
-
E
i
y
. y
-
-
y .y y y . .y . --
.
qE.. y. ..- g E-
!-
( .- ilj ! E.
-. .
-E
qjjjjyy
.
-
yy . y .
! y.y . . E.y. E q . q - y tid-r'jlqjj.. y.yy.-.y. y...... y ; y.;. g yy -
. -
-
.-
!q
ii
.
-
.,.
.
-
.
j .
,
-
j ggj. yy .
. g
,,.
.-illiiiir':;:,.
.-;:ii!:7
dt:;i!r
.-ilr:::'':,r:-
'i,,.-,-
,,
::::::::::
:::::::::.
.
-
::::::::::
24
Secton 1.d
Chapter 1 Brownian Motion
So Z@) is
a set of measure
In
zelo.
.
..
g
.
=
u(z)
(3.8)
.J%(Tc< t)
2#(,(#f
=
> a)
Intultlve proof We observe that if Bs hits a at some time s strong Markov property implies that Bt .5(L) is independent pened befpre time Tc. The symmetry of the normal distribution c) 0 then imply =
Intultion Since D(z, J) C G, Bt at Bs. When Bs = p, the probability hOW B %0t to y. Proof
of exiting
F
G in
,4
E
To prove the desired formla,
erty, (3.7),to
qrst exiting Dk, J) is nyj independept of
1(s c' Ek)
.
'
Markok propr
strozk
Will apply the
we
=
,
exit G without
cannot
(3.9)
.&(T.
.
.
.
. . . ..
.
'
J%(T
(
half space .I zd (z E principle reflection (3.8)implies =
=
.
P n(Tc
t)
tz)
2
=
J
.
Here, and until further notice, we are dealing with a one dimensional Brownian motion. To flnd the probability density of Tc , we change variables z = 1/2c/s1/2 1 dz = -tI2a(2s3I2ds to get 0
(4.1)
Jb(Tu < t)
=
)
t
=
0
d. (-t1/2c/2s3/2j
(2*)-1/2exp(-c2/2s)
2
(:
A's
:3
-1/2
)
a expt-c
2
/ :s) ds
=
=
supt
K1 in Ekercis
Cr
%s) .POIf.
.
. . .
(
1 =
.. .
.
:
(1
u.)
-
x
t-
,
az
?n u -
..
au
to intuitih,
=
-
Q
> 1
s) dz
-
zr
=
2 1
= =
0
Co
-
-
g' 1
1-,
'm
1-,
co
1-a
:$)-1/2 r s/tr
(sr
z exp
+
s)
(2:rr3)-1/2exp(-z2/2r)
(-z2(r + dv
r.
N
=
'
-
(t(1 -
A'
0
t))-1/2
jbr 0 < t < 1
.
is symmetric about 1/2 and blows up near 0 and 1. This is one of two arcsine laws associated with Brownian motion. We will encountr the other one in Section 4.9. The computation for R is much emsier and is left to the Teader. ( (
.
'
.
:
.
. .
Exercise
'
..
: .
Show that the probability
4.1.
,.
density for R is given by
:
P 0 LR = 1 + t) Notation. PT t) =
.
.
.
' . .
=
=
1/(*1/241
+
j))
for t k 0
ln the last two displays and in what follows we will J(t) as short hand for T hms density functio ft). .
.
t
.
Fl
drdp
s)/2rs)
d dr
often
write
.
'
' ' '
' As our next appllcation of (4.1)we will compute the distribqtiop of Bv where r = inftt : Bt / S'J and 11 (z : zd > 0J and, of coprse, Bt is a d dimensional Brownian motion. Since the exit time depepds only on the lmst coordinate, it is independent of the flrst d 1 coordinat aztd we cart compute the distribution of Bp by w'riting for z, 0 E Rd-l ) p 6 R .
*
(sr3)-1/2 ()
to get
the density function of L
-
exp(-z2/2s) (2xs)-1/2
z
.
arcslnjvsp
=
p,(0, z).&(%
=
112
=
N.
Using the lmst formula we can compute the distribution of L Bt = 0J and R inft k 1 : Bt 0J,completing work we startd 2.3 and 2.4. By (2.5)if 0 < s < 1 then =
.!actz, t) =
pith t
,
V
2
-
dv
t))-1/2 dt
(t(1
()
The reader should note that, contrary the lmst 0 before time 1,
.
dz
and thep again
'm c
'
(2>)-1/2exp(-z2/2f)
above
-
';r
1.4. Flrst Forrnulas
s
(r + s)2 '
-
-(
1/2
s) a
*%
1 =
-
r s
1-,
27
to convert the intejral over r E (1 s, x) emsier + slz#r, so to make the calculations
(r +
Changing variables ms indicated
.z;
=
t:r
1 -
a /2f
-(2c-/)
s)
=
get the joint density r(,yy
13
.h
see that 2c
=
B, to rwrite
maxcs/i
=
f%(2c
=
shrink to z
Mt
First Formulas
P (r,&)'B
r
=
o 0)) '
dspn,vtr
=
'S)(2=f)-(d-1)/2e-Ir-#IQ/2J =
0 =
# (2T)d/2
-(ja-#j2+pa)/2a
-(d+2)/2
0
ds s
e
by
(4.1)with
c
p. Changing variables
=
Jc -(Iz ).
#
(2T)d/2
so ve
Section 1.4 Flrst Forznulas
Motion
Chapter 1 lrownan
28
s
-
#1a + !/a ) dt 2t2
-
pj2 + y2)j2t giyes
(lz
=
(a+2)/2
2f pl2 +
lz
-
p2
By induction it follows that n
n-f
Pr,vlBr
(#,0))
=
=
( lz
pjajz
.j.
vqdjgj gd/a
r(a)
=
.L*pc-le-Vdg and :
Proof
.'
The first
c
k 0Jhas
T:
o
-
Tc)
T5
=
-
Tc
'
then
a < ,
(3.7)and
the strong Markov property
so if 1 is bounded and measurable, lation invariance imply
ll'b
0v.
,
...
independent increments.
stationary
is to notice that if 0
nd c-# = c: so c: above apply equally Sipc tlie arguments s, so Ca has a Cauchy distributin. well to Bt1 cBta ), whre c is an# onstant, we cannot determine lc with this argument.
-C,
-+
,
-
zi-z )
.
Efrtzt-v
-
z--z)lJl--,) .
E .
n-1
1-1hzt
(Ca
.5,
.-+
n-1
E
Under
The scaling relation
The fart that Cs
=1
FI'fizti
.
=
-
=E
D
::i
,
.
conclusion.
=
< t ,, and 1et h, 1 K i K n be bounded Let t () < t 1 and using the hypothesis we have FtvvConditionin: on .
jj
independent increments.
Then Zt hms stationary Proof
.
be useful later.
will
which
trans-
:
f2k(Zf;-t;-z)
=
(1.2)intplies
The scaling relation
Exerclye 4.2. Prove
0r. 1.T.) .E/(T-c)
The desired result now follows from the next lemma (4.5) Lemma. Suppose Zt is adapted surable function 1
Z;-,)
=1
which inplies the desired
is the usual gamma function. = probabilists should recognize this as a Cauchy = 2 0, z When d the glance, fact that Bv hms a Cauchy disyribution might flrst At distribution. be surprising, but a moment's thought reveals that this must be true. To explain this, we begin by looking at the beh>vior of the hitting times (Tc a k 0) s the level a varies. where
-
=1
y #jz
-
rl
/(Ztk
E
have
(4.3)
-
Zf-,)
29
E1Zt=-t=-j
=
-xlpl
i.=:
..:
'
E
,
'
=
'
.
.' 30
Section 1.4 First Formulas
Chapter 1 frowzlan Motion
Exerclse 4.3. Suppose w(s)is real valued contirmous and satis:es w(s)w(t)= ln(w(s)) to get th additive equation: #(s) + ps + t), w(0) 1. Let #(s) for equation = #(t) to conclude that #(m2-n) m2-n#(1) #s + ). Use the @4(1).. continuity extend and this then use 0, to #() to all integers m, n Exercise
since
=
=
hy)
=
0 for y
k 0. Combining this
E=JBt)
=
)
7% >
=
Adapt the
4.4.
for po (s)
argument
.lclexpt-r,zll
=
( . . :
..
=
yo shpv
E (exptC4l)
=
The representation
The lmst formula generalizes
exptc-cxi)
Exerclse
4.5. lf u
'p))
Tc is discontinuous i
('tz,
Cs is discontinuous in
(u,r))
:i=
1.
W here
=
1.
rxerclse
Eirt
'tz
-
,
. . .. . ' . .
.
.E..'
.
.
..E ' ' '
.
.
.
'
'
..
The discussion above has focused on how Bt leaves H. The rest of the sedion is devoted to studying where Bt goes before it leaves H. We begin with the case d 1. =
(4.9) Theorem. where
If
z, y >
0, then P=Bt #)
#f (z,
(2A) -1/2
=
t)
y, Tn >
=
pjtz, y4
=
-
-p)
pftz,
-(p-.)2/2f
d
The pqoof is a simple extension of the argument we used in Secti Proof wlth .J(z) when 1 3 to prove that Pa(L K ) = 2#c(#, k c). Let J k 0 Clearly 0. zK (
f7.(T(.&f);Tn
>
t)
=
ErlBt)
-
.
.E%(.f(lt);%
S f) nd
If we let J-(z)= T(=z), then it follows from the strong Mrkov property symmetiy of Brownian motion that
>Q(#(.R);T'tlK f)
l
E=LEz1Bt-n)3 T'n S fl ' .Eb(.E/'(.t-v.c);T0 ; t) = .E%(/(#t);Tn E=1-Bt44 K g.
=
=
(
=
: . .
.
.
. . .
'
' .
'
' .
.
.
=
#rl.R
-+
-+
Exercises 4.5 nd 4.6 Hint. By independent increments the probabilitie scaling implies that they do not depend on the but then only depend on size of the interval. 'p
Let r
that
of the Cauchy process given above allows us to see L and s Cs are very bad.
paths
)
-
the frst
equality
E=hBt)
jptz,-pt(z,-p))J(p)#p p)
'' . .' .. E j. . .. .
. ..
(4.10)Theorem.
its slmple
E=1Bt
=
with
y- =
(p1 ,
4.7.
.
inft =
.
,
pd-l,
Prove
:
(4.10).
to d k 2.
Btd = 0). If z, y E H,
ll, r > tj
-pa)-
.
easily
=
pttz, #)
-
ptlz,
#)
shows
r
.
' ...
.
:'
..
.
' . .
.
.
. .
2 Stpchastic
Integation
J!
.Jfadx, To motikate Ip this chapter we will deflne our stochastic integral h o at time s and ffa price of stock a the developments, think of X, ms being the short). The which negative number of shazes we hold, may be (selling msthe relative proflts at rfresents time t, the et to our wealth at itegral h then of the inttal of chae check this note that the inqitesimal rate iime 0. To of shares we stock of = number of chge the the rate times the dIt Ih d-'' d ( hol jyjtjoduce the integrand H aj the p rdttable ( It th flrst setion yj# e w jj) version of the nti PrOCeSSeS, tht ttke number of stia#es a nxa ihematical stock and not on the future of past behavior the held must be bmsd on the will nd sectiori oi intidu intgiatis X the we perfoymance.I t h sec Irtinales COIl fair Ihtuitikely local iiiitingales.'' are tinus games while t tlocal'' Tequkqments reduce refers integrability that fact the the we to the to admit a wider clmss of examples. We restrict oui attention td the cae of simpler continuous paths with have Xg pprtingples thep'y. a to t =
.
=
q
.
E
N
,,
,
1
,
-+
.
2.i. .
Integrands:
'' . '
.
' E' ' . '
Pred 1c table Processes EE E .
:
'
.
.
.
.
. .
To motivate the clmss of integrands we consider, we will discuss integration Here, we will assume that the reader is familiar w.r.t. discrete time nartihgales. With th basics of martingale theory, as taught for example in Chapter 4 of However, we will occasionally presept results whose proofs can Durrett (1995). be fOMn d there Let Xg n k 0, be a martingale wa.t. Fn If Il, n k 1, is any proess, we define tan .
.
,
n
.
.l
'
X)a
Hmxm
=
-
Ak=1)
m=1
q
th lst formula and the restrictioz we ar abot to place on the will with example. Let &,fa, consider be indpendnt cocrete H mi we =t) k-l-a. = = = symmetrlc = 1/2, = and the 1) P let Xa Xn is P &+. , &). slmple rnndom wnllr and is a martingale with respect to Fn = o'(1,
T intivate
.
.
.
.
.
.
.
34
Capter
Section 2.1 Integrands: Predictable Processes
2 Stochastic Integration
If we consider a person iipping a fair coin and betiing $1 on heads each time then Xu gives their net winnings at time n. Suppose now that the person bets an amount Hm on heads at time m (withHm < 0 interpreted ms a bet of -Hm on tails). I claim that CH X)a gives her net winnings at time n. To check this note that if Hm > 0 our gambler wins her bet at time m and increses ltr 1. fortune by Hm if and only if Xm XmThe gambling interpretation of the stochastic integral suggests that it is natural to let the amount bet at time n depend on the outcomes of the flrst n 1 flips but not on the flip we are betting on, or on later flips. A process Hn that has Hn E Jk-1 for all n k 1 (here (0,t'1), the trivial c'-fleld) is said since its value at time n can be predicted (withcertainty) to be predictable at time n 1. The next result shows that we cannot make money by gamblint on a fair game.
Elnmple ar random
Npdop
1.2.
(
F, (f), with
P) be a probability space on which there is defined T PT < t) t for 0 K t K 1 and an independent = with P 1/2. Let 1) = P4 q= =1)
Let
variable variable
35
=
=
.
X
-
.-f'a
=
Let Xn be a martingale. .X')n is a martingale.
is predictable
If Il
.
..
.!tl
aj
u..
l
yt
) . . .
'
.
Proof lt is easy to check that CH Xla 6 fn The boundedness of the implies E 1(S' X)a l < (x7 for each n. With this establiqhed, we can compute conditional expectations to copclude .
.
=
k
'
-
z.=
=
-
since .Jfrj+l E Su and .E'(Xrz+z Xn 1.Tk)
=
'
0.
,
t
The last theorem can be interpreted ms: you can't make money by gmbliik on a fair game. This conclusion does not hold if we only assume that Hn is optional, that is, ll 6 Fn, since then we can bmse our bet on the outcome of the coin we are betting on. Exnmple 1.1. lf Xn is the symmetric and Hn = a then
ll sincefm2 1 =
.
.x)s
=
(2
=
.
j'
1
=
.
.
To check this note that $he memsure dX, corresponds to a m>s of slze >t T and hepce the iniegral is times the v>lue there. Npting nw tltat 'Fn = 0 while $ 1 we see that h is not a martingale. o
simple random
walk considered
above
n
)-7
m
.m
=
'
The problem wtt te lmst example is th same as tlte problem with the our bet can depend on the outcome of the event We one in discrete time are betting on: Again there is > gambling inteypretation th>t illustratrs what fter thewheel is spun and the ba is wiimz. Consider th game of roulette. rolled, people can bet at any time before (ns we should say locally a submartingale but we will continue to use the other term.)
Why local
Section 2.2 Integrators:
2 Stochastic Integration
Capter
38
(2.2) Theorem.
martingales? working
for
are several reasons martingales. with
There
with
local
k 0). is
'
'
a martingalek
show: (
.
(2.3) The Optional Stopping Theorem. gale. lf S K T are stopping times and Xht 1. t en
Proof (7.4)ih Chapter 4 of Durrett (1995)shows that if L and times kaa is a uniformly integrable martingale w-r.t.
'
ftyzTlfzl
,
is meaningless, since for large i X' ( i pot However, it is trivial to deine a local mart gal y:
then the concept 'r deined on the whole space. there are stopping times Ts
1r
so that
.
.
.
'
.
.( .
. .
'
.
=
.
i .'
(
' .' . .
.
E'
%M n
are stopping
then
'Fz
To extend the result from discrete to continuouq time 1et Sn ((2n51 + 1)/2n. Applying thp discrete time result to the uniformly integrabl marting>le Fm = (x$ Xzamz-a With L = 2 Sa and M w s that =
=
times a (iii) Since most of our theorems will be proved by introducing the proofs ar no to reduce the problem to a question about nice fnartigales, harder for local martingales deflned on a random time interval than foi ordinary martingales. stopping
'
..
Lei X b a continuous local mrnis a uniformly iptegrable martingale
ot mrtihgale
< x,
39
.
(i) This frees us from worrying about integrability. For exanple, let X$ be a martingale with contirmous paths, and let p be a convex function. Then pXt) is always a local submartingale (seeExercise 2.3). Howver, w can cnclude that ph) is a submartingale only if we knoW .E'lw(.X)l < (x7 for each 1'$'a fact that may be either dicult to check or false in some cases.
(
,
First we need to
th
rather
martingales'
(uYv(f),.T-v(f)t
Con tinuous Local Martingale
Remson (iii)is more than just a feeling. There is a construction that makes r) it a theorem. Let Xz be a local martingale deflned on (0, and let Ta 1 r be a sequence of stopping times that reduces X. Let To 0, suppose Tz > 0 a.s., and for k k 1 let
(
'
'
.
.
.
.
.
'
.
'
.
'
.
X'xr llsa) = Xsa Letting n
(x7
..--.y
expectans
.
r
;
the dominated convergence theorem for conditicmal
and using
((5.9)in
.
.
chaptert
te (1905))
of Durrtt
foltows.
resutt
(:I ,
-almost
=
n'(f) To understand
=
t
t
-
n
(
1) Tk-l + (k 1) K t + k 1) t -
-
n
-
s s
this deflnition it is useful to write it
1
t
2
-/(@) z2 -
-
out
.
for 1-
=
'
f(-Ygkar'I.'G(,))-'G(,)
=
=
. .. .
.
,.
'
. .
P roving the desired result.
1, 2, 3:
The next result is an ing local martingles are
/,,n
(:l example of continuous.
the simplcations
that
come
j
rotn assum'
(2.4)Theorem. lf X is a continuous local martingale, we can always take the 1> n) or any other sequence sequencewhich reduces X to be Ts = inftt : 1-'V T < Ta th at h as T 1 cxnas n 1 x
o
IS71+ 2, Ta + 31
.
. g.
. ,
. .
.
:
.
In words, the time change expands g0, onto (0,x) by waiting one unit of time each time a Ts is encountered. Of course, strictly speaking y compresses (0,x) to be deflned for all t k 0. The reason onto (0,m) and this is what allows .;G(f) change with the time for our fscination can be explained by: 'r)
=
f(Xv(t)I.Tk,))
+ 1) (T1 + 1, Ta + 1) go + 1, + 2j (Tz + 2, Ta + 2)
T'a
=
1)
zk+ k
(az',l
f n t -
=
K Tk + (
Proof of (2.2) Let n + 1. Since v(t) S Trz A n, using $he optional (@) gives Exnhn lFv(t)). Taking conditional stopping theorem, (2.2), Xv(f) expectation with respect to ?v(a) we get
Proof
optional Ex
that reduces X. If s < 8, then applylng theorem to Xy at times r s A Tm?and t A Ti gives
sn be stopping Let
a sequence
=
A Tm' A Sr:)1(sa>p)l.T'(s A Q A &))
=
Xs A Q A &)1(sa>0)
the
40
2 Stochastic Integration
Chaptr
Multiplying by 1(tm>n) E Fo
Fs h
c
Section 2.2
TmJ
Continuous Local Martingales
Integrators:
41
2)p + (d 1) > 2). i.e., p < #/(d if To see that Xt is not a martingale we begi by noting that Bt z) t12(Bi z) under Pr so as 0 i probability. To x, 1#tI (:0,z,k>c)l-'/1.s A TCA Sn )) )'
As n
=
Xs h
TmJ
A &)1(sa>0,m>:)
..--y
'->
-+
-
-.
h Tm'A Su ) 1 Ss
1 cxl Fs ,
h Tm? ), and
E ir X f
-Y(r A Tm' A &)1(sa>c,tm>0) .X'( A Tm')1(n>t) -+
for all r k 0, and 1aY(rATm'A,$k)I1(s.>(),rm>n) %rr, so it follows from the dominatpd convergence theorem for conditional expectations (see(5.9)in Chapter that 4 of Durrett (1995))
K (2 0 it follows that E,X, is not constant and hence Xt E1 is not a martingale.
s
-+
'
.
E
''''''''
''''
E (-Y(t A Tm?)1(v',k>n) l.T1s A Tm'))
.
Xs A Tm')1(:m>f))
=
.. .
proving the desired result.
(:l y...
.
ln our deqitio n (2.1), as m inga le (wr t jkmhf. t k 0)). Wedid thi with the proof of local mrtingle
;.
..
.
'
f
(2.4)inTamitl exercise shows The next that we get the same deflnition if we assume Xj js a martingale (w.r.t.(J5, t k 0)). mart
.
Exercise
Fghs, t
.
.
a ,
6f
.
Let S be a stopping time. Then XgS is a matingale 0, if and oly if it is a martingale w.r.t. Ft t k 0.
w.i.t.
2.1.
k
,
In the flrst version of this book I
.
.
': .
...
.
fr'ny
'
:
..
2.2. .
.
'
.
(2l E
for each t then X2 is a
y j-(d-2)pdy < x ..
1
.
q
artd
r
d-1
no
dv
trouble
'
with
the
'
.
convergepce
which is bounded near 0 and
< x
lA%I
< cxl
Cleazly our mssumption implies .71A'tI < cxl for each t so all we have to ECXL 17,)k X.. To do this, note that if Tn i? a sequenc pf stopping ti.mes that reduces X Proof
checkis that
;we
txpIaa,ul
.7 then we let n tional expectations. -.+
(
sup
osasf
submartingale.
=
=
'
.
='
.
2.1. Let Bt be a Brownian motion in dimension d k 3. In Section 1/l#f Id-2is a lcal martingale. Now that X
,,
.....
by
,
.
As many people have pointed out to me, there is a famous example which shows that this is WRONG.
E l.X'f jr
...,
is pvided
3.t * will lt2 that it Bt is twtjdimnsional Show E 1.X':IF < o z.z: lok IAI is a lcal mrtiklet LS'Xg i:xnso X i ot a mrtiltgle. .
.
(2r5) Theorem. . .. .
t'You should think of a local martingale that would be a martingale if it had E l.X I < x.:'
3.1 we w11l see
Irt jtion fnotion the Xt > < x but limt-x'
lxikcls .
copnterexampl
worse
said
ms something
Exnmple
An eve
ivi
g.
jryijjg..!,l
('
x
k x-
the dominated convergence theorem for condi(:I See (5.9)in Chapter 4 of Durrett (1995). and apply
shows that the last result holds for supermartinAs usual, multiplying by gales, and once it is true for supem and sub- it is true for martingales (by pplying the two other results). The following special case will be important: -1
(2.6) Corollry.
A bounded local martingale
is a martingale.
Captez
42
2 Stochutic
Integration
Secton 2.3
Here and in what follows when we say that a process Xt is btmnded we mean there is a con>tant M so that with probability, 1, lAhl K M for all t k 0 theorm fpr lpcrl l!!qpp irumstanes F Fill ned a pnvergp marsimple result is sucient tingales. The ibllowing for our needs.
(2.7) Theorem.
Let Xt be
a
local martingale
E then Xp
=
Proof
Lt dc
=
0
limu,r Xt exists a.s. and EXz
ajFa-l)
-
S(Ak2j.'&-l)
=
-
(since
.a
yla-l
-
.
that
ro
.s.
-
-
da
Brt
-
Zirclse 2.3. Let X b a continupus local iii>qtlnal u function. Then tX. ) is a local subinartinale.
,
1rt
and
.
Exerclse 2.4. Let X be a continuous lotal maftingal stopping time. Then K = .Xk+, is a local martingale
k
b
gkk (
) (E
:
'
Xk
lt
ad with
ckl
to
respect
b
G
=
.
S(.n
=
and it follows by induction tht
=
.
y
FR+.
.y
has the desired properties. so To see tha d is unique, observe that if B is another process with desired properties, then pl.n Bn is a martingale aud da Bn 6 .Tk-1. Therefore
.
'i
=
-
.
= 72n-1
=
::
43
k 1
Xai y + E (.L2j.'Fn l ) A1
=
Exj
Proof Let y be the time scple of (2.2)and let X .Xv(). X is q.pprtingale. upcrossing The inequality and the froof of th rik r ti/ale conkigit c therem given in Sectipn 4.2 of Durrett (1995)generalize in a Ayrqightforward way to cptinus xitj a,.s. and time >nd allow t!s to Eclude th>t y limjx oncluion *e not X. limf.r Xg exists prove the lst XZ and cvefkence therm f'l then use the the drinatd t convergenc conclllde EXz = fF: E Fx = EXv E) y .q.
and Covarlance Processes
From the de:nition, it is immediate that W,z .Tk-1, 6 d,z is incremsing Xj is a submartingale), and
on (0,r). If
EX.
deflrte for n
.s
sup 1A% l < cxl 0KJ < r =
and
Vrance
= Bn 1.7k-1)
-
-
.s
Bn
=
Jlc
.s-l
-
Bz
-
fa-l 0.
=
ID
The key to the uniqueness may be summarized t'ay predictable disaq crete time mrtingale is constant.'' Since Brownian motion is a predictable maringale, the lmst statement fails in continuous time and we need another
Msumption.
'
(3.3) Theorem. Any continuous local martingale Xt that is predictable loally of bonded variation is constant
and
tintifne).
roof By subtracting v' we may suppose that Arou'z 0 nd prove that X is identically 0. Let F( be the variation of X on (0,t1. We leave it to the rader to check ,
Exercise 2.5. and EXz < x
Show that if X is
then Xt is
a continuous
local
a supermartingale.
martingalewith
X3
k 0
yl
2 3 Varlance *
@
and Covariance
Processe
E .
The next deqnition may look mysterious
: :
r
' .
.
' .
.
. .
tj
3t1r lf X is o tirmous and locally Exe*cl is continupk
of
bounded
variation
thn
r-y
14
( . . '. .
but it is very important.
s ciu inqs : Mtk zf). sincet / s irplies I.X I < Jf (2.4)iplies . Defln that the stopped rocess Mt = Xt A 5') is a bounded martingale. Now if < s t usitk well known properti of conditional expectation e.g., (1.3)and'other (see resultsin Chapter 4 of Durrett we hve '
)
(3.1) Theorem. If Xt is a continuous locat martingale, then we defne the predictable increasing (.X)f to be the unique continpous variance process martingale. and makes local = that hms 0 process A XI A a .a
=
(3.1)
To warm up for the proof of
we will prove a result
in disrte
tim,
,
-+
(3.4)
S((A1
=
Mal 2l.'Fa) y
.E'(M2j.F ,
) L 2M a F(A1'tl.'>-,)+ a ostvj ata) stsy.z y =
t
-
an equation ments since
that we will refer to the orthogopality y is the fact that the key to its validlty
(and hence EM,(Mt
-
G)) = 0).
Mi
E
,
..xyajs,l
...us
.
(3.2) Theorem. Suppose Xa is a martingale with EXn2 < (xn for all ?!. Then = - Wa is a there is a unique predictable process Wa with 0 so that Xxn martingale. Purthermore, n da is increasing. .c
(1995))
ECMS
,
of martingale
(Mt
-
.6)1F,)
incre=
0
Section 2.3
Chapter 2 Stochastic Integration
44
Letting (j
.
.
=
.
(0,t), the orthogonality K (Ef lcf1)sup.f lcjl, and the
subdivision c?&
< ta t be a tn < l martingale inequality increments, the of fact that Mas %ff imply =
E
of
E Mj2
S
E
=
M2
M2jm-z
-
.
m=1 rz
M,--,)2
-
m=1
<E
sup lMzm
v's
Mt--z
-
m
< ff E sup lMm
E ((.X
I
m
'
=
=
.
since
..s.
-
n
'
-+
=
-+
.'.'
.. ' ..
..
..
,
.'.
ECXL
Writing
Xs 1.S) 0. g Qj Qsa + (Qja =
.
.i
A
Qt
A
Q,
-
=
=1
-
Witi aie tw ppiessef ln (3.1) Suppose A >nd Proof of uniqeness z4' martihgate tht is is a contlnuous local the desired properties. Then Jk Sine Jlc = X p 0 it lpc>lly of boundd variation and hence must be constant. (:I q follows that d( for all ./4(
s
,
,
) (
.
-
tmpprtynt, ;
,
.
.
.
,
.
tet
=
.
Qr (X)
=
puxk
k=1
E
-
Xtg- ) +
(.;L
-
As(,)
)
,
y
k
. . .
,
..
k(,)
Xtk-
-
k=1
)a +
t-X
.;V.(.))2 (X,
-
-
hu
-
Xtkfst)a
-
-
2
Xtpta))
.Xp-,)2 + (.X
Xtkfk))2
-
The next step is to take conditional expectation with respect to Fs and use the flrst formula with r = k(,) apd t tk(a)+1 To neyten up the result, deflne 1 sequence a t for (s) K n ; (t) + 1 by letting uk(a)-1 = s, i(s) K i < (), aud tik(f)+1.= =
'tlrj
.
'tzf
-
,
=
.
Exz
t
4A(x)Ia)
-
.1
=
EXlLFsl
-
Qlxj
-
E
(t)+1
)C (v'u
-
=k(,)-f1
Xui-
)2 & '
j
k(z)+1
'W
= E (.V1.S) Q) (X) -
(f )
the difterence
out
.
-
,
.)
.1
jkorkjng .. ..
=k(,)+2
'
E
-
x fk-z )2+ txj xj )2
p
IA).
.
.z4
(x
(f)
,
The ln (3.1) when XL is a bounded martlngale Proof of existence uniqunrq. complicated ipjpiqe onsiderably To its th:rl xistenc pf is more reader ?orttte battle to come, we would like to point out thyt (i) Fs are the ey oo about to prove the special case that we need of the celebrated will construction lpfql prvide infprmation, and som: decopposition (ii)the Tltough and (3.8)are quike the dethlls of thk (3.1) (7.6). e.g., ah>d skip reader boldfard TEp squeamish the ts tp can pr9o are not, so the ahead. Pzoof'' flve q of Existence pages r wtth ltms = = partition < < h x, we t,, Given a ta ( to () = supt : tk < ) be the indk of the lat point before time @. (To avoid confusionlatr, we emphuize now that k.ltj is a number not a random variablt) quadratic variation by We next deqne for a protess X, an approximate ,
(3.4)one
'
-
+
.&
.
.
= (-X.(.)+z
-
.
of
is a
-
a an (j Qa )
-
k(a)
we can do the
.
-
(z)
E
=
(3.3)established
Q/(X)
=
-
E
With
-
A-a)2l.T'a)
-
-
-+
-
E ((A-,
=
the remson for
in the proof
as
45
.X'r)2 - 2(Xa .X'r)J2(.X'j A'alfa) + (uL = E ((.m X,)2IF,) + (.X', aYr)2
Art = (0 g < h1 ;'. k @'i(rz) tJ in which take a sequence of partitins the mesh Ilal = supm Ifnmtnm-ll 0 theu continuity implies supm I.&lk ; Mfz-z j 0. Since the supm S 2ff the bounded convergence therem iiplls 0 a.sk In th last Mzz-, I 0. This shows EMI= 0 so Mt fsupfw lMtz arbitrary, probability with have Mt 0 foi all rational cohclpsion t is one we so El continuity. result t. r'I'he desired now follows from
lf we
-')2l?,) -
then Xf2
martingale
-
l
Mm-,
-
(see(3.1))then
/oof To prove this, we flrst note that remsoning can conclude that if r < s < t then
= jgtwzvE
If Xf is a bounded continuous
Covarlance Procsses
and
If the reader recalls what we are trying to deflne the dehnition is explained by (a) Lemma. mrtingale.
n
Vrfance
= x2.% Qhx) -
4
-
E
J-) f=(a)+1
.L2y -
Ak2y., y-a
46
Section 2.3
Chapter 2 Stochastic Integration
(3.4).
where the second equality follows from
E1 . .
.
.
.
sj. .. .. .
'
.
g . .
.
.
j
Looking at (a) and noticing that Q)(.X) is incremsing except for the last term, the reader can probably leap to $he conclusion that we will conssxuct the desired in (3.1)by taking a sequence of partitions with mesh going process and proving that the limit exists. Lemma (c)is the heart (ofdarkness) to 0 prpof. the of To prepare for its proof we note that using (a),taking expected value and using (3.4)gives
Varlanc
Summing the squares gives h Qr ' (Q ) S Qraa' (.X')suptlapx.z
wherejkj
=
+ Xsg
k
suptj
Covarance Procepps
and
2.X,(.))
-
47
2
tj %skl. By the Cauchy-schwarz inequality
:
.
Eqtxt
(b)
Q)(-Y)IF,)
EX3
=
=
ztxt
To ispire you for the (c),Fe propise routine. of rest of the proof (3.1)is proof of
-
-
that
.
hh'
(e)lf IX 1K M
$
.
the
3
Th ot
valu
A? are two partitions, we calt Lf the partition obtained by Proof If L taking al1 the points in A and in L'. If we apply () twice and take diferepces M2-QAA QhX)-QL'Xjt is a martingyle. By definition t (y) t we see t ha t yi = is a martingale, so A' a' (A-))2 = f'Fr 2 = ('F) Eor h (-Y)
Proof of
(e)
If
r
=
in a moment, we will drop the argument drop the r when we want to refer to the
P rocess t
-+
Qgh.Since
(c+ )2K 2c2 + 22 (since2c2 + 22 (c+ )2 (c
for any real
(3.5)
=
-
Qr
.
J
(F)
-
)2k
'
A
)+
K 2Qr (Q
Q we
.
A'
L'
6Qr (Q )
(d) if IAI+
IA'l
0 then
.
EQ7L'Q*4
. .'
.
.
. .
(e.1)
0
sa < sl
=
.
Recalling
-
Q)k
=
(xaa+, -
.X,)2
-
(.X',p
-
.X,)2
-'Lp)2+ 2(a'Lp+, -L.)(A%p = (-Yak+z 2.X,) (X,s+z .X',p)(-Y,k+z+ X'u -
=
-
-
n
.-
-
.X,)
< sn
r then
=
1
J7l-Ya
2m=1
+
-
.
X,m-,)2(Qrr(A-)
QB a.
-
(X)) .
To bound the frst term on the right-hand t then some arithmetic and (3.4)imply
side we note that if IX
IK M
for all
n
-
X,m-z)*
Slx,m Xam-z)2
S (2M)2f
-
m=1
m=1
(e.2)
.-.o
. ON = 441
-
x-x
.x.2
&,
A ,.
-
.aa..,
va
m=l
K 4M 2EX2r
%4M4
-Y,m-z)a E Fam then use (b) ' botmd the second term we note that (Xa. and IIr Xanxl K 2M to get (X) #((aLm Xawk-z)2(Q1; QCa.(X)J 1.fa#a) . lzstt:rg(.x)z :r,.(x)) jau ) (xa. (e.3) #,m-,) a Exr v'Lm)a IF,m) = t.X'a.,v -
-
=
-
.
m=1
-ux,..z
Qhk+
.
-
-
< sk-v K tj.j., we have
.
A-,--z)2
-
-
To do this let sk C AA? and j 6 A so that % K sk the deinition of Q,a (X) and doing a little arithmetic
does
5-2(x,- x,--,)4
.
0.
-+
J'-!(A-,-.
.
.
that the bond
clear
2
that it is enough to prove that
see
,.
-+
. '.
and
.
it
n
.
X is bounded
n
E V(X,m
E ' .
4
'
LL' is the partition
=
=
. .
since
2A)J(,))
-
m=1
b
0) we have
LL'
Combining the lst two results about
c apd
numbers
but it does mak
a
Qr(-Y)2 r
EQr
For remsons that will become clear from Q) when it is X and we will
S 12M4
r
12 is not ifnprtant on r, A, or L'.
Edepend
and
'
EQhL'Xj2
for a11 thn
-+
Qr
E suplAlk..z + X'k
-+
-
-
(X) 2 J
xalzlsa) once (c)is established
a
'
l/2
(Qa )) K LEQ.
.Y.71.'&)
(c) Lemma. Let Xj be a bounded continuous martingale. Fik r 9 0 and lt r) Svitlk mesh of partitins 0 z g k ty < tnk czi r of L n be a equce 0. Then Q1=(.X') converges to a limit in L1. llal = supk Itnk nk-ll .
'
Whenever IAI+ I'l 0 the second factor gos to 0, enough ti it is to prove nuous, so con
,
-
Eov a *
-
-
K (2.u')2(x, m
-
2,.-,)2
48
(e.4)
-
)2(Qr(-Y)
-x
,m
am-z
Qram (A-))
-
r
m=1
s s
4M ZEXZ r
am
-
m=1
%4M4 the piof I:j
Let An be the partition wtth mqrtirtgay (by(a)), 4 of Durrett (1995))
E
(f) From this and
(c)it
StlP
sr
Am
IQ
-
:Aaj2
s
t
4sjt.)l,a r
-
:Aa r
E If (Z1' 1F,)
j2
Z,w(&)
so that
Q
a limit
-+
uniformly
..
on
g0,r).
Since Q) a convezges in L2, it is a Cauchy sequence in L2 and we can pick an increasing sequence n(1-) so that for m k n()
Using
(f) and
-
Qr
n(p)
l % 2-k
.P sup
t Kr
lQ,
Aa(p)
-
Q,
l
1/j:2
>
sstjgy
it follows th>t ELZ, 1.F,)
E
(3.7) Lemma.
Then
(xYT)
=
jpja;
Zs
=
az
and
Let X be a bounded martingale,
Let Ta be a equence of
< k42-
'i'n
side is summable, the Borel-cantelli lemma implies the only fails finitely many tims and the desird result inequality on the right ' ' ' ' '''''' . . . .. . . El follows.
Since the right-hand
E .
Zt 11,)1'
sjzy
' :
.
.
the proof is
=
XT=
.
(XT)2
-
(vY)T
is
,
...y
complete.
a
stopping
times incremsing to
1(z a >o)
is a bounded martingale
cxl so
time.
that
By the previous xesult there is a unique continuous predictable increasing prois a martingale. By (3.7)Wy = .A7+1 for K Ta so cess Wn so that (Xn)2 we can unambiguously deflne (aY): = z4y for t %Ts Clearly, (X) is continuous, predictable, and increasing. The deflnition implies Xy t.X'lzaa: is aj 1(z.>a) (.X') is a local martingale. a martingale so Xl E1 .y
-
.
-
Tt
-
End of Existence
Proof
-+
-+
(:I
local martingale
--+
-->
()
(:)
.
we By taking subsequences of subsequences and using a diagonal arzment each Q? is continuous, all Since N. Nj for uniform convergence on (0, can get a QLh is not increqing. A is, as well. Because of the lmst term (.;% .X()) dkz-ar is increasing. However, if m k n then k 'n is incremsing, so k Qka-ar continuous that it f6llows and is t arbitrary A is incremsing. Since zz is t A -
y
.gg
and T be a stopping
(X)T.
Proof By the optional stopping therem so the result follops from uniqueness.
2
Chebyshev's inequality it follows that Aa(p+z)
LP
-
.yj
Proof of exlstence in (3.1) when X ls a loal martingale Our flrst step in extending the result to local martingales is to prove a result about the quadratic variation of a stopped martingale. Once one remembers the deflnition of XT given at the begirining of Section 2.2, the next result should be obvios: the quadratic variation does not incremse after the process stops.
Proof
f lQrim
E (.Z 1F,) T = ELE Z1
-
g
Taking limits in
follows that
(g) There is a subsequence
&,
Proof The martingale property implies that if s < then .E'(g1tIFa) = Z1. Th right-hand side converges to Zs in LP. To check that the left-hand side cont ELZL we note that linearity properties of and Jensen's inequality verges to 1.,#-a) for copditional expectation (seee.g., (1.1a)and (1.1d)in Chapter 4 of Durrett (1995)) imply
.
.:2
2).
,
. i? a z-nr with () s k s 2n. sinceQtnx inQ).' poipts Chapter maximal inequality (seee.g., (4.3) usingthe gives -
(withp
check =
-.+
,m-,
.
It only remains to put the pieces tgether.
is to
49
(3.6) Lemma. Suppose that for each n, Z.J'is a martingale with respect to and that for each Z; Z$ in LP where p k 1. Then Zt is a martingale.
-X1
completes
Varance and Covarance Processes
The last detail for the cmse in which X is a bounded martingale yl. is that Ml a martingale. This follows from the next result
we
.
4M2E EX2
(e.2)into (e.1)proves (e)which
Plugging this bound and of (d)and hence of (c).
(3.4)ms before, n
l
g(x
f
over m, and using
(e.3),summing
Taking expected value in get n
Section 2.3
2 Stochastic Integtation
Capter
Finally, we have the following
extension
of
(c)that
will be useful later.
Chapter 2 Stochutic
50
In tegration
Section 2.3
(3.8) Theorem. Let Xt be a continuous local martingale. For every t and of with of subdivisions mesh IAn Aa se4uence (0,) l 0 we have
Proo
Vrfance
and
Covaziance Processes
51
Since
-->
sup ast
IQ)n(X)
(vY)al
-
Q)n(X, F)
0 in probability
-+
Proof Let J, : > 0. We can flrtd a stopping time S so that X( is a bounded coincide martingale and PS K t) % lt is clear that Q*(.X') and QLLXS) n (0, From the deinition and (3.7)it follows that (A') and (.XS) are equal on i (0, 51. So we have
tis
follows izmediately
1
=
a a(X (Qa
+ F)
an y Qa (
-
from the deflnition of (X,
.s))
#;) and (3.8).
E1
.
,%.
Our next
is useful for
result
(X, F)k
computing
.
E
sup IQ)(X) a
-
K + P
:
-+
lf X
and
'
(-Y,1')
as
(c)implks
0.
(3.9) Defnltion.
two
are
'
that the lst tern joe to 0 as (:1
locl martingals,
continuous
1 k((X + F)z
=
> e (-YS),I
-
X -
are continuous local martingales. (X, F)f process A that is locally of bounded variz A a local martingale.
.
Since X-ne. is a bounded martingale
lAI
sup
lQ)(-YS)
(3.11) Theorem. Suppose Xt apd is the unique continuous predictable ation, hmsJlc = 0, and makes a'V
ArZ'i
Welet
(.X' 'F)t)
-
Proof
From the de:nition, it is -
(X, 5')z
=
1((,Yz+ K)2
sy
tb se th,t .
(aY+ F)
-
=
.
((
-
K)2 = (X
-
F)))
is a local martingale. To prove uniqueness, observe that if Jk and /( re two processes with the desired properties, then Aj .A( (vY! (A'i$ yk) locat martingale that is lotally f bottdd is a cntinuous varition ad hence H 0 by (:1 (3.3).
-
.()
-
-emark. If X
and F are random
variables
with
mean zero,
,
=
-
-
-
,
covtx, F)
=
EXY
1 k(f'(.X + F)2
=
1 = ztvartX + F) so it is natvral,
I think, to call (vY,F)
vartx
-
the
EX
-
-
Fjzjz
From follows, X,
F))
-
covariapce
of
X
and
=
.
k()
(A'k
=
-
Xp-zlt'Ft.
-
.X,(,)
)($
-
Exerclse
3.2.
(-'Y+ F, Zjg
Ekercise
3.3.
(uY
Exercise c = b, X
Kk-,)
=1
+ (X
')t
=
1
(m Zjt
=
(F -Y)t )
-'''
--
. .
+ (F, Z).
=
.
-
get several useful properties of th covariance. I what and Z are continuous local martingales.First since V'VX= Xx, (.X
=
Q)(-Y,F)
'
Y.
Given the defnitions of (.X)t and (vY,'Flz the reader should not be surprised that if for p, given partitipn L (0 p < l < t2 .J with lims tn x, we let (t) supt : tk < t) and deflne =
(7.11)we
5$k(,))
Exercise
-
a'V,Z)
=
(X, Z).
lf c, b are real umbers then (c.X', F)t abLxjF). Taking F, and noting (yY,yYlf (.X')f it follows that lc.Xlt = c2(A')t
3.4. =
=
=
.
3.5.
(.XT,
FT)
=
(A-,F)T
then we have (2.10) Theorem. Let X and F e continuous local martingales. and sequence of partitions Aa of (0,t) with mesh llal 0 we have
r every
t
(x,'.FT)
=
(A-,5')T
-+
sup
lQ)n(Y,
') -
(-Y,F),
l
-+
0 in probability
Since we a<e lazy we will wait until Section 2.5 to prove this. We invite the reader to derive this directly from the deqnitions. The next two exercises prepare for the third which will be important in what follows:
52
section 2.4 Integration
Chapter 2 Stochastic Integration
then
Exercise 3.6. lf Xt is a bounded martingale integrable martingale.
XI
-
(c) (ff X, J'f 5')t
a uniformly
(.X') is
-
In this step pake thir flrst
and S is a stopping time then Exerclse 3.7. If X is a bounded mrtingale = with (X)s+t-(X)s. aud (F) = respect to Fs-vj Xs-v -Xs is a martingale 'I
3.8. If S constauton (S, T1.
Exercise
KT
are stopping
times and (.X')s
(X)z, then X is
=
=
-
.Jfaff Jnf
a
Bounded
w.r.t.
S
T are stopping
times and X is
constant
n E
=
2.4. Integratlon
E
we will only prove a version of The other in Step 2. Before plunging into appearance
(a).
two results will the details recall th t a t the end of Section 2.2 we deflned Ht to be bounded if there was an M so that with probability one Ht K M for all t k 0. (4.1.a) Theorem. If X is a continuous martingale and H # is bounded), then (S u'Yltis a continuous martingale.
(ff r
ln this section we will explain how to integrate predictable processes w.r.t. inbounded continuous martingales. Once we establish the Kunita-Watanbe continuous able general local will be equality in Section 2.5, we to treat a martingale in Secion 2.6. As in the case of the Lebesgue integral (i) w will begin witi te simplest fupctions and then tatt limits to get th general case, and (ii)the seqvpce of qteps used in defining the integral will also be useful in proving results later on. Integrnnds
tJ) We say Ills, (J) is a baslc predictdble process if Hs, of predictable = basic processes. where C E Sa. Let Tlo the set aud X is continuous, then it is clear that we should deflne
=
1((,,514s)C@) If .I 1(.,)C =
-
aYlz H
Hs dX,
know
we
..
.
E..
.
.X')lF,)
.A-)a
..
EH
=
= = where the last two equalities
-
CECXL
/ (c, ) and
for t
-
=
-
follow from C E F.
Exercise 4.1. lf 5$ is constant is a martingale. t K then
CH .X),l.'&) u'L)1J7,) .X',)IF,) 0
.-Y)t
ECCIXL
and
Euj
1.f-,) X, =
.
= .E'(5$1..#'a) F, whep c
We say Sts, tJ) is a slmple predlctable 8an be written ms the sum of a llnit nuzber is easy to see that if Il' e II1 then H can be '
.
''
''
.
'
.
Hs, of
)
.Ar)t,
written
as
i =1
the next thrre '
.
.
.
.
...
.
.
.
.
where
tl
0 we say subsequence passing probability.'' By to a we can hve in
1(#n .X')
u r xllz
-
4jI(.Jfn A')
.
E
.x)
tst#n
sup
4.4.
Exerclse
4.5. lf X is a bouned
II II
to
0
(# ..X)t
2.5.
rlnlte
ll#
X112
=
.
Ineqallty
local martigales
are
1/2
co
Isffa IdI(.X',F)1,
where dl(vY,F) j on (0,).
lastands
and
H and ff
surely
0
two
1/2
x
H, d(-Y),
K
are
ff, d(F),
0
for dk:t where
16 is the total
variation
of r
(X, Flr
-+
'''''''
and
(ff
.
H, ff
e
1I2(X)
.X')f
that S'+ff 6 1L(Ar). The triangle inequality for the norm 11 Proof .IIximplies n and n ffllx with 0 Let Hn ff e llz llff K Ilx 0. The triangl llffnr&) implies ll(JJrz CH Jf + )llx 0k (4.2.b) + ff inequalityimpes -+
-+
-
-
,
-+
-
'''''''''''''' r
.
Remarks. d1(.X',F) 1, s and (i) If .Y y and d(vY)a s then d('), (5.1) reduces to the Cauhy-schwaiz ineiuality. ()i)Notice that H ad ff r not mssumed to be predictable. We assupe only that Hfs, (z2) and ff (s,tJ) are mmsurable with rvspect to .R. x F wheie 7 is the Biel subsets of R. ht tlat tt notiop of nrtingqle ttyin this leyel of generality we ca doespot ntr int the fiof arfter the first stef. =
(Gn vY)t .
-+
(G vYlt in 442 ivhen .
Cl
=
=
ts
(
.
Proof Steb : Obserk tht if s S @,(#+F, lf Welet (., #)f, z (M, Nj3 (M, #),, then .
(
'
'''
'
'
.
.
and use the fact that
=
rraszt
:
.
and H 6 lla(aY) then
martingale
Knita-watanabe
(5.1) Theore>. If X and F mesurable processs, then almt
0 El
.
,
-+
Ip this section we will prove arl inequality due to Kunita >nd Watanabe and afply this result to extend the formula given in Section 2.4 for the covariance of two stochastic integrals.
a.s.
continuous.
((# + ff ) .aY)t = CH .X)f +
=
llznll llzIl.
0 then
-.+
-
.
+
Now 1et n x G H, ff .I + ff
llzn zII
and
-
(x)
(4.3.b) Theorem. lf X is a bounded continuous martingale, then H + ff e H24A') and
-+
1111is a norm
lf
--
The fact that H.X E M 2 is automatic from the definitiim. lf H e TIl Sine Xlt is contipus. 0 then (4.2.a)implies have Ilffn Jfjjx maxiral L2 inqulity nd the using lll#n -Y) CH -Y)II2 0, chebyshev .
Exerclse .f
Proof
=
will be given in the next section. To develop the rsults there we will need to prppexty extend the ismetg from 5111io H24.X'4. To do this it is useful to analysis. rall some of your undergraduate
.
'
''
'
X+yl
'''
.
.
E
.
''''''''''
(.X+F,X+'/),. '''
.
.
-
0
K (aY+ F, X
= (.X',x)t,
-
+
ylz
-
(.X'+
2(A', F), +
2
F, X +
(F, y)
'F),
.
60
Capter
Secton 2.5
2 Stochutic In tegration
umber of lf we llx s and t and throw away a countable for al1 s, t, and will with hold for all probability inequality last the of 0 then one sets measure rational Now a quadratic cz2 + z + c that is nonnegative at al1 the rationals and not identically 0 hms at most one real root (i.e.,52 4cc K 0), so we have that ((-Y,5'),)2 K (aY,-Y),(F, 'Fla .
.
-
'
< ta be an incrmsilg sequence of tims,' let Step 2: Let 0 tn < tl < hi, ki 1 ; K n, be random variables, and define simple mauiable proc =
.
.
.
Te Tfunlta-Watanabe Tnequallty
61
To see that this exists, note that J, is the Radon-Nikodym derivative of d(X, F), with respect to d1(.;LF), 1.Now apply (5.2)to .Jf, = Iffl and ff, J,lff, l.. El =
With (5.1)established, from Section 4.
(5.3) Theorem.
we are now ready
If H E 1I2(X)
n 1la(F) then .
...
.l (A- + y))t
,
=
.
business
H E l1z(X + F) and
'. ... .
.A-)t + CH
..
of unflnished
to tpe care
.5')z
n
.Jfts, tJ)
f()1(t-z,.j)(s)
=
= n
ff (s ,
(J)
Proof
First note that Exqrcije 3.2 implies
1 . -ft$d
)1(t i-
=
i=1
..
.
z,
f)
(s)
.
g . .. .. . .
;)
q
and the Kunita-Watanabe
the integral, the resplt of Step 1, >nd the Cauchy-schwarz From the deinition that inequality it follows of
n
S'aff ,d(X,
'),
Ifl
n or IXtI> iz1, it suces By stopping at Tn inf ( : that if X is a bounded martingale and H E I1a(#), then CH .X')f is a cpzinlou. martingale bt this follows from (4.3.a). (:3
ff ) + ffa2 where
.rf'l
H, Clearly, H i mplies
.
X)
=
1
(ff
=
'.F)j for
.
(Jf2 .X'),
al1
(.Jf 2
=
.
and it follows from tr Combining this
.H'4 ltasz)
=
a
Jga 1(asz)
=
t. Sipce Sa2 .xla
3.8 that (S2 .X')f the qrst result and using
xercise with
(Jf
=
.
conclusion.
K T, (5.4)
((.Jf + K ) .X')t
2
.
.X'l
.
=
0 for t
k-
(4.3.b)gives the desired
14Jf 6 IIa(X)
f3 1L(F)
-(ff
extend
the integral Sa
%Tp
inf
=
let Sn be
now,
(t
IA'tl >
:
>
times
of stopping
a sequence
with
.Jf,2(f(1), > nj
n or
0 '.
k
and ' Sa 1 n. tt. Sl' z ffaltassa), and observe that if r? n, (6.1)Jmplis that Hm Xlf V: Hn X)3 for t K Sm so we can defln H X by setting CH X)t (Vn Xl fOr t S Sn To- complete the definition we have to show that if Rn and Sn are two sequences of stopping times K Ta with Rn 1 G) and Sa 1 (x7 then we nd p with the same (# .X')t Let SJ be the stopped version of the process deflned a t the beginning of Section 2.2, let Qn Rn A S,z and note that (6.1)inpli E
.
.
=
.
..
xg
.
,
CH .X')f +
=
(s Proof norm
.(x
+z)),
(s
=
.
aYlt
.A-),+
u
Tp provy the flrst formula we note Shat th trt>ngle inequality for the 1 11Ix implles H + ff E l1a(.X). Stopping at .
Ta
=
inf
t
't
IA) I
:
,
t
H,2 dt.X'la or
,2
Jf d(F)a
,
0
>
nj
0
reduces the result to (4.3.b) For the secohd formula we note that #he argument in .I e Hat.X' + 5') and by stopping we can rectuce the result to After seeing the last two proofs the reader
can undoubtedly
.
(6.5) Theorem. K e 1Ia(F) ther
lqcal martingales,
.
lf X, 5r are
continuous
(5.3)shows (5.3).
imyrove
that E1
(5.4)to
(IIR,
.X'),
.
=
(SW..
.X'), for
s
% Qn
#,ff,d(X,
=
.
.
chosen. The uniqeness
If X is a H$
.
X
result and
continuous .l
=
.
X)V
(6.1)imply
local martingale =
of stopping
H
.
XV
=
HV
and .
XV
H E IIa(aY) then
e
l1a(vY) >nd
F),
0
Using Exercise 3.2, we
Since Qu1 x, it follows that (.Jf.J)t is independent of the sequenc
H
j
LII X, ff F)z
=
(6.2) Theorem.
(ff
.
'
.
tires
q
,
then H E 11a(X + F) and
=
-
locq) in>ytirlgales. If .J ff
.rj,
El
=
'
Let X and F be cpntinuouq (6 4) Thorem. l1a(-Y) then H + zf 6 IIa(x) and .
0 for s
=
6.1. Extend the proof of (6.1)to show that if X i a bonded Execise continuous martingale, S, ff E l12(-Y) and H. A-, for S K s K T where X,T stopping (S .X), (S.X)s thn (.Jf .X), .Xjs for S % s i T. times are
To
.
for s < T
0
=
.
ffa2
=
=
to show
can
generalize
(6.5)to sums
of stochmstic integrals.
E/n=I JfJ Hi X and y Yi where the X (6.6) Theorem. Let X = E'C=1 Yi continuous local martingales. If Hi e l1a(.Xi) and ffJ 6 lIa(5Y), and arq then Hjlf'is dxi FJ), (.;Lyl = t) ij =
.
.
j
,
l
Chapter 2 Stochastic Integration
66
Sectbon 2.6 Proof
We leave it to the reader to make the qnal extension of the isometry property: '
.
Exerclse 6.2. If X is a continuous .I X e 442 and llS Xlla = IlSllx. .
local martingale
.
.
.'.
.
:
'
We will conclude this section by compttting some stochutic integrals. The first formula should be obvious, but for completeness must be derived from the deflnitions.
Ill#n=
H4
.7111
=
Ilsn
J.f11i K Mlsup
=
as n
-+
tx
by the bounded
For the rest of this
Jfa dXs
section,
(7(.Xy
=
zka zz (0
az
-
=
Exercise
local marting>le
6.4. If X is a continuous
2A%dX' .. .
.
.
< fnkta)
...
. =
..
.. .
tJ ls
0. Our a sequence of partitions of (0,f) With mesh Ir,I = supf ltJ' t7-1I theme here is that if the integrand Ih is continuous then the integral is a limit of Riemann sums, provided we evaluate te integvand c f8 lefi entlpcin of e intervals. -+
-
'
(6.7) Theorem. If X is a continuous local martingale then as n cxl
and
t
t -
-Y(f7))
#, dX'
-+
()
i
Exercise 6.5. Sltow that if X is a at the right end point then 1-12A12 zlX(f2+1) +
-
.X(f7)l
X
-
(.')f
local martingale
continuous
2X4 dXa + 2(X)t
-+
0
i
-
=
A)2
and we evaluate
-
XJ+
.
-+
.&'t7l-Y(Pz+z)
XI
=
then
(.X')z
Comparing the lmst two exercises you mightjump to the onclusion k if we evalute at the midpoint we get an integral that performs llke the calculus. However, we will not ask you to prove this in general.
Ht is continuous
--,
0
.
=
be C for
< cxl
0
t. Since s Jf, is continuous o (0, t) we have fox each ta that
2a
=
'
dimensional
motion zrownian
starting
1
J'-)2#(( k=0
,
a one
that in
one
+ 1)2-ntj
+ 1/2)2-nt)(f((
-
#(l:2-nt))
-+
,
sup l#,n
-
H, l
-+
0
The desired result now follows from the following lemma latr.
which
will
be usefpl
' .
(6.8) Lemma. Let X3 be a contirmous local martingale with (.X')t % M for all lf .Jfn is a sequence of predic#able processes with lSn I K M for alt sLw and . with supa lS,' .Jf, 1 0 in probability then Hn .X') CH .X) in Mz. -
-+
.
-+
.
converges in probability
Jc2f4
to
dBs + t
=
B t2
.
integral, is more conThis approach to integration, called the Stratoovich venient than It's approach in certain circumstances (e.g.,diflksion processes on manifolds) but we will not discuss it fvrther here. Changing topics we have of (6.7)that we will need in Chapter 5. one more consequence Exercise
6.7. Suppose /t
:
(0,x)
-+
,
is continuous. Show that /12ds.
a norma 1distribution with mean 0 and variance
Jpf ,
J
,
dB' has
68
Section 2.7
Chapter 2 Stochastic Integration
2.7. Change
Varlables,
of
It's
Formula
Comparing
This section is devoted to a proof of our flrst version of It's frofrmla, (7.1).In with a slicker. Section 2.10, we will prove a bigger and better version, (10.2), proot but the mundane proof given here has the advantage of explaining where the term with T' comes from. (7.1) Theorem. Suppose X is a continuous continuous derivatives. Then with probability
y(.m)y(.x)
and
+
:
.J(.t)
(7.2)
f has two
/*(dn)
(
1 -
a) 2
f''Xs4dLXLs
t)
=
variation,
=
.;Vy+, ) when
,
s 6
(f)'t;%1), ,
(.m+, xlz -
X
$
i
so that
t
-X7)2
-
G1d.'',
=
and what we want to show is
to
ot
(7.7)
'prv
any a and
(Izj
.-+
g
1 (c(c,j;;
j
Jnx, )g(x),
To do this we begin by observing that th uniform continuity of T' implies that uniformly in s, while (3.8)implies that .An, as n x we hav Gnr y&t.X'aatl probabillty to (X)a Now by taking subsequences in we can suppose converges that with probability 1, Wn,at converges weakly to laYlaaz. In other words, if (.X').at as distribution functions, then Wna and s td and regard s we fixssociated memsures converge weakly. Having done this, we can flx LJ and the deduce(7.7)from the following simple result: -->
-+
t
't
(?J,
.
=+
-+
'j'(
+
show
-+
.n
0
ajfas
-
to
69
0
=
=
t
in probability as n x. (7.5)follows from (6.8). To prove (7.6),we let Gns gl @) fncxj: Gn JN(.;V)for s k t and' 1et
:,P(*)(A7:.+,
By stopping at Ta; inf (t : 1AR Proof or (X)z k MJ, it sces lcalculus and result when the we know that for l.;Vl (aY)t < M. From there is a ctc, ) in between c and b such that ,
laj
0
j
i
want
Formula
fxsjdx.
-+
1
-.+
-
,
,
-
Vaiables, It's
=
As the reader will see in the proof of (7.1),the second term coms frolit the fact that local martingale paths have quadratic variation (.X')t while the 1/2 in frpnt of it comes from expanding f in a Taylor seriesk
J()
.w(*)(-'L;'+z A77)2
fAsjdAs
=
-
1 -2
=
Remarx. If Jk is a continuous functiop that is locally of bounded and f hms a continuous derivative then ('ee Exercise 7.2 below)
-'Ly)
-
(7.6)
of
becomes clear that we
T'(.XI-)(-Yz:-+z
j,' rtxaldxa!./' r'(x,)d(x),
=
-
local martingale 1, for a11 t k 0
(7.4)with (7.1),it
Change
'
Let t
tg < ty
.
.
.
msl
t =
.t
'are
m
.1
-
(t.0) Quadrtlc Varlation. Sppose X ad 'F gales. If rj is a sequence f pititions of (0,t)Svith
my:
.yd
0
dxkit
X,klrnl
=
0
Summary
,
,
z,z
n+1
n
By deflnition,
(jjxl)
k(m)
X, ,f
1,
,
i
1aw (9.6),
associative :
.'
j
1
+2 i -
m=t
m#
n-l-
of the proof of the multivariate With the completion fprm of Is's formula, we calculus enough stochastic have to carry us through most of the applicationsin the bookj so *e pause for a moment to recall,what we've just dtje. In Section 2.1, Weintroduced the integrands ff, the predictable processes nd in Setion, 2.2 the (flrst)integrators Xt the contipuous local maitingals. with p, local In Section 2.3 we defned the variahce process assiated map covariance of and (.X, generally the two local marting>leg. more tingale, Theorem (3.11)tells us tha t t h e covar iance cpuld hye been deflned as the uniqu continucms predictable process dt that i loca ly of boundd vari>tion, The next result gives a has dtl is a martingaler 0, and makes u'V'i pathwe interpretation of (.X,F)t :
X' ktrnl dxkij
=
=
(7I
=
Applying
79
.'
'
.
Summar.y
y: , j .
Tl
Xs (m) J.X'k(rl+1)
I'I
m=1
,
Taking X time t.
:::F:'.F we see
that (aY)zis the
tquadratic
variation''
of
the path up to
82
Section 2.11
Chapter 2 Stochastic Integation
used only in The developments described iu the previous paragraph are material in Sectioh (9.3) of Chapter 4 and in Example 2.1 of Chapter 5. The would like used until 5.5. We Section 2.12 concerning Girsanov's formula is not of Chapter 3. beginning to suggest that the reader proceed now to the
To dal
extension of It's formula dtte to Veyer (1976) ln this section we will prove an Loosely speaking that hms its roots in work of Tanaka (1063).See (11.4). functions. of conkex two it says that It's formula is valid if f is a diflrence and cuntable hke a t Since such functions are diFerentiable except at a generali/>tim. modest signed memsure.. this is a second derivative that is a nice Brwnian general: if Bt is a staudard in do can best the this is one HoFqver, di the seminpytingale be mpst # t en f renc of motio and Xj ::p fBtj is a and Jacod, protter, (1se). conyex iunctiops. see (j this section follow those in Sections 4 and 5 of hapter IV o
slkazpe
cinlar,
two(1990)
developments in but things are Protter is step
since
here
simpler
we
.-,
X be
a contiuous
fvt
Ixzj
Here f') = limhtntftz) convexity and f is a z by -
(z continuous -
'
.
.
'..
.
By be the decomposition of the semimartingale. Proof Let Xt = Mt + and N N, that l.Xtl S l.l 'stoppingwe can suppose without loss of generality witlt compact upport in (-zx,0) function be all C* Let g a (M)z % N for and having j'gsj ds = 1. Let .lf
.
Js(z)
(a) Then
(b)
and
Jn is convex 1uXj4
-
j J (z +
-
Jn(A)
,(p) dp
0
Jn'(X,)
.
sets. Since
on compact
in t
difl-erentiate
(a) to
get
.X.)gy) dy yztz) n .y
:
'
tu
.
J
integrals (Eyrcise 6.2), imply E
(sup(z;'
.n)2)
zhu zlz s xsup t
-
-
'1
Kln'
4s
by the bounded subsequence we
/,
(-Y,)
-
T'(A%))2 d(M),
theorem. (Recall (M)x K #.) improve the last conclusion to
convergence can
sup
(Q* -
ft)2
almost
0
-.+
=
lJ:
(f)
sufp
-
Jt I
s
.
-
T o take the limit of the third note that (b)implies
and
0
surely
t
j,- 1Ja/(x,)
-+
By passing to a
/ 't ?' Let J1 /' (JL) dAs ln this L (-Y,) dAs and h from the bounded convergence theorem that for qlmt every o
g)
.
=
(z +
we
83
J'(x,)l dA'
final term ffy
-+
() 2 Jf
1
=
case
it follows
0 .J??IX, ) d(X), r,
we
')
C* Using It's formula we have t
the right,
=
is the left derivative which exists at a1l adapted increasing proess.
.
y?
=
uniformly
uniformly
-.+
dA', + Ik
0 .))/
on
Fomula, Local Tfiaae
' t X( co Now if 1: = s n x 4 ) 4 and z = () J'x , ) d.u , then tli; Z2 maximal inequality for martingales and the isometry property of stochastic
(e) 1'X't
=
y,a (z)
'rke
t -
(d) '.
.f(z) -+
Iht
'->
the irst term
with
hav no jupps. Our flrst
R be convex and let Let f : R (11.1) Theorem. and semimartingale f(X) is a semimartingale. Then
A(z)
fnv)
Local Time
Formula,
2.11. The Meyer-Tanalta
interval, it is easy to see that l.Xtl f N, it follows that
The hfeyer-lnaka
dxs +
'j
t
(j. 0
fn''xsl d(X),
co in (b)to get the desired Our strategy for the proof will be to 1et n continuous on each bouuded formula. Since a convex function is Lipschitz
,
ff
''k
=
fnu(.Xt) 1nu(.X0) -
-
()
IL (Xa) dX, g
Combining (c),(e),and (f) we see that almost surely the right hand sie of (g) converges to a lint uniformly in t, so th left hand side dots also. Since Ikn* is continuous adapted and incremsing the limit is also. E) If
-+
ated
we with
fix X then for a given f we call ff the increasing process associf. The increasing process associated with Iz cl is called the local =
Sectfon 2.1 1 The Afeyer-Tanaka Formula, Local Triae
Stochutic In tegation
2
Capter
84
that justifles this
tlme at c, and is denoted by LI. To prove the flrst property we need the following preliminary name, (11.3),
stopping we can suppose I.X I % N and (-'Y)t < N for all t. Having done this we can 1et N J(z)
1et fff be the increasing tl with begin hh-h = Iz-cj,so ff/+ff = processmssociatedclaim Jf Wesince is that h Ja = z c we have ffl fff2 0. To .! Our second 1#X, o ' let g) = < c which has g?(z) = 1 and note Xt A'c = see this ( the associated fft must be H 0. Let
A(z)
(z
=
(z)+,
-
c)observing by
A(z)
.
(z
=
-
,
and
(11.3) Theorem. Ljt' on l y i ncremses when X t ft' be the memsure wlth distribution function t ( : Xt = cJ.
= -+
E
. '
E
'
il . ' .
: . :
*' li c r t be' preis a sqpported by LI, then is .
'
. . ..
'
'
tf ' .
,
.
=
=
c)+
Xs
-
c)+
-
=
r
1(x.>.)
dXs +
1
'j'(f.1.
-
.q
#'(z) Starting
with
the deflnition
j
j.1
=
S
'((f
cl
-
-
and then integrating 1/2 p(vY)
lA%
cornplete.
N
Lx Jztdtll
.(A%)
-
-
Since LI for g and
We are nop rvady for our main result. Let (11.4) Meyer-Tnnalca formula. Let X be a continuous semimartingale. aud of derivative left be the functions, f of f' two convex f be the diference distribution functin = y in the sense of distribution (i.e.,# has suppose T) Then t 1 I'X'j #-Y, + /z(#c).U Ixzt
r?
0
cl c'=
-
=
p(A%) p(A%)
:
=
c)
-
signtxa
c)
-
dxs + Lta
'j
j'
Proof Since f is the digerence of two couvex functions and the formula is linear in J, we can suppose without loss of generality that J is convex. By
1
1
gives ' . . .' :
/.N
j.x
ptdc)
N
y
rt
j
signlx,
-
( . .
. .
c)
dXs
..
. .
' .
.
ydajL;
-N
(11.5) and (11.6) below will justify interchanging the two integtals in the flrt side to give right-hand the term on
=
-
-
signtz
the local time
of
-
/(m)
yda)
-N
+ =
.
N
't
1A)
1)
Zj. side vanish so L The left-hand side and the integral ou the right-hand : Xt k = and > S rational inf when T with H S holds any (t q Since this q a((t argument : Xg < c)) = 0. A similar c 1/n) for any n, it follows that c)) and a)+ : Xt > shows the proof is using (z a)- instead of (z -
cl
-
.q,
=
-
pztdcllz -x
,
a so LI Intuitivly, I-'V cl is a local martingale except when Xt Proof emsier is it to wlzen however, use the constant To Xt a. prove (11.3), is # alternative deqnitions in (11.2).Let S < T be stopping times so that (S, T1 C pplying (11.1)to /'(z) = (z c)+ we have (t : Xt < c).
xr
1 2
-
J(1
-
-
-
=
=
-
-
-
.
=
gz) Since (J g)H z a + bz for 0 on (-#, Nj it follows that fz) IzI/ N. Since the result is trivial for linear functions, it suces now to prove the result tor which is almt trivial. Difl-erentiating the dqnition of we have -
Proof
85
=
0 for
=
j
t
g'xsj
dXs +
> N (recall l-'Ll K N and 1a1 the proof of (11.4).
completes
use
1
CN
y j.x
Jtldclf4
(11.3))this
proves the result E1
For the next two results let S be a set, let c be a c-fleld of sgbsets of S, artd c$-). let ;t be a nnitemeasure on (,% For our purposes it would e enoug totake S R, but it is just as emsy to trat a general set. Tlle first step in justifying the interchange of the order of integration is to deal with a measurability issuq: if we have a family .Jf/@) o? integrands for a E S tlien we an deqne thq integrals .Jf: #X, to be a measurable function of (c,t, td). =
,
,
semimartingale with Xo (11.5) Lemma. 0 and Let X be a continuous let S/&) = Stc, =) be bounded and S x 11 measurable. Then there is a =
,
Capter 2 Stochatic Tntegrafpn
86
Z(c, t,)
e 't
x 11 such that for p-almost HI dxa
ts
J;
versionof
. @,tJ)
a Za,
every
is
a continuous .
.
Mt + Jlf be the decomposition of the Proof Let Xt = stopping we can suppose that l.l % N, lMtl S N and (.X')t a11t colusi Let 7 be the collection of bounded 11 E S x 11for which the Theorm of (2.t) the Monotone Clss hOlds Wewill check the assumptions check (i) of the MCT suppose in Chapter 1. Clearly 1'f is a vector space. To (t,tJ) where ff (t,td) e l'l and fa) e bS. In this case #tc,t, ) Jtclff =
,
.
=
t
'1 .f(c)
=
0
0
=
i.:zz
:/
.
Js
Z /z(da)
ff s dxs
TI ?t. Taking /' and zt-to be ipdicator functions of sets in H). = (X with B E i S, B of checked MCT x e Tespectively, we have (i) te with Hn 1 H and H bounded. To check (ii)of the MCT let 0 K Hn E of the The L2 maximal inequality for martingales and the isometry property stochmstic integral imply u%
so Jzf e
.4
.
sup It.Jfn'c
.f')
-
.
CH'
.
't
M)tI2 g 4sup
EkHn' (Jfn,c
= 4E By passing to a subsequence
we
sup lt.Jfnk't' variation
To deal with the bounded ..),
ltzznz'z ssup f
.
almost
have for p .'l
-
.
(ff
'
-
.
Mjt
-
ll''
Hall d4M),
-
.
M)l2 O
-+
'z..)I s
Af)t
l
Js
Iat
/2tdc)
Tf
l
A'lt
'
CH -Yl
=
.
so f ff e 7t. Taking f and ff to be indicator functions of sets in S and 1-1 respectively, we have checked (i)of the MCT with = (.A x B : 6 <S,B G 11/. To check (ii)of the MCT let 0 S Hn E 'S with Hn 1 H and H bounded. Letting 11,11 be the total mmssof pj applying Jensn's inequality to the probability measure Jz(.)/IlJzl), then mltlplying each kide by we have ,4
,1.
11,11
1 s(/
sup
s 11/211
lzlz'a z/l
ydallz
-
s E
sup t
S
Izfn'? z/ 12gda) -
Using Fubini's theorem, the L2 maximal inequality, and the isometry property the right-hand ide
0
-+
=
Izz;''c H:6 dl.lf -
0
'
every c
part we note that
Ej*
1L .X')t
=
(l/s T(') pdas =
and
'//
E
Proof Let a'V= M3 + A be the decomposition of the semimartingale. stopping we can suppose that l.Alz< N, lMl K N and (.X'): = (M)f S a1l As noted in the proof of (11.5), the result in the special cmse X .A just Fubini's theorem from memsure theory, so we will suppose the proof that X M. Let 7 be the collection of bounded H E S x li for which the conclusion holds. Again, we will check the assumptions of the Monotone Class Theorem, (2.3) in Chapter 1. Clerly is a vector space. To check (i) of the MCT, (t,Y) = where ff (t,kJ) suppose fftc,t,l Jlalff e ll and .J(a) e bS. In this cmse ZI = J(c)(ff .X')t so .
By (Af)z K N for
semimartingale.
Hs' d.X,
'
'
Section 2.1 1 The Afeyer-Tanal'a Formulat Local Time
-+
0
s 4
S
Izn'
fsup
1
7/12 gda)
-
.i'latn'a
s
sup
-
j
gda) Z.112
Cr
= 4
E
t.Jfan'l
) s by using the bounded convergence values inside the integral we have
zlz,c,(dc) stsypl/s
/-/-u:
dx- ,(dc)
-
/'/ utyda) -
-
Hslll #(;),
-
plda)
theorem three times. 2
-
j zC,(dc)I)
s / sup lzJ',.)(
dx,
0
.->
-
J
S
'
zplptdalhz ,-,
J
()
88
Secton 2.11
Chapter 2 Stochutic Integration
Taking H1
js tff'''
=
nk we have that with Jlldcl and pmssing to a subsequence Zlk'' /.lldtzl converges uniformly to ZI lztdcl. ..X') =
Js
Js for then it The lmst detail is to check that #n X converges to .I X in proprty, the isometry we cap followsthat Zf llda) = CH Xlt ln view of ffllx 0 but this is routine. showing that llffn by detail lst the complete
probabilityone LHn
.&d?k
.
.
h
1jj E 11#
j
x
0
.
.
-+
-
js
#,nztlgdaj
-
2
J Iha gldaj
ys KE
(14.Ar),
We can now
-x
(.J6
-
S:)
#(.X)a
ydaj
-z+
(:1
of L3 ms the local time at c. with
-
.
1
-M 2
(.;% c)+
=
-
(.X
-
Ald (vY): i N '
'
c)+
-
1(x.>c) dXs
-
-
-
(A% c)+ is continuous, the stochastic integral
so we only have
-
of
0
and 1et f G C2 Suppose flrst that g is continuous Proof compaTing with It's formula proves the identity in quesion. this cmse (11.4) continuous function, using the Monotone Class Since the identity holds for any (:1 memsurable Theorem proves the result for a bounded fH
=
g. In
.q.
When X is a Brownian motion the identity in t
da
=
(11.7)becomej
R
0
Ia as a mapping from R to C4(0, T!, R), the Fix a time T < x and regard a real valued functions continuous on (0,T1, which we equij with the sup norm view of ln Kolmogorov's contlnulty teorem (41.6) in IIJII supcysz IJ(s)I. Chapter 1) lt suces to show that for some a, p > 0 we have -.+
=
E 11.f?f.511/f Clc -
11+G -
.
< CE
-
There is a version
2
yz
.0114
.E'1lf.J
J
j
(11.8) Tlzeorem. Let X be a continuous local martingale. L3 is continuous. of the LLin which (c,t)
dX,
1(x.>u)
=
Suppose without loss of generality that c < One of the Burkholder Davis Gndy inequalities (45.1) in Chapter 3) implies that
#(X,) ds
0
spent at a up to time From this, we see that ? can be thought of as th time occupation time measure function for the density @,or to be precise it is a lA(.) = lx (Xa ) ds. The There are many ways of defining local time fpr Brownian motion. above is famous for making it easy to prove that there is a one we have given of c and t. It is somewhat version in which L: is a jointly continuous function martingale and it remarkable that this property is true for a continuous local is not any more dicult to prove the result in this generality.
-+
lm1K N
c
c)+ (X It is clear that (c,t) continuity to investigate the joint
'
that
we can suppose
.
-.+
.(X,)d(JY)a
=
-+
Proof As usual, by stopping for a 11t By (112)
local time LI.
with
-x
=
-
s
0
f0:(c)da
fC#(J)
=
-->
-+
j
x
=
'i$
'J
Let X be a continuous semimartingale (11.7) Theorem. function then If g is a bounded Borel memsurable co
=
,
X
the identi:cation
complete
89
Exnmple 11.1. (11.8) is not true for Xt lftl, which is a semimaztingale by (11.1). lntroducing subscripts to indicate the process we are referring to, we clearly have f,1 (t) 1,1 (t)+ f,s-'(f) for a > 0 and fg1(t) 0 for a < 0. Since 0s() / 0 it follows that a j-(t) is discontinuous at a 0. One can complain that this exampl is trivial since 0 is an nd point of the set of possible values. A related example with 0 irt the middle is provied motion. by skew Browninn Start with l# l and flip coins with probability of 1/2 and > 1 + to assign signs to excursions of Bt i.e., maximal p o p time intervals (a, ) on which lft 1> 0. lf we call the resulting process then f,e (t) 2ps0 (t) as c t 0 and f,e (t) 2(1 plso (t) as c 1 0. For details see Walsh (1978). -
bounded convergence theorem three times.
by using'the
Tlle Meyer-Tanaka Forznula, Local Time
/0
To bound the right-hapd side we note that using Schwarz inequality and Fubini's theorem:
=
E
=
(
j -
c)
b
2
L dz
j
b
d(.X'),
1(u<x.s)
(11.7)and
&
.'&,
a
in the lemma defines the restriction of Q to .F for < frz, Wf oh C, lt t1 < ta n). Deflne the i Jlf i for be Borel subsets of Rd and let B (1 : % K e fnite dimenpional distTibutions of a measure Q by setting
is a
if and only if as
(local)martingale/o
(local)
Proof The parentheses are meant to indicate that the statemept is trt!e if ltd X q Fs. X/W, b a martingale/o, s d t the two locals aie removed. Let Z dQf. So ir Z dfl, and (ii) Zat dfl = if Z e .'Ft, then (i) Z dp and martingale/o, have we is a (i), (ii),the fact that (ii), (i)
QBt
.
.
ttfl
=
1
semimartingales and th: In this section, we will show that the collection of equivalent affected by a locally dennition of the stochwstic integral are not canonical probability change of memsure. For concreteness, we will work on the coordinate generated by the maps Xj @) space C, C), with h the qltration said to be locally Two memsures Q and P deflned on a qltration & are . equivalent, and i.e., Pt are equivalent if for each t their restrictions to Qt = dot/dfl. The Temsons In th cmse we Yt a mutually absolutely continuous. for our interest in this quantity will becom clear as the story unfoldsy
X is
The last equation
any t. To see that this deqnes a unique meure
ormula
2.12. Girsanov's
(12.1) Lemma. martingale/'.
a martingale/o
(12.2) Corollary.
and (.X')t by by the isometry property and the bounds we have imposed pn lxl heed to clzeck th iqulity i kolmogrk' stopp ing. This gives us what we proof. and completes the continuity theorem
M + A. X) The proof above almost works for semimartingals Remrk. t argument the = dAs 1(x >a) and then = J/ dMs 1f we deqne f/ 0 1fx >.) t t) 11 is continuous, so we wl11 get the desired conclusio above shows that ((z, JI is continuous. if we adopt mssumptions that imply (c,t)
91
This shows af is a martingale/P. lf F is a local martingale/o then there is a sequence of stopping times T,l 1 x so that Xv' is a martihgale/ Q and p thorem implies hence aykagk is a martingale/P. The optional stopplng that and magksarr'a is a martingale/' it follows that at is a local martingale/'. To prove the converse, observe that (a) interchanging the roles of P and and applying the lmst result shows that if pt dpt/do, and Z3 is a (lca1) Q martingale/P, then ptzj is a (local)martingale/o and (b)#t = th-', so letting result. Zt mK we have the desired C1
f1,.
of
To bound the sup, we recall the deinition
Gfrsanov's Formula
=
t
.
'
dP
B
whenever t k fn. The martingale property of tyt implies that the qnite dimensional distributions are on?istent so we have desned a unique memsurq on (C, C). Ea We
are ready
to prove the main
result
of
the
section. .
r'
(12.4) Girsanov s formula. If X is a local martingale/p f aa (a, Y) then Xt Wt is a local martingale/o. j
Jo
and we let dt
=
-1d
.z
, ,
-
'i
J
=
'
afykd.p
.
f
tnrjx
=
dPt
'Fido
=
A
W
W
=
Y,dQ,
a,F, dP,
=
.4
f
J
,4
a,Y,d#
= yt
.
.
..
'
..
..
Proof Although the formula for z1. looks a little strange, it is easy to see that it must be the right answer. If we suppose that .A is locally of b.v. and hms .zl.n 0, then integrating by parts, i.e., using (10.1),and noting (a,.A)t = 0 since W hms bounded variation gives =
(i2.5) tvj(aL-z4t)-pX:
't
t
(X,-u4,)d(,+
=
0
asdxs 0
t -
0
a,d.,+((,
A'lf
92
Chapter2
Section 2.12
Stochutic Integration
(12.6) Theorem. The quadratic variation ?')f under (aY, is the same P and Q.
made in Section 2.2 that our Eltration At this point, we need the assumption conyinuous only >dmitp martingales,so there is a continuous version of t:e to apply formula. iptegrationzby-parts which we can our martingales/p, and tlze fifst two terms op the right local (:4 Xs are Since view of (12.1), if we want Xt Wj to be m (12.5)are local martingales/P. Irt Jl. the choose that need martiugale/o, sum of the third and to so we a locll fourth terms H 0, that is, .
.
=
(Ax,
= a, d.Aa (a,-Y)
.
n=1J
(t
xla
(12.7) Theorem.
is to prove that the integral that deflnes If t K Trj then
The last detail remaining : at Izet T,z inf < =
tu-ldta,
=
E
E
yk exisss.
E
(3.9). To
< f2n
.
=
t) is
prove the
a sequence
-
x,7)2
(A')t
-+
lf H E fll
then
(#
.
.X')t is the
under
same
P
and
Q.
Proof The value is clearly the same for simple integrands. Let M and N be and B be the locally bounded variation parts the local martingale parts and of X under P and Q respectively. Note that (12.6) implies (M) (#)t Let
.
.4
,
.
and
.
in probability for any semimartingale/P. Since convergence in probability is not afected by locally equivalent change of measure, the desired conclusion (:I follows.
that it is neces-
(9.6),it is clear
.
-+
t
F'rom th last equation and the asspciative law that sary and sucient t
and hence the covariance
P-oof The second conclusion follows from the first frst we recall (8.6)shows that if Aa (0 tg < tkt of partitions of (0,@)with mesh llrjl 0 then =
-
0
(;)t
=
0
' dI(a,-Y)I, n s tx
.5
.
'
djta,xlj, 0
< x
Tn
T,z then A is well by the Kunita-Watanabe inequality. So if T = limn-ztx, = Et, defined for t % T. The optional stopping theorem implies that Safwza have > Ts noting t we at on ataza so =
.E'(; Ts K t) Since at
k
0 is
continuous
and
=
; Trj i t)
Ean
lf H G f11
and ffz
Letting n
-.+
x,
K .E'(&t;Ta S f)
we see that at
=
=
.E'tag'aiTn
0 a.s. on (T
-
=
f(a;T
But PL is equivalent to Qt so 0 = f$(T %.t) #(T < x) = 0, and the proof is complete. ,
S t) K 1/n
(12.4) shows that the collection of memsure. Our next goal is to locally equivalent to P, and H : under P and Q. The flrst step is
% t). Since t is
arbitrary (:I
is not aflcted by change Q is that if X is a semimartingale/', l'l, then the integral LH .X'l is the same
of semimartingales show
.
0 and
lll k
or
(4.5)we
j I#r
n)
can
flnd
simple
Jf,1 d(I.I+
-
z4))
=
Hm CN +
Hm
lfI)
so
that 0
-+
.p))t
.
(the left-hand side being computed under P and the right that for any H e fH the integrals under P and Q agree (x7 the proof is complete. n is arbitrary and Tn
S t4 = 0 PIT
l-l
(M),
.
allow us to pass to the lirnit in the equality
Hm (M +
so S @J,
=
-+
-
-.+
Q(T K t)
:
0 for t k Ta then by
S'IIN 11.2= SIIAT, 111.,',1
These conditions
T's K T, it follows that
=
inftt
=
.
f'tcet;T S t)
93
E
'
t,
Girsanov's Formula
Q) to conclude to time Tn Since
under up
.
E1
.
3
'
..
.
Motion,
Brownian
'
'
11
,
In this chapter we will use It js formul to deeperi ou j un tj ers ( a utjsg nian motion or, more generally, continuous local martingales.
oj
syow.
and Translence
3.1, Recurrence
f lf Bt is a d-dimensional Brownian motion and B1 is the th component tll B3 If i # j Exercise 2k2 in Chapter 1 tells us that is martingale with (#)f = Bi.1Bi't is a martingale, so (3.11) in Chapter 2 implies (# Bijt = 0. Using this R is C2 then infoimation in It's forzula we se that if f : Rd l
.
1
-+
i . :
:
..
' .
E
. ..E
f(Bt)
E
.
.
JBn)
-
'
the Laplacin
:::'z
Dl, of
.
J, we
1Bt4
(1.1)
.
.
,
0
=
quation
'd VT(#a) dm
:
j.
. i .
%(
E
.E
(
DiifBsjds
g
of
rceht
'
0
(
'
the lst
write
cn
.
E
DLILB, ldsj +
by) fbr tlw
1Bz4
-
.j .
=
i
Writik vy
'
()
i
J, ftd l i:z8 Ef.l natlf ws mre i
1
+ ;;
Lbiil for
Lfmtds
0
of two vects hd tv fo# the inr prdt i ti qrst ieim td equation. given previous mening precise of is in the that the nmctions with AJ = 0 ax c>lld harmonl. (1,1)shops that if we pmltion iult BioWian with i a lcl irtartihgale. hrmonic the futi pos a The next reult judicious choices of hl4monic functions) is the key to deriving pxoperties of Browian motin frop It's formula.
tlot
H,
(nd
inft : B j G). If Let G be a bounded open set aqd r closure continuous of G, G, then for tlke and o AJ 0 in J is k=fB.). hake J(z)
(1.2) Theprem.
J e :2 z
E
and
G we
=
=
,
=
suptlz 1. Let ff < *) Ottr qrst step is to prove that '> zl B / and then of ff G. lf z E G 1#1 z, y e G) be the diameter r < 1. Thus .&(r
Proof
=
=
=
.&(m< 1) k .P.(1f1
-
zj > ff ) =
.b()#1j
>
ff )
=
> 0
#1: & and -
Capter
96
rzowzlanMbfon,
3
Recutsence and Aansezice
Section 3.1
.II
Exl holds when k 1. 'To prove th.e lmst This shows supy k ) K (1 y) ptz, is the transition probability result by inductlon on k we observe that if for Brownian motion, the Markov pzoperty implies .&(m
=
-
(1.5) Theorem.
For
all z and y,
< x)
/.IT:
97
1.
=
/.-:,(:r < x), it suces < x) Since to prove the result shows reqection intended) A little 0. we can also suppose when y (pun (M 1)/M, and the right-hand side > 0. Now using (1.4)/.41L < Twz.) z D approaches 1 as f oo. .p.(4
Proof
=
=
E
(
pzlz, 2/).P!,('rk
.l%('rk ) S
1)
-
ty
'
:
E
=
l
-
where the lmst two inequalities follow from the induction fact that 1#1 zl > ff implies B / G. The lmst result implies that x) = 1 for all z Q G and moreover that '.
'
.
.
.
'.
.
.
'.
'
.
(1.3)
.
.
Errp
sup
'Ikansince
and
Recurrence
0 for
=
all z
motion will not hit 0 at a positive time even if it starts
k 2 Brownian
=
z
We are now ready to use (1.2)in d k 2. Let Sr Since (p has L 0 in G (z : r < lzl< J0, r < (1.2) implies .&.
=
=
Exerclse 1.3 Use the continuity of the Brownian path and P=Sz to conclude that if z # 0 then P.IS, 1 (x7 as r 1.0) 1.
inft : I#fl r) and and is continuous on =
=
,
=
Enpmj
=
whrr w(r)is short for the
epwsc
5)
r
-
Prsr
(1.8) If we fzx r and 1et R
(:x)
-+
in
msv
n1/2 for a1l t The strong Markov property implies =
a' d repeatlng
the
q
in the sense
Twdimensional Bropnian motion is recurvent (1.9) Theorem. PrBt G i.o.) > 1. that if G ls any open set, then
lf
z#0
we llx R, let r
-+
0 in
Pzuk
0 Sa)
lim P=% K r-w0
< S.a)
=
:
Bt
=
0), then for
=
=
0 for some t
k
e)
=
.E'c(#s.(To
ll 0 for z # 0. To extend the lmst Sa 1 cxl as R 1 x, we have Pcsz < result to z = 0 we note that the Markov property lmplies
.Fl.8t
P
0
(:xl))
k Snj and
note that Sa < co by
(1.3).
.
=
0
ho inqnitely time Sn and
.this
nxcKkz
.4n)
< co)) UaO=N
=
(nl/2/n)d-2 --,
0
Xa has
k limsup #(.a)
often the Brownian path never returns implies the desired result.
=
to
1
(z : jzj K n1/2)
after EI
hav proved the following result about how Dvoretsky aad Erds (1951) fast Brownian motion goes to cx) in d k 3.
Caper
100
i.o. as t 1 x) J%(I.IK #(t)V@
accordingas
section we will investigate the any z
Suppose g(t) is positive and decreasing. Then
(1.14) Theorem.
#(t)d-2p
)*
dt
1 or
=
concerned Here the absence of the lower limit implies that we are only ttnear shows calculus that x.'' little A behavior of the iutegral X
(2.1) .Pa,
x or < x.
=
t-1 log-c @dt
:::
the
with
(ii) -P.(.
k 0) H 1 if and
0 for some
=
>
0)
H
we
have fouud is that
j
if d
only
'
/(z)
(
. . . .
..
(i)
w(z)
cxl as
-+
(ii)lw(z)l -.+
(z( -+
co as z
3.2. Occupatlon
-.+
.
.'.
.'
.
Tk
=
inf (t >
are
x if and only if d
1, so Bt goes to x faster than 81/2.81 according ms a K 1 or a view of the Bmwniarl scaling relationship Bt =d that in any a > 1 Note The lmst rsutt thau V@. sensibly exct p e at a fmster rt e we could not shows that the escape rate is not much slower. Revlew.
Occupation Times
Section 3.2
Drownfan Motfon, Tf
3
p
j j pttz,
E
p)T(p)dydt
y(p) dy 5
V
tor
/ 2 t is the transition density Brownian (2xt)-d/2so if d 2 then Jpttz,yjdt x. J
s
=
Sectlon 3.2
Chapter 3 Pzownan Moiion, 11
102
When d k 31 changing
variables
lz
=
(X
(x7
pflzl p) dt
0
t
=
1
-Ip-zl d/2 e
0 (2.mt) () s
=
Flz
(2.3)
l =
z
# l2-
s
0
r(#2 1) = 27+/2 lz
whererta) 'i'
.
z'r
=
4.1.
' '
1k.1r
y4
(?(z ,
:
g .
-
'
''
''''
''
.
.
* pttz, v)dt
j,
=
1, we let
=
ds
c
tu
then in d k
#
'
z. /-
(2.4)
2.mc =
..
.
'
..
'
G4, p)
t)
1 ,/-2
=
J 0
,
1p
To complete the proof of (2.2)now we /(0, r) lxld changing tp polar coordiuates G(0,p) D
(fp
r
=
=-
(2.5)
that taking f
observe
s
0
cd
s2-d
c
ds
=
extend
.r
j
is
(#,) ds
=
.E
Ip
zl
=
since
W
co
ds :() .
,
s-
1/ 2
(y
N
1/2
'u
z)2/23/2
-(g
ew
-
-
as
c
.sj
x
x
c=
2 z 'u
du
,,-a/ztu
-.ds
z
u
.s
ds e
z) a
-(!/
z )a
-
e
c
W
-
X
j
z
-
-
zl
2
j.)
-u
,/a
ds e-'s-
ju
2
a
-jp
=
zl
-
0 co =
dv :-
r2/ 2
a
strong Markov motion starting at 0
j
lp
) ds
m
K Eo
j
the
7-2 =
-
1 1-2 2z. 2 .
y-
=
The computation is almost the same for d 2. The only thing that changes pt(0, 0) again, then for z is the choice of at If we try c y the integrand and the integral diverges, so we let at pt(0, d1) where 0 as t ew el (1,0). With this choice of cf we get
'
'
''
.
-f-1
'
-+
=
=
(
lo
=
.
C(z yj
(fa) ds
kez-nel, because G(., yj is tht electrostatic We call G(z, yj the potential for more (1978) potentialof a unit charge at p. See Chapter 3 of Port and Stone approach another take to In d K 2, j') pf (z,#) dt H (X) so we have to on this. useful G: dennea
,
t:o
1 F=
zg )
(e-lT-#I a /2f jzt
x
j
p)
=
0
(pt(z,
-
ct)
dt
(2.6)
.
,
= 21 c -
Gz' p)
1)t-1/2 dt
-
=
applying
property at
lo with D
x
2W
y
-
< x
2
-
=
-y.r2
d-1
j =
.
the last conclusion to z # 0 observe that the exit time from B, lzI)for a Bzownian rotational symmetry we have using the antl G) (r T
Tp
=
a
(e-(p-''r)/2t
-
(e
';r
/c(z,,).f(,)o
-
*
1
=
and the integral converges, since the integrand is K 0 and /2zI gives y z) a -+ Y. Changihg variables t
y, and
.f(s,)a
p,(0, 0). With this choicek
=
-
:3, G(z, y4 < co for z
..:E .
.
When d
gamma functipn. If we deqne
'
'
(d/2)-2
0
pl 2-a
usual
is the
s-e-'ds
2s2 -3
/c(z,,)y(p)oj- ,'..f(a)a
cu
G(z, g)
-
n*
-
&
*
2,0/2
pl 2
-.lz
-,
d
-
J
dt
d/2
#l2
-
a /at
103
where the cf are constants we will choose to make the integral converge (at least when z # yj. To see why this modifled deqnition might be useful note that if f(#) dy = 0 then (mssuming we can use Ftzbini's theorem)
:12/2,: gives
-
Occupaton Tmes
1 = 21 -
=
-zg
1
jz-pja/af
-
e-1/2t) j-lj f-ldj
e-' ds
l/aa
x
ds e-3
()
jx-yja/z,
1 t- dt
=
ds c-4
(- logtlz
-
pj2))
-
=
1
logtlz
-
p1)
''
Captez
104
Section 3.3 Exit Tlnies
3 frownlan Motion, 11
To sum up, the potential
are given
kernels (r(d/2
by
1)/2xd/2)
jz
p12-d
dk 3 d'H (2.7) Gx, #) = 2 (-1/ 1Og(I i/l) = pl d 1 -1 Iz p1) where Ctpllz The reader should note that in each cse G(z, p) v is the harmonic function we used in Section 3.1. This is, of course, no accident. Gz, 0) is obviously spherically symmetric and, as we will seei satisfles z G(z, 0) 0 for z # 0, so the results above imply G, 0) + Sw(Izl). The formulas above correspond to W 0, hyhich is nice and simple. But what about the weird looking B s? What is special about tm? The answer is simple:They are chosen to make laLGz, 0) (a point mass at 0) in the al sense. It is emsy to see that this happens in d distribution' 1. In that cmse = then P (z) IzI 1' X 7* 0 (P?(z) 1 z < () Bpnzj yeaders n check that -2Jn. Mre sophisticatd if B then so pageq 96-97.) this is also true in d k 2. (See F. John (1982), To explain why we want to deflne the poten tial lternels, we return to Brownian motion in the half space, frst considered in Section 1.4. Let .I (g : yd > 0), v inftt : Bt / H, and let -
-
the formulas for G imyly that IG(z, #)T(p)ldg are qnite, so the last two equalltles are
compact
J
of
support
1 and
-
=
-
-+
=
=
.
.rztz,
,
=
(
=
=
# be the
.
,
.
-+
so Gxz,
yj
=
If z
e
H,
; k
0 has
Gxz,
pd-l,
(g 6
of p through the plane
zqection
(2.8)Theorem.
then
.
Rd
:
yd
=
and
support,
compact
0J (z
.f(z)
:
>
0J
11*
/c(z,,).f(,)o s-(/-y(s,)a)/c(z,,)y(,)o-
J
0
Proof Using Pubini's theorem which is justiied since f k 0, then > 0) C H we have fiofn Chapter 1 and the fact that (z : .f(z)
E. J
J T(f)
dt
=
J
0
/
> f)
EslBttr
pl +
-
p)
-
,,
j- j (pt(z,,) j- j (pt(z,#) -
=
(z + (z +
y)
=
y4
=
2p 2z
when
0
when
z < y
p)J(p) dy
-
< y < z
so we can write (2.9)
y)
Gn,
=
2(z A y)
for
al1 z, p >
0
Gn, yj is constant 2z for y lt is somewhat surprising that y a11points expected occupation time! is, y > z have the same =
-+
k
z,
that
3.3. Exlt Tlmes
(3.1)Theorem.
H
J
p) + z) +
-
-
atttytdydt atttytdydt
-
Gx,
p1. Separating things into cases, we see that
ntz, f))J(p) dydt
H
0
-(z -(2/
=
#1
-
=
=
(pt(z, j /0
(4.9)
p)
lz+
-Iz
=
In this section we investigate the moments of the exit times r = inftt : Bt / G) forvarious open sets. we begin with G (z : Izl< r) in which case r Sc in thenotation of Section 3.1.
0
*
=
dt
using
introduced, we invite the reader to pas Gx, yj looks like in one dimension. If you will probably Iot guess the behavior The corhptation is easier than guessing
-+
-lz
7/).f(p) dp
Gxz,
=
Witi this interpretation for GH for a minute and imagine what y you don't already know the answer, for small tallt. x. So mh ms y the answer: t7(z:#)
-p4)
(p1 ,
J(ft)df
g
=
=
,
T
E=
=
=
E1
S'ekpected
-
-&
=
E
The proof given above simplifles considerably in the case d k 3; however, part of the point of the proof above is that, with the deflnition we haye chpsen for G in th #eurrent case, the formlas and proofs an be th yame toi al1 d. p) = G (z,p) f?(z #). We think of Guk yj s the Let time at ocqpation (density) y for a Brownian motiori staztin: f z and killd leaves The rationale for this interpretation is that (forquitlble Jj it #.'' when
=
-1
lG(z, yjlyjkdy Jvalid.
'
=
'
.
.
The and
105
J
G(z, #)/.(! dp
Proof
If
1zIs r
then Ecsr
The key is the observation
l.512 -
(r2
=
-
that d
dt
lzl2)/(I
=
i =1
((lj)2
-
)
106
Capter
3 Prowpan
foton, ff
Section 3.3 Exit Times
being the sum of d maztingals is a martingale. Using the optional theorem at the bounded stopping time Sv A t we have
.
lzI2 =
Em
stopping
llfsratlz Sr A t)dl
(3.3) Theorem.
-
(1.3) tells us that Er,h < cx), and we have lfsra 12S r2 so lrtting t Srd) and using the dpminated convergence theorem gives IzI2 Er implis the desired result. ,
(r2
=
-
,
Exercise
3..
tet
b > 0 and T
c,
These local martingales are useful for computing sional Browniau motion.
infft
=
:
whici
I::l
q
)). slzowEzT
/ (-c,
Bt
x
-+
=
ab.
To get more fomulas like (3.1)we need more martingales. Applying lt's and formula, (10.2) in Chapter 2, with X3 X3 a continuous local martingile, X,2 = (.X)j we obtain =
,
i
lXt, (-Y))
J-%, 0)
-
. ..
jvblh + jo + jo =
,
From (3..2) we see that if () D11 + Da).f tingale. Examples of such functions are
J(z, y4
z,
=
z2
z3
y,
-
t
1
z4
3zg,
-
-
fh
(X)t) is
,
6z22/ + 3:2
local mar-
a
1Z
n
,
#)
Cn
rn2*-2FN
#
,
-
-
so that D=/2 of the mth term is the (n/zjthterm is 0.)
=
EzfBp-hl.n
-
(a
6./aaza
.
Ta) + 3('q
..
.
(/'n)
J
.
' . :
(1.3)and 2 the dominated2 cppyergence Usij (i)and c 4 ttzEora +3.Eq
=
-
.
-c#4t+#2j2
-cj3
js a
j
.
. .
theorem
a
.
.
we
ieranging (:I
(local)martingale
Our next result is a special cmse of the Burkholder bavis Gundy inetualities, (5.1),but is needed to prove (4.4)which is a key step ih qur pot of (5.1). :
.
.
.
.
If Xg is
a continuous
E
FFI
K n/2,
cs,o
=
1
and
for 0
%m
sp
local martingale
44
lJU/ qF3/l(Jf)/
zz
6>l(JUt2(Jf)t) % 6(1lJF/)1/2(/;(7f)/)1/2 '
.
-
(.X)f re
Using the L* mxximal inequality fact that (4/.3)4 K 3.1605 < 19/6 E satl X1
(/.3) in Chapter 4 of Durrett we have
S (4/3)4fX1 X 19
LE
X1(
syuy
1/2
:
. .
.
ard (1095))
'72
E (.1)7 )
the
Section 3.3
108 Capter 3 rrownanMotion, 11 Since 1.X,1< M for all s to get
divide each
we can
X14ll
by ftsup,
=
Zg exp((W)z)
0
and
k0
-av
1/2
a (21rs )
'
transforms, you
(3.8)
can use
.-aulzs
ds
ce
=
-
..'
.
.
.
.' ..
.
E .:
'
' .
.
.
=
Ez expt#fnasa '
.
-
#2(L A Sa)/2) .
..
. .
.
'
.
.
.
. .
l
If 0 k t h q 4i ght-hand side is K expt#c) so letting n -A (x) and using the 02Taj2). Taking 0 = bounded convergence theorem we have 1 = Eo exptpc result. 2 now gives the desired (:) -
12
'i$
#2/2) and Srt % n be expt#f #nIT. < x) = 1 by (1.5).Let XL Proof stpping times so that Sn 1 (x) and Xthsn is a martingale. Since Sn % n, the optional stopping theorem implies 1
by
Then for
If you are good at inverting Laplac (4.1)in Chapter 1: J5(n
( -
Let Tc
1
PIIrPOSeS.
exptzxf
that
we see
Ia
=
(3.8) Theorem.
Jc
=
Chapter 2 now,
'
=
=
Kt
Eetting 0 E R and setting Xt 0Bt 'in (3.6),whre Bt is a one dimensional 01tj2). These Browrian motion, gives us a family of martingales exptpf mrtingales are useful for computing the distribution of hitting times associated with Brownian motion.
exptyk), which for a given continuous function a(t) has unique solution J() assumptions) suitable Jk = c(s) ds. It is possible to prove that where (upder details. for solution of Dolans-Dade Z is the only (1970) (3.6).See The exponential local martingale will play an important role in Section 5.3. Thn (audnow) it will b useful to know whvn the exponeriql loc>l martipg>lr and (1.15) is > martiug>le. The next result is not yery sophisticated (see(1.14) enougi results) for is and but of better for Yor (1991) in Chapter Vl1I Revuz
-
(2.5)in
's
=
d.;L Tis property of X As in the cse
=
(3.7) Theorem. Suppose X is and A = 0. Then X explA
.'
-
diferential notation, that dls X the right to b called the martingale exponential ordinary diferential equation
Ol1r
.
we have
.
0
i
or, in
that
. '
1F,1
Remark. It follows from the lat pr o f that if X t ls tingale with 0 and (X) % M for a11 t then 5$ mar ti/ale ih M2
..' ''''
-
Using
lsup
k
,
'>Kt
.
therem 2
sup y2
E
4e An
=
.
convergence
109
inequality
,
S 4e ut Ezthn)
a K 4f5$avk
Mt
If we notice that T(z,yj another useful result. we get
Zg the L2 maximal
of times that reduces
Supxagu
E
Exit Times
110
Section 3.4
11
rrowmlanfoton,
Captez 3
Change
-2/.1/12
then is a martingale. If 0 = is a local martingale. Repeating the proof
Let
a
inftt
=
:
Ez expt-a)
Then for
2e -cWI/(1
=
c >
k 0,
and
0
+ e-acW.)
Exerclse
3.3. Let T-c
Proof Let () Ez expt-L). time a (anddropping the subscript
inftt
=
Applying the strong Markov property at a to make the formula emsier to typeset)
=
la
c).
j#f I k
JtT-c
Zt
:
of
-a).
111
0 and exp(-(2p/?)Zt) (3.8)one gets
-
=
If c, ;t > 0 then
=
< co)
Time, Lvy's Theorem
020.2/2
-0y
(3.9) Theorem.
of
exp(-2(w/J2)
=
(
g ves
>otexpt-r); Bp + fntexpt-'rl/zctl;
Ez expt-T'c)
=
-#z).
Us tng te
xpression lbr
1
,/.()
given in
(3.8)is
of
Another consequence in Section 7.2.
a
-c)
Br t::z
3.4. Change
it follows that
e (-c, c)
Llvy's
Theorem
'
re.ult.
the dsird (3.8)now zived bit of
thatwill
calculus
l
in handy
come
.
,
lf 0 > 0 then
(3.10)Theorem.
of Tlme,
ln this section we will prove Lvy's characterization of Brownian motion (4.1) artd use it to qhow that every continuou local martingale is a time change pf Browhian motionk
expl-r)
(1+ #2.()).E
=
c)
Since Bp
Symmetry dictates that ('r,Bpj =d ('r, and we have r and Bp' are independent,
#.()
=
'
,
(4.1) Theorem. If X is a continuous local martingale (.X') % t, then Xg is a one dimensional Brownian motion. Proof
(4.5)in
By
Chapter 1 it suces
with
(4.2) Lemma. For any s and Xa.k.t X. is independent normal distribution with mean 0 and variance t.
of
-
/,
J
.za
/ zj e
va'
1
.#j
gj
a
w,
Changing
variables
=
=
t
* 0 1
20
1/2s, dt
=
2.: 2x X
()
z a4 C
B-
the integral
=
#/ 24
-
1
y-ie27.s3
ds 2s2 p/2a
above
8
.
.
.
#X1
e-a
a J.
:
1 = io
'-
e#x'. dl?
0
.
ps
t
2
'
,
cfpxv' du
)
g
.
.t
,
.
,
,
'
and integrate over
. ..
..
.E'tKavki.Al since Fo = 0. So
we
=
E CE lKaau
( .
fle
fGlavu
lh'4 ;.A)
.
.. . .
The deqnition of conditional
expectatin
=
.E'(y;.A)
.
.
.
implles
0
=
have
=
exptplzz
'z
o,
=
'.?
=
02o.2tj2j
-
'rhe Erst termon the riat, which we tet Fp' a+r and let e F. = W ill ar11 martingale with respect to .F* i a tocal To get rid of tat termlet .j Ta 1 fx, be a sequence of stopping times that reduces K, replace i by t h T,,, : ::.:.z
The exponential martingale can also be used to study a one dimensional c's + pt where c' > 0 and p is real. Brownian motion Bt plus drift. Let Zt c2 with martingale (-Y)t = so (3.7)implie that Xt Z% pt is a pt)
(7.9)in Chap-
..$.0'.
.
-
hms a
,
= z2 and consulting and the remark afe: (3.8) Using (3.8)now with c = expression is equal to the lmst of the T. we see that for the density function D right-hand side of (3.10).
-
and
F,
Proof of (4.2) Applying the complex version 9f It's formula, ter 2, to X./ = Xs-vv Xa and fzj cii e ie= we get
'i
-
0 and
-
-dsl2s2
Proof
z jyyj
oxp ( .j
=
show
to
,
t:o
a'
.)
;
.p(.,4)
-
=
()
-
tan
02 -E -
Z
a
e
ioxt
v
tfzj..x ,
112 Caper Since
gives
Ie#'l
=
1, letting n
.p(z)
.g.
t
:2 2 :2
--E
=
-
=
(ef #X
E
, v
;
''
it
PI.A)
--.
0
are
J('z)dn E
..
.
:
. .
.
.
:2/2, Since we know that IJ(s)l S 1, it follows that I.(t) y('tz)1% If conclttde and equation continuous diferentiate last j is to the is can we so j diferentiable with -p2 T Oge ther with
=
.P(.A), this
itj
2
shows
.
that
#(.A)e-#Qt/2 =
:-#2z/2
#x:.z)
j
Exerclse 4.1. A 0 and
F(p(X()l.'S)
that
'
EgvM
i
P(A1 q B)
=
expectation,
.
.
and
we
*1 .
.
.
:
have proved the desired EI
Suppose Xgi, 1
with
local martingales
are continuous
S Kd
=
(-Y then Xt
or
dp
=
,
we see
dp .E'(1s(A1)1.'S)
=
by the deflnition of conditional independence.
=
xt,
.
.
,
(0 f
XiLt =
if
=
i
otherwise
Xjdj ij a d-dimensional Brownian motion.
j
.
An immediate Ee
e B)
'
.j()
=
integrating
'tzl
=
.j(0)
and
113
A monotone clmss argument now shows thqt the last conclusion is ttue for any bounded measu.a ble g. Taking g = ls and integratig thJ lst eqality ovyr E SLwe have J7(.)#(aX1
..z-
=
(4.3)by e(-#)
Theorem
Lvy's
of Time,
.4
.
.. .
J'(f)
and hence
t
02 =
.A) du
says
=
each side of
Change
.E'(:(X1)1.T-1)
the two measures
and
../()
-
Multiplying is a constant
dp; .A
()
-z
-
epX.'
by Fubini's theorem. (The irtegrand is bounded zex, ;z), the lmst equality writing
luite.)
theorem
using the bounded convergence
cxn and
.-,
i8x1; )
Ee
Section 3.4
Afolon, 11
3 frownan
(4.1)is:
of
consequence
Every continuous local martiugal Tvith A% s 0 and i>ving (A'). > (xa is a time change of Brownian motion. To be precise if we 1et y(u) inftt : (X)f > ul then Bu = Xvtul is a Brownian motion and Xt #(x),
(4.4) Thepreim
W
=
Since this holds for
all
.A E F it
f8Xl
Ee
(4.)
ibllowsthat
=
characteristic conditional or in words, the variance and with tribution 0 mean dis
'. ' '
'
E
.
.
. (
:
Proof Since ;((X)j) t the second equality is >n ippedtate consquence of Bu To the first. prove that we note Exercise 3.8 o f C h ap ter 2 implies tt Bu2 and maytingales. yhow lpcal continuous, B%q that it is to are sp suces .
e-#2t/2 lyl()) az
.
=
'u
of VY(is
function
that of the normal
@.
Tp get from this to (4.2)we flrst take expected values of both sides to The .concludethat X; hms a normal distribution with mean 0 and variance condttional X't constant charcteristic suggests is tat function th tht a fact is idependent of FL.To turn this intuition into a proof let g be C fhctlon nd let with compact support,
-
0 is
(4..5)Lemma. Bu .Fv(u), u ,
a
.-+
'tz
lpcal martingale.
.
E .
)
'
.
j.
Proof of (4.5) Let Ts implies that if u < then
poj
#(2)
-
1 2*
1Ah1> n.
The optional
=
wherewe have used Exercise 2.1 in Chapter to replce F-tulhn by .'G(t).To let ?, (x) we obsvrve that the 12 maximal inequality? the fact that X2v(v)A v,a , imply is a zpartinggle, an the definltipn of tXlvtvlaza .2
dz
-
more than enough to conclude that
E sup X 2(vlaza n C
W(-
theorem
l.'G(u)) Xvlulac-,k ,(z))
ior
stopping
'
-+
mssumed
be its Fourier transform. We have p is integrable and hence =
j
e#'plzl
:
Exvjhn
' =
inftt
=
'p
o)
tyz
v
k 4sup EX2vtvlav'a '
E
-
Fl
= 4sup .E'tA-lvtvlwza s n
41
Section 4.3
Chapter 3 Brownian Motion, 11
114
theorem imply that as n The last result and the dominated convergence expectation L2 for t conditional is a Since in Xyt) Ivttlagk x, lt follows that f'txvtvlagu in Ehv) contraction in l.'G(u)) I.'G(u)) E1 and the proof is complete. -+
-+
'tz,
=
'N complete .
'
''''
'''
(4.6)Lemma.
(4.5),the optiozal
Proof of (4.6) As in the proof of that if u < then
-+
deqnitionof
x
'y($
(X)v(v)an lFv(u))
-
that using
we observe
then
theorem implies
stopping
2
Xvtulavk 2
(c + )
-
2
(X)v(vjAn1
-
A
c
,
(3.4)j nd the
+ 2.E'(X)v2(v)
a'vvlagk
Let Tk
inftt
(X)t
:
(X)tAn
=
'
.'
.A
=
((X)x
B f+
K J) # >(X
' ,
2
>
2#(x
)+
>
#(y
(5.5)vith Z -. K &zu't EvXI24
arld using
=
>
:
Jj
?'Jr
1)-1h-iS(#(r h S1 A T)2j A Sa A 1a)2 BLv = (#2 < x) s (#2 1)c1-2(J)2.p(S1 1)-12.p(g.1/2 > ) % p2
so
that ff :2
:-1.
Fom the
K KNEVCY)
Evk-lnq
-
..
Ev(YI8)
,
-
J)
>
X/2, X, Yl6
1 and then pick N k 0 so that 2N ihat growth condition and the monotonicity of it Pick J
-
and using
) #w()
>
. . .. s
Integrating dpj
A T)J
j,
=
-
using the stopping
119
lqumption it follows that
From or
(r A & A /l
f((r
Replacing by a nonnegative Rzbini's theorem gives
(5.5)
in the proof, it is easy to check t it -
Adapted to Brownian Filtrations
Mavtingales
'X'
:
.P((r A 5'2A T) 2
h
('rA t) 1/2> ) : (r A t)1/2 > #) : l#(r A t)l > J)
inftf
S
>
Section 3.6
Motion, 11
-
since 1f('rA T)IK (J)2 proving(b). ;
this ombining
E1
'
.
.
=
=
=
again gives
E (X)
'..
It wlll take ohe more lemma to extract (5.2)from (5.3).First, we need moderatly lncrasing if tp i? a > deinition. A function p is sajd tc! be nondecreasing fnction with w(0) 0 and if there ts a constant ff so that w(2) K ff w(). It is emsy to see that w(z) = ZP is moderately increing 2F) but (ff w(z) ea= 1 is pot for any z > 0. To complete the proof of (5.2)now it uces to show (takep 2 in (5.3)and ote 1/3 / 1)
Solving for EpX4
Evxt
=
w
EvX4
S CEeCY)
It is enpugh to prove the result for bounded p for if the result holds for v A n for a11 iz k 1, it also hols for p. #ow p is the distribution function of a measure on (0,x) that hms Proof
e()
=
j
h
dwt)
Jcxl
=
j
1(?,>) dwt)
381 Revlslted. 16. Taking 8 i:
=
%ff 82EpX) + ffN+1Xw(F)
; fffw(X/2)
now gives
-
(5.4) Lemma. lf X, F k 0 satisfy P(X > 2, F K 4) S &zPX zk2) for l1 8 k 0 and p; is a moderately incremsing function, thrn there is a constant C that only depens on the growth rate ff jo that
inequality and usipg the growth cpnditipn
with the previous
ff N+1 jaf w(F) ff
K1
E1
-
To compare with (3.4),suppose p = 4. ln this case ff 1/8 to make 1- ff J a = 3/4 ad X :z 3, we get a constant
164 4/3 .
=
87, 381.333
=
24
=
which
''
.
.
.
Of course we really don't care about the value, just that positive flnite constants 0 < c, C < (x7 exist in (5.1).
3.6. Martingales
Adapted
Flltrations
to Brownlan
Let (f%, t k 0) be the fltration generated by a d-dimensional Brownian motion with Bz 0, deqned on some proba bility space (f),f, >). ln this section Bt will show (i)all local martingales adapted to t k 0J are continuous and we =
t1% ,
120
Capter
Section 3.6 Matingales
3 Brownian Motion, 11 variable
(ii) every random integral.
2(f1,
X E
Bco #) can be
ms a stochmstic
written
,
121
1). Repeating the aud it follows from step 1 that 5$ is continuous aron (0, above and using inductiim, it follows that th result holds if F gpment hBtt lkBtuj where tl < t2 < < tk % n and h, fk are bopnded =
-
All local martingales
(6.1) Theorem.
Adapted to Prownan Filtrations
to fBt, t k 0)
adapted
.
.
.
.
.
.
.
continuoustunctions.
are coninuotls.
Proof Let X3 be a local martingale adapted to fl'h t k 0) and 1et T,z % n to show that for be a sequence of stopping times that reduces X. It suces enough wprds, other continuous, it is to show that Ta) in is each n, X(t A or whee = of .E'(Fl1%) martingales form result the 5$ holds for the e Bn We steps. of three generality in build up to this level ,
'
.
,
3. Let '.F e Bn with E 1y1 < x. 14follows from a stand>rd application class theorem that for any E > 0, there is a random vartable the zpozlotoze considered Fl < 6. Now in step 2 that hws E l.XT of the form Step
of
Xf
-
.
Step 1. . Let 'F = fBn then X /(#n), so t Markov property implies .->
=
is a bounded contitmous function. lf t k n, is trivially continuops for t > n. If < n, the .E'('jl%) = (n t, Bt ) where
) where f l
K
.ls, z)
=
1f(A' , 11'lt)f.( jz ls t ij -
s stjx
yjjsj)
-
and if we let Zt = .E'(lXf s F111%),it follows from Doob s lnequality(seee.g., that (4.2)in Uhapter4 of Durrett (1905)) j
-
-jN-.ja/a,y(yy; 1 e a/a (2vs)
=
'
dy
sup t Kn
Z)
K EZn
>
=
Slxe
'Fl
-
Now .X'f(t) = ECX' l1) is continuous, so letting e ot continuous functions, uniform limit is X(u?) so a td,
ts,
&(s,z)/ -
so the dominated convergence lBns. Step 2. Let F contlnuous.If
=
=
'
Remark.
=
dz
I
like to thank
would
above.
a
Kn
and
1 n, Itn
-
t, Bt4
f Ja are bounded ,
-.>
and
VichaelSharpe for telling
.f1(lt,).E'(.f2(fta)ll%z)
on
(t1,co).
.f1(ft,):(ft,)
=
On th
.
.
.
(6.2)Theorem.
For any X
with
e
fa2(f),lx,
P) theye
,
d
=
Ex +
J'l =1
x
are unique
a.e. I:I
the proof
me about
let t1't,
x continuous
s
We now turn to our second goal. Let Bt (.R1, Btd) with Bo t k 0) be the qltrations generated by the coordinates.
11BtlEhBtzl3Btt
=
0
and
Hi E 1la(ff)
H1 tllj
0
We follow Section V.3 of Revuz and Yor (1991).The uniquenesq is immediatesince,the isometiy property, Exercise 6.2 of Chapter 2, ifnflis thr is ozly on way to write the zero randoz variable, i.e., using Hi (s,1) > 0. To prove the existence of I%, the flrst step is to reduce to one dimension by proving : Proof
'
where :12)
z./.;)
theorem implies that as t
step 1 implies that K is so the argument from then if Yi = .E'tlz 11) aud haud, other %t1,
'tz
+
ivez
=
where tl
-
.
. . . .
-
E Ei. . ' i . E . -E .-
.
z)
,
.y
. .
.
-' . . E.
(.R,
.
. .
yilyE
-
-+
,,...,........,g,,,,
,,.,,,..,,.,,,
.:. : q. ij,.,,,,''j'y,y,'qE,q'y.r
. .
-
n
-+
-
,,-;jjj:(jjjjg,,,.
.
qjjrlyiyt jtyE
-
,
.:2
-+
,,,;jjjqjj;jjr,...
. -
. . - .
-
j
,jjj;;-
E
-
j-)
E
.yyryq, jy . j k;Ey'E;!E;jlEl)))j!l$!1jljl!i
:
.
-
-.
yE -
. . - .
.
-
.
E
-
.
.
-
-
'.,t,yy,.r
Proof of (6.4) We will show that the meuure I/Fdp (i.e.,the memsure p to show th>t To do this, it suces with dpldp = *F) is the measure. rero Bt,. ) for any nnitejequepc: 14:dp is the zero measure on the c-fleld c(.&, < a. Let j, 1 % j ; n be real numbers and z be a 0 tc < tl < 02 < complex number. The function
(y;))I
-)t..t.(..,.y...(.,.
-. E i.
-
.
be
a
,,rjjr':l,'r'.k.
.
-i q . .- . .E E... . y .E'Eiitp' . y. .r.E...'.'.,.',.r@;)Ejij.I@,pl .,.y.(yj..(....,).yqy.,..;.y.rq
can
.,.;j!,:i;jiq'''.
Ak)2 . ;i jj jjjji.. i. .... 0 and EXn EX, it follows sucexn x ju jmplies Exn .. k.'4)!..';r:..!:.(t': yj'yj j(jjj yy. :;!E!Ilt-. jjjjy,rjj., '..'..j(.jj;jjyygii:)j!(j;jjj'.j..( j....yy..jjyyq.yy.j.y. jjs jyj jj. ... j.jp'. 1i111'1)!t ...jjj.jlj..., i. The completeness of 112(.5) see the remark at the beginning of Step 3 in ..EE)i,.(.i!.j,l;:1)l.,)..II (( pj:;Iy 0. . . . r. .... iilkrEi',lydjilEllijjl'j'j'jliyljilk,jljjtjl section2.4, implies that there is a predictable H so that Il#n #lls . j of ; jy; y isometry implies that property Hm H B B y converges the to ; use .. (ililt')i)jljttt . jg.j.j.yygy..j.....j wuother . . .. in M a va ksg jjmjts irj (6.6)gives the representation for X. This completes r;r(i(E;iy).yy(!)j(!)t.j!j!Tj)())jE( k . tljy i('jyj yy; jy'l IZI tjje proof of (6.5)and thus of (6.2) ' i )'iiT'l(jkk7jt1 ! .!. :@ . k.;!!.;:j.j.(.j)jlijqrj-jjjj.jjyy.y y.y...(yy.(.;y;.j).j!(.q;.y..y.(..y.j .. y(.(jy(yrjyj:j,jkjj . . ; j! j. yj. yyy... j .yy..y j. . yji .!;r!ry).j;t'yj !.gl)jj.y jj'. yp-i -. .qj ry.)jjy..y..j.. . ;y().;.!yjjji.jy.jjty ( )); .j ( y g j . . j ; y ...q'i..i.ijjy..r)'.. y j . )l;y; (jqyy. . jj.jj.;jjjj.y.;rjj.y.y.j.jjjyy y!gj;j;'. .jy. ' ..' .r.j..lq ));. .y....y(.g(y.jq;yl(jjp:y'. i. (;.yjyg;y. ..g.yy.( l . y j .. .;.(.)l;gj.y!;y)y..yj, rjyyi :jj.!q..j...jy (y .. y y.. '... q;(j)jjj!g;.tk'rrj...y.jr...j((j.yj;y.yj)ryy.yj.yyjy(y..jy...yy .; j .j . ktr . y..q.E.gy;.)kijyjjjyjjyjjjjryj'. jy;.j.;j(ygj.).yyjjjj.y.y..... jj. )) (jy(y j.. y . ; q y . . )'. . ; ... ! j!;;ijjy;jjjy lj; j(y.y.jr.yy;.;yjyjyyjy.yqyjy.g.y..y...gy j!jjyj'. y ; j .'. l'(j.q(qj.yjq.!.lyjj.pjjrjjyjj..jjj...j.!.y(j;.ypjjjjjy.g! . . j. .j!.yjyy.y.(y.....g ..i!!. yj;! .j'('!:. .g. jy ;'(4!tj. j'j. t'j. r... yj .... .. .-...'y)..E..'.y!(.j(.iy...jyy(y;y(j'!. yjjyyjjjyjjj.y.yyj.y . rr'. j yj.. . r . . y jyyy..j.rj.j..yjyj;(y.y.y.y ...yj . ......!.l)(.;ry)yi:j,yy,jy'. .-.:.j .y(.i.kiy(yjjj.;y.j(y.jjjIjg.j)yjgjyj.yj.jyq.yjqyyjgyy.!.;y.;(:j.j..q..(j..j(..yr .. y. (ry.jyjj.yj). .....;;'ry:..j.jjr.jyjjj'. jj.,$'jj'. y .jy j yy. .. yjy .j.y.y j..,j . E
that
-
()
y
E' ' EE i .. ..-'
.
-
......,..,....
.
--
then Wf H 0.
.'
zxg sau))2
.xm)
-
.
.,-;1!:111(j;,..
.
y - y : y. y . E.i.i E. EE. . Ey y . yE.. -.! y. y . . . . '(E.'''Ek';l:Ei'i)))'.k'j'jtyi('tj';jk'y .
' : .
,
sttxs
.
!- E q
.
-
to show
a1l
0 for
=
-
-
=
sx.)2
.
.-
. .
r.......i,i;..)E....(;.i;.,);.k.,
it sues
-
.
. : . E .Ejy,lij,t,jjtjjjyjyjyjyyj . E. . . ,,jy,,.jyyyj, y -
.
...
f
z
-
-
collection random variables X of Then is closed subspace of L p
E lF' exp
s(x
E i.EE,,E,jE(!gj'!.!jl)rl)ll!i y y E,. y
.
.
=
-
.
..
:
.1,2
w(z)
-
-
:
(6.2)now,
If #F e L2 hms .E'(g#F)
.
.
(sj, (w) sj, (w), sj.(w) sja-,(w))
. .
'.) jjj;ryjyj'j ))jj yq y'y .. y . .... k ;j. .
E
.
w
-..
.
.
=
,
r ; jijy.j,ypiljijyylj.tiqitjyjl ; .. .. )y r)(;jij .jy'jjj . yy .
'liilil,lll'i.tiijyltyqyj
i.e., the-desired representation holds for .' = (f(ff B)t : .Jf e 1, t k 0).
x
,
...y..(.)....(.:(
..(
.
.
(6.4) Lem-a.
123
-1.
EE,jijpjlyrlyjlltltjrt.yrjj
-
(r =
Filtrations
theory tells us that p must vanish identically. In The lmst equality implies that the image
lEiE .illiltj,jtll)l ; jj.qE.
'.l..'jj:);)t!),jy))(yj,
1. Since Z3 E
Aaztlrgales Adapted to lzownan
kllkyyrtyj.yjylytjjlq.j,Etiy 'y
'
E
0,
=
section3.6
y..
,,,y.i,jjttily
0
'Fn = 0 and (F)n follows that
. . - . .
,t'.....).'..'
'iqliljjljjt.jljEri.EEE
X in Using an fact suppose that Xn ..........,,g..y,.y.r('(.pj!jgyi.yy.j);j;jjj.;..jj.yj.jjt,.r.j..y...y..,..y,..,...y..y.y j illiy.Ei'rliljllrkl.jtl.ljtttpjyyl and of stpchastic
Zs S, dm
=
..'
.
.
. ; -
-
y,yy.j,.j,yyyyjgy,r,j.yjyg...y,j (!. .rjy.yj )jyjy jy . .E.! .;rr.jtrq'lt:!)ri k. ..y!. . .y .. .y...(.(y..(jy;yy rjj):iy'j y-tj jj.. gyyyyyg.yjj.y;jyy . j ('-. y j!y.yj.rjj.jy...yygjyyjy.jy..y.yyjyg.j.jj..j... .j. Iyyjj.gy jyj. ....;... ( !jt,jjjt.jrt); . y jj ! . k j jjj.)j..j(.yr((.y qj..ryy.r(..(.g (ti yjj. yyj.y. jj (yjjy . .ri (!i!!j.jrj! j yjjyjyy .yy..... ... tjjr y ( j lj j j j I j . g . k yjj-))y'. j..y. ;jjyyjj)yjyyy..j. .....ii..yj.(j.!..lkl .jjyj. j.yyy..j yyy; y. jyyyy-p jy .(.r.(g;).).I);rytjjjt'. y yg jyj. ;.jy;, y (jyyjy .
.
(: I ; . i p'ji. . ;jyt)': .
-
E
. .
-
q
.
q .
-
.
-
q
.
.
-
-y
. !-y
E . E
,
-
jy -
y .. y !-
-
-
.
- . . , -
-
-
,
,
-
-
q
yE -y... -q ! E y j . gEi- E.q(-- y-y. . .. ! yyy j-; . y . y.-.(yyy- y yyy.y - . y , - - j y-y -(gy y. y.- yy - - jg y -. y y- -yy yy yg.--. --(.; .
.
.
i y i .(.E i. : ,;,;$. ( ( IIi
-.
. -
-
-
--. . . . E. .
!.. .
-
.-
(. i :. q: g y . ..y. gy. yy... . y ! E g yy yk , y y. -
ttE
EE . i E; . E.q ( q E . .: E!(E E.Ei.-EEE ! E - -E r E E
..;
-
.
,
-
.
.
.
.
.-
.
.
.
.
4
Equqtions
P ar tial Digerertial
Paraolic
Equations
In the flrst third of this chapter, we will show how Brownian of the followihg equations: solutions usd to contruct (clmssical)
uf
motion can be '
1 -.AiI
=
2 1
-l
uf
=
+ g
2
y
g
u in of
+ cu
-Au
:
2
x Rd subject to the boundary condition: (0,x) Rd (0) x and u(0, z) J(z) for z 6 Rd. Here,
u is continuous
at
ch point
=
Au
(.)21Il =
a +
oz
.
.
+
.
:2
'tzd
t'lzda
and by a classical solution, we mean one thyt hms enough derivatives for the equation to mke snse. , That is, u e C y a th functions that have oe cont j neach of the zi. The uous derivative with respect to t and two with respect to continuity in in the boundary conditipn is needed to establish a connection IR.J . and u(0, z) c.z: J(z) which between the equation which holds in (0, ) x holds on (0) x Rd. Note that the boundary cpndition canpot possibly hold R is continuous. unless J : Rd W will :ee that the solutions to these equations are (undersuitable assumptions) given by '
,
'tz
-+
s.y(s,) + /' gt s-(y(st) J
-
s,
0
(/,' z-(z(s,) exp
ct
-
s,)
ds)
s,s-)
ds))
Capter
126
Partial Diential
d
In words, the solutions may be described (i) To solve the heat equation, run
(ii) To introduce a term gzj
add
a
s
= .X'0 >
follows:
Brownian motion and 1et u(,
the intgtal
of g along
z)
path.
the
=
N
(X i xJ) )
'tz
continupus
Proof
lf
'tz
in
(0,x)
at each point
satis:es
Applying lt's
.
.
of
(1.1a),then
Ma
:
E '. . .
. .
.
Rd and u(0, z)
utt-s,
A'tIt
dBr
.
'
Br) dr
r,
-
.
t
=
otherwise
0
'
.
'i
..
:
If there is a soltioh
..
.
.
.
' '
.
of '
.
.
=
sihce
JLu
+
-.'tzz
B,) is a local martingale
0 and
=
E1
theorem.
(1.1)tht ..
.
oudd
i ..
'E
.
.
the it must be .. .
'
.. .
.
EcfBt)
>
use n for a
'
'
.
.
: .
.
. .
.
..
('
.
. .. .
.
..
is bounded, then M,, 0 K s < t, is Proof If we now assume that martingale. The martingale convergence theorem implies that
/'(z).
the
while
j Sd
equation
.
(t.3)Theprem.
j. . . . . .. .. . .
=
r, #r
-
,
1%
r
Our next step is to prove a uniqueness
''
x Rd.
(0J x
Tnt
(1.2) follows easily from the It's formula the second term on the right-hand side is ' local martingale. '
yzy
#r) d
r,
-
Here H means that the last equation deqnes r. We will always generic soltztion of the equation and w for our special solution.
from the fact tht if the This equation derives its name, the heat equation, units of measurement are chosen suitably then the solution ult, z) gives the temperature at the point z E R d at time t when the tempratpre proflle at E time 0 is given by J(z). rhe frst step in solving (1.1),ms it will be six times below, is to flnd a local martingale.
(1.2)Theorem. on (0,8).
E
'
.
.
r
y q uajou
and .Xr0has bounded variation, Brownian motions, so
-dr
=
.
r(@,z) equation:
t
-w
=
()
To check this note that duho X%vwith 1 K % d are indpedent
Equation
ln this section, we will consider the following
(1.1b) is
Bz)
u(0,
-
ea j
d gives
the killig rat Jvhn th Articl particle survives until time t, or
is at
;g
Sectim Y.1 Te
Equations
'u
Mt :'
H
1imM, ,1
a
bounded
exists a.s.
this limtt must be JBt ). Since M, ip pntformly integrable, lf v s>tisfles (1.1b), it follows that 1?(t, z) ETMZ utt, z) I::l EZMt =
. .. . '.
=
=
.
Now that (1.3)has told us what the solution must be, the next logical step is to flnd conditions under which v is a solution. It is (andalways will be) easy to show that if is smooth enough then it is a clmssical solution. 'v
formula,
(10.2)in
Chapter 2, to
ulzo,
.
.
.
,
zdl
with
(14) Theorem. .
supposeJ is bounded.
If
%'
e c1'2then
it satisqes
(1.1a).
128 Proof
Chapter 4 Partial DiRrential
The Markov property implies, >Q(T(ft)l.T-,)
=
Section d.1
Equations see
Exercise 2.1 in Chapter 1, that
fb.(T(#t-,))
'J(t
=
s,
-
1?(t s, Bs) -
'p(f,
,
=
-
0
1
(-'pt+ jAr)(t
-
=
#,) Dfpttz, y4
The left-hand side is a martingale, so the right-hand side is also. If then repeating te calculation in the proof of (1.2)shows that .80)
(2,rt)-d/2c-I.-,Ia/2t
pttz, y) here calculus gives little 5V
r,
1)
e
=
-(zf
C1,2, Dipttz, p)
=
(z
p)
=
(z
Dpttz, p)
=
Diilhz,
Brldr
Writing D
.
yi)
-
t !n)2
-
0l0zi and Dt
=
129
0l0t,
=
pttz, p)
t ,t(z, p)
-
a tytzj a
-
ffeat Equation
Te
yj ) pftz, p)
-
t -(f/2 + Iz p12 2t a t -
+ a local martingale The left-hand side is a local martingale, so the integral on the right-hand side and locally of bounded variation, is also. However, the integral is continuous and L'v are surely. Since H 0 almost must it be by in Chapter 2 so (3.3) l.aL'v H 0. For if it were at 0 ontinuousy it follows that + some point # (, z), then it would be # 0 on an open neighborhood of that point, and, hencej (:1 with positive probability the integral would not be H 0, a contradiction.
lf
1 is bounded, then it is
j
'pf
to
easy
see
ID.pt(z,p)
pttz, p)
that for a
=
i,
,
or
t
J(g)Idy k cxl
-kn
lt is emsy to give conditions that imply that keep t e exp sition simple' we nrstcosider the ,
(1.5) Theorem.
If
f is bounded
Y-@1/2# Bt Proof 0 and variance 1, so if tn implies that .&)
-
-+
and
continuous,
where N and zn -+
satisfles (1.1b).In order to situation when f i bounded. 'p
then
'p
satisfles
(1.1b).
hms a normal distribution with rfiean 0 theorem z, the bounded convergence
and is continuous in Rd, so (1.6)follows from the next restlt on diferentiating Jmder the integral sign. This result is lenjthy to state but short to prove since work. Nonethless w will :ee we mssume everything we need for the proof to that this result is useful. (1.7) Lemma. Let (S, 3, m) b a J-qnite memsure space, meuurable. Suppose that for z e G an open subset of Rd have:
(a)u(z)
(b) Jf (z* + ltei 'zf(z)
is a solution is to flnd conditions that The flnal step in showing that smooth. is ln this it that case, the computations guarantee are ot very difficttlt'. 't/
Proof
If
f is bounded, then
'v
E
6'.1,2and hence satisfies
By deqnition,
1'(t,z)
(1.1a).
an
tj
=
p)
ErlBtj
=
Jpttz,
y41y4 dy
Gx S
ff (z*,y4
-
-.,0
jark (z. og
nnloziexists
-+
=
mdyt
R are measurable
is continous
yyjgtyjj
ogj
go mtty;
at z* and equals
w(z*).
.y.
,z(z+)
with
y G
at z.
%.
1 and A(z) = K (z,n) gives the following. Note that in (a) att (c)of (1.7)it is implicit that the integrals exist, so here in (a)and (c)we mssume that the sums converge absolutely. =
(1.8) Lemma. we have:
Suppose that for
z 6
G
an open subset
of Rd and some
0
>
'u(z)
=
and
-+
*)
-
=
'tzf(z)
j
and(d)E,zJ-p! o
nulozi
kT:
&yith
n :
'tl
-
is continuous.
/'(z).
J is continuous. e C1,2unless
Here gt, z) '
,
=
=
a is a solution of the equation with 1 = A and pj so we can restrict our attention to the cmse J H 0. Having made this simpliflcation, we will now study the equation above by following' the procedure used in the lmst section. The frst step is to llnd an asociated local martingale. =
.q
If
u satisfles
=
(2.1a),
Ethen
4
(*
j
M, is a local martingale
u(z*). .. .
E
' ' : .
:
(
Proof
'tIt
=
s, Bs) +
-
.s'.ly(.st)I -
/
Izl-2log-b I.f(z)I -+
0
as z
Replacing the bounded convergence theorem in convergence theorem, it is not hard to show: (1.9) Theorem.
If
J
is
continuous
4.2. The Inhomogeneous In this section, to the equation
and satis:es
1ttt
-
s, Bs)
'tztt,
Bz)
-
!
=
0
'tzt
=
lal'tz
' .
co
(+),then
+ g in
'p
satisfles
(t.1).
happens when we add a function lmst section. That is, we will study
(0,x)
x Rd
.
.
(1.5)and (1.6)by the dopinated
gt,
(2.2),since
1
(-ut +
jAtI)(t
Tut
r, #r)
'
r, Bc)
-
dr
a local martingale.
Azkin the
hext
0
-
.
dBr
-1.:
+
-eu
''''''.
Ly
-g
=
.
.
apd the second .
.
the .
g
(
right-
'
result.
.
>
E=
.
t
a solutin
gt
'tJ,
.$1t
gt
q:z ()
(2.1)'that is
:
s,
-
,
) ds
M,, 0 % s
1.
: ffa if p < 2.
The special case p
< x
The Felman-ffac
1 is the Coulomb potential
=
which is importgnt in physics.
Proof If t:v < 1 then changing to polar coordinates is largest when z 0 we have
and noticing
the integral
=
E z.a
fa
jo
tf, ) ds
0
-+
su '
If we put absolute varllzes ipside the integral and strengthen #he last result to uniform convergence foi z E a then we gt a deqnition that is essentially due A function is said to be in ffd if to Kato (1973).
/' I,(s-)I#s)
sup t0 lim
kJo
r
By Fubini's theorem,
the
we can write
liy tt
itz,
sup a,
where
above
-
ctt
r
when p
=
0
jw-pju.
2. In d
s ,
-r,
0 and
Iz -
g12/0
-+
-(d-a)
c,
= p1) I(p)1dp 0
d a r 2 log d r :(r) d 1 r of The equivalence (+)and (++)is Theorem 4.5 of Aizenman and Simon (1982) or Theorem :$.6 of Chung and Zhao (1995).Section 3.1 of the latter =
-
z
=
=
which
holds in d
# 2, then
the
r1-F
2 Fhen p
Let g
=
c(t
-
s, l,)'p(t
-
s, Bs4
The Markov property sup f. .'r
.
.
ds
.
0 < z< ,s
.< .-
a a
0 : Bt / GJ. Tp see the similarities to the solutions given for the equations in Part A of this chapter think of those solutions in terms of spac-time Brownian motions B, t s, Bs) run until time = infts : Bs / (0, x) x Rd). Of course, f' *
have
.
1 n
.
.
t
1
u
.
.
w
C
Pcm
6 G for some s E
i
(:,@))
is continuous for each .7 > 0. Letting e t 0 shows that z K ) is an incremqing limit of continuous functions and, hence, lower semicontinuous. (:) .!%tr
Proof
> s)
(-r
implies that on
The Markov property
Ez (T(f,.)1(m<x)1.F,)
(#)
=
-+
'
'''''''' '''
The lefhand side is a local martingale on (0, so (#,) is also. lf then repeating the calculation in the proof of (4.2)shows that, for s E r),
i7(#a) -
1
'p(fo)
=
j'
'pt
6 C2,
If y is regular for G that if zn v, then
and
t
j, then fjt'r K i)
>
1, so
=
-+
from (4.5a) tt tollows
'19.9'.
.
(0,r),
liminf n-x
.
.
.
.
. ..
''
Lb.B
r)
dr +
it is easy to complete the proof. Sini f is boundd With this rstablished, contiuotts, it suces to show that if D(p, J) (z : lz pl < J), then
local martingale
a
=
The left-hand sid.e is a local martingale on (0,r), so the integral on the rightand locally of bounded hand side is alo. However, the integral ls continuotls variation, so by (7.2)in Chapter 2 it must be H 0. Since A is continuous in ' G it follows tht A'J H 0 in G. For if A # 0 at some point we Would have a El contradiction.
(4.5b) Lemma.
If y is regular for G and zrt Prn ('r
0
-->
Bv E DD, f))
1
--+
1
Up to this point, everything hs been the same ms in Section 4.1. Diferences condition longer (4.1b),sin it is nosatisfy appear when we consider the bouudary continuous. must a G set and The open sucient for f to be bounded regularity condition. Deflnition.
.
1 for all t > 0'
('rK t)
.&n
'
.
.
A point y 6 DG is said to be
polnt
a regular
if #vtr
=
0)
=
1.
(4.5) Theorem. Let G be any open set. Suppose j. is bounded and cohtinuous fy). y, then :(zn) and p is a regular point of 0G. If zn E G and zn -+
-+
Proof
Let
6
so small that
0 and pick
>
Jb Since
/ tr (r S ) r&
-+
1 ms za
-+
sup
0s,st
-
J 2
0) 9 Pzlv
rtlzli
conditions
will
-
=
=
L.
=
.
.
.
..
.
.'
.5
.
.
.' :.
.
.
Tllorn.
c)
(z : z
=
Let d k 3 and let
=
y+
Now that we have defined a is spherically symmetric,
ov +
e 7(p,
Clnlm. lf
cn
(x7
term is the
t 0 suciently
single
(0J.
point
PnBt na K liminf tt0 *
=
In
=
(z : zl e (2=',2-'+1) .
)
za
=
za
.1.
z
and
'?p,
Izl
s
j
1#, -
gl > r
0
=
Combining the lmst two results with a trivial inequality we have
fmst, then 0 is not a regular point of 0G.
Pzlhl, .R3) (0,0) for some t > 0) Proof Brownian motion Bt starting at 0 will not hit
(0,cr),
y, polnting
vertex
where ea is a constant that depends only on the opening c. Let r > 0 be such that F4p, c) n D(p, r) (: G'. The continuity of Brownian paths implies
,
=
0G
0
an r >
the rest is easy. Since the normal distribution
cone,
PyBt
z) where
lim Py sup -.0 where the n
D
show that if Gc is too small near y, then y may be shows that if Gc is not too small near y, theu y is
. .
'p,
'
=
Lebesgue's
4.2.
irregular pint.
.
'??,
Exnmple
t is an
1/2 and
.
.
-
'
.
''
7'4g;v
Disc. Let d k 2 and 1et G H:z D Exnmple 4.1. Punctmed (0), wheie = x) f%(Tn 1 then D = (z : 1zl< 1). If we let Tp = inf (t > 0 : Bt 0J, regular of point 0D. One can get an by (1.10)in Chapter 3, so 0 is not a D ff phere example with a bigger boundary in d k 3 by looking at G = ff (z : zl z2 0J. In d 3 this is a ball minus a line throgh the prigin. .
n=1
Pxoof The frst thing to do is to deflne a cone with ln direction with openlng c as follows:
give two examples.
=
.j
,.
.
' .. . .
. ..
y.
fo a point to be regular?
ln order to answer the flrst question we
...
< xl s y'latrz'r, s1
,)
(4 5c) Cone Cpndition. If there is a cone V' havin: vertex y nd such that F' n Dy, r) C G', then y is a regular point.
Do irregular points exist? What are sucient
'
=
The lst tWo exmples irregtlar. The next result ... .
it is sucient From the discussion above, we see that for to satisfy (4.1b) situation necessary) regular point. each point of almost OG This is a that (and questions: raises two
=
Co
' . . ..
't7
147
probability
so with
0,
=
1,
'a
e
GC)
K lim/tr t0
% ) K .J$(,r = 0)
and it follows from Blumenthal's zero-one law that =
.
.
.
=
a
=
0)
Since Bt is transient in d k 3 it follows that for a.e. u2 the distance between (#, : 0 K s < x) and % is positive. From the lmst observation, it follows
.' ..
'
. ' :. ..: ..' . .
: . .. .
.
' . .
.
(.
' (4.5c)is ,
Exerclse
for
=
0)
'
.
=
1.
.
for most
ucient
4.3. Let G
each p E
./5(,r
(z : #(z)
cases.
< 0) where g is a C1 function 0G. Show that each point of OG is regular. =
with
Tgyj
0
Chapter 4
148
Partial Dirential
Section 4.4
Equations
However, in a pinch the following generalization
be useful.
c>n
'p,
gzt
.
. .
.
' ('. .
..
E
.
-.s
.
.
k '
.
.
E .
:
. .
:
.. .
..
Let G
e any
(4.1a).
open
i
.
then
ounted,
set. If J is
. 'p
i'
E
.
=
.
.
and,
So
is a
.
. .
.
.. . .
; ' . .
jo jo
times g
cpnstant
conclude
.ts(,,r;
(z +
dv rd-1#(r2)
hy)
=
/z(z)
and
(z)+
tnf
.
+
yi
'tavexaging
E
i
.
(4.6a) Lemma. Let G be an open set property the averaging
(z) when z
e
G and J
0 so that D(z, J) C G. If we let o' = Proof details Markov' that implies property more J)) strpng D(z, the then Bt 4 (for See E X ample 3.2 in Chapfei 1)
+ z) dz
hence C 0 we note that the multivariate Taylor's theorem implies that if Ig zl K r then
ro
.
.
.
C
-
dv rd-lktra) J
.'
.. ..
. . condltion,. . we noF jurp Having our discussion of the boundary to determining when ip is smooth. As in Section 4.1, this is tte our attention under minimal assumptions on J.
(4.6) Theorem. hence, satisfes
=c
=
-
.
completed
=
J
When d = 2 this says that if we can find a line segment ending at z that lies ifi!z each potitt t he complement then z is regulr. For example this ifnplies that the boundary of the slit disc G = D (z : z: k 0) za 0J is regular. .
-
#(IzI2)( Js(,,,)
-
.
149
Note that we use l!/ zl2 = E(!n z)2 rather than j!/ zl which is not mooth at 0. Making a simple change of varibles, then looking at things in polar coordinates, and using the averaging property we have -
tz) of 74#, b, a) to b' th intfection 4.4. Deflne a flat cone y, : 0 E contains 0'v the line (# + 1 dimensional hyperplane that RJ. ullat ucone'' cone.'' replaced by is Show that (4.5c)remains true if
exerclse with a d
The Dirichlet Problem
.
.
bj r2 and letting ,
. .
.
-+
r
.. .
0 it follows .
.
.
c)
Unbpunded G. As in the previous three sectins, 4ur last topic i to wliat hpbens when somethinx becozes nbounddk Thi time tve $v1ll d ignjre J. combining d tocu p (4.s),(4.5), (4,6)we haki
diss
(4.7a) Therem. Suppose that P oint of OG is regular. If for al1 lttn of bndd
(4.1).
.f
z
is boupded and contiuous 1, ttten e G, .J$tr < x) =
'p
d that each is the uique
To see that there might be other unbounded solutions consider G = J(0) = 0, and note that u(z) cz is a solution. Conversely, we have =
(4.7b) Theorem. Suppose that f is bounded and point of OG is regular. If for some z 6 G, P.('r < x) (4.1) is not unique.
continuous
0
=
show that /z is a solution
The last two observations completes the proof.
.
By working a little harder, we can is the only way to produce new boudel
that
show
(4.1)with f
of
&.&('r
adding
=
H
0, which I::I
x)
to
solutions.
(z)
-
< x)
Erlmtkr
=
+ tJ%('r
We will warm up for this by proving the following special
(4.7d)Theorem.
'
:'
....
..
:
..
'
.
.
. .
.
cmse
If u is bounded and harmonic in Rd then
in
.
=
Rd.
o(z)
ppxr
=
boundary function
.f
(4.1)with
=
x),
e 0. boundary
follows.
k 0. theorem
the
on
martingale
hBtj
at
=
/z(z) -+
=
.2.((.B);m
(z) k
cxl proves
t)
>
-MPrr
=
1 -MJ(r
> @)
x).
''
..
'
.
'''''
:
'
. ..
r .
.
.
..
.
.
,
'r)
'tz
is
constant.
imply tlzat uBtj is a bpunded Proof (4.2)above and (2.6)in Chapter 2theorem &; implies thay s t martingale. So the martingale convergence u(#t) Ufn Since U is memqurable with respect to the tail c'-fleld, it follows from (2.12)in Chapter 1 that P.( < Ux < bj is eithi H 0 or H 1 for any The lmst result implies that there is a onstapt c independnt of z so that a< values it flloWs that ,i(z) c E=U. k c. E expected Ku,z c c) 1. -+
-+
.
.
,
'akig
1z(#f) is > bounded localrs y (4.7c) From the= proof of (4.7d)we see that x4, hk U, iii J( r) so i-) On (i. < lipnyr w n (0, s U. frprtingay of tlt st i a cstat Wa ce ifdepekt h t we needthatto sh + is that ot th co). this i Intuitikely, qen #r a o (r pointBo so msymptotic c'-qeld, but the fact that 0 < Pztr = cy?) < 1 makes of the triviality extract tltl. in Chapter 1. from (2.12) tt difftc ult to around identiqed in the previos p'ra raph, Fe 4i1l the diculty To get ' in doing this aie t steps whole space. The two extendu to the EcfBpj (a) Let #(z) = utzl r < x). (4.6)and (4.5)imply that # i with boundary function ). H 0. bounded and satisfles (4.1)
Proof of
(4.1)with
satis:es
To do this we use the optional stopping time r h and note (#z) 0 to get
.
G
1t?
Equation
Because of the Maxkov property it is enopgh to show that for all z and t we hake tptzl S .E'.o(#t). Sinc w k 0 this is trivial if tr / G. Tp prov this for z E G we no te that (i) implies J/l = wBt) is a oundect local martingale on (0, and I'Ip'g 0 on (r < x), so using the optional stopping theorem
'. '
which
to G,
's
(iii) 'tp(St) is a submartingale.
x)
=
(ii) w
Letting t
'tz
'z(z)
(i) When restricted
.'
Suppose that f is bounded and coninuops and that each (4.7c) Theorem. nd tisfl (4 1)''i G , th thie i PO '!1t 9f OG i? regular. If is bounded stht 4 such that
i
To complete the proof now, we will show in four steps that from which the desired result follows immediately. The proof of (4.7b)implies that Mprr < x) satisfies function f > 0. Combining this with (a) the desired relt
E
.
Poisson
=
o(z)
=
.Q(I#,.Af)
(iv) There is a constant
p
=
.E%('tp(ft); r
so that o(z)
=
t4 K .#Q('tz?(ff))
>
ppzr
tx).
=
Since w is a bounded submartingale it follovs that as t s+ x, wBtj to a lin'lit W'x The argument in (4.7d)implies there is a constant ftW'x p) 1 for all z. Letting t x iy .
=
=
converges
#
so that
-.+
'tp(z)
=
ErwBt
);r
t)
'>
'tz(#t).
'titlk
'thet
arid using the bounded
convergence
gives
(iv).
(:I
=
'
'
'
-
(b) Let f
=
llllx
and 1D(z)
look =
at
(z) + Mpzr
(c
=
x)
z z
eG s cc
4.5. Poisson's
Equatlon
In this section, we will see what happens when we equationconsidered in the lmst section. That is, we
(5.1a) u E 6'2 and lzLu (5.1b)At each point of 0G, =
-g
add a function of z will study:
to the
in G. u is, continuous
and
'u
=
0.
As in Section 4.2, we can add a solution of (4.1)to replace 0 in (5.1b)by u = f. As always, the first step in solving (5.1)is to find a local martingale. 'tz
=
4 Partial Dilerential
Capter
152
Poisson
Section 4.5
Equations
's
Equation
153
side is also. The left-hand side is a local martingale on (0, so the right-hand If ?? 6 C2, then repeating the calculation in the proof of (5.2)shows that for s C (0, 'r),
j
tfk nBt ) + =
is a local martingale Proof
(,
or
v(s,)
E
yj 'u(#c)
/
=
-
%nB,)
< r.' hand side
The next step is to prove
(5 3) Theorezn. .
(5 1) that is
.
result.
,
re bottnded
. '
?
If tht'e is a soluti
f
w(z)
H
. .
.
.
.
.
.
gBtldt
Ez
'
'tl
I-fl K Ilullx+
rllpllx
(2.7)in
'
Chapter 2 and
.
H
tztzl As usual, it is (5.4) Theorem. then it satisfles Proof
emsy
=
lim Mt
,(#)
o
=
)
EzM;
=
(J* J 0
implies that on dt
l
j
J
J-
('r > sJ,
gntt
J 0
If
'p
E
di E
E
-
the conditiops
corfl
rtded
ms no suipris.
to
.
-+
We begin by observing: (i) lt foliows from (4.5a)that if e7> 0, then Pcn (r > :) s+ 0. (ii) If G is bounded, then (1.3)in Chapter 3 implies C supz E=r t')
'
(i)and (ii)proves (5.5)since
e: is arbitrary.
E1
Last, but not lemst, we come to th question t smoothness. Foi thee developments we will assume that g is deflned on Rd not just in G and hms Recall we are supposing G i bounded and notice that the compact support. values of g on G are irrelevant for (5.1),so there will be no loss of generality if we later want to suppose that dz 0. We begin with th case d k 3, because in ihis case (2.2)in Chapter 3 implies
J#(z)
=
Co
(ll
Suppose that G is bounked and g is continuous.
gnt,
.q
rpartingale
tter tlteektensive
.
to show
The Markov property
+
The left-hand side is a local martingale on (0,r), so the integral on the righthand side is also. However, the integral is continuous and locally of boundd vaiation, so by (3.3)in Chapter 2 it must be > 0. Since + g is continuous in G, it follows that it is H 0 i G, for if it Were # 0 at some point then we Fpuld have a contyadiction. I::I
dt
=
t1r
ErMZ
a local
Tht'
(5.1b)imply
T
Mv
,
Proof
'tz
is integrable,
(.j1A.p
/'
=
f
Proof If satisfles (5.1a)then Mt defined in (5.2)iq a lpcal martingale on (x) r). tl G. < then in Chapter implies for If G is bounded, 3 Err : z (0, a (1.8) E nd g are bounded then for t < r If
side
/,.
4v
-+
.
0
Since the right-hand
' gBr,
.
'
:
.
..
T
-
+
Jlr
I:I
a uniqueness
Suppose that G and must be
/
Lumjds
the flikt term on the right-
and
-g
=
pg
J 0
o
(0,r).
bounded,it
.
.
laLu
This proves (5.2),since is a local martingale on
j
:;-
dB, +
.
J 0 )'
for t
,,?(s,)
-
r).
formula ms we did in the last section gives
Applying lt's 'u(#t)
'r),
g Bs ) ds
1(z)
=
Ez 0
:
Ip(f)I
0 : Bt / ff) then .
f
the right-
.
deverything
exptcrl)
EzfBr)
E
l
k: 0
.... .. wtt . .. Exampte j.i)) tht the trouble x/2 then c H y is too large, or to be pmise, if
(agatnin o(z)
We will not do this, howeyer, because the folloFing simple this result is false. 1, G
.
see betow
;
,
''
r(w)
sine
'llrst
At this point, the reader mlght eypect that the next step, wsitr hps been fve times before) is to mssume that everything is bounded and conclude that if there is a solution of (6.1)that is bounded, it must be
=
.
ab > x/2
ds
E
6.1. Let d
.
:
for < r. This proves (6.2),since d% = c(#,) #s, laLn + cu term on the right-hand side is a local m>rtingaleon (0,r).
'p(z)
''
exlmpl is that if c
j
1
' ' . ..
'
Wewiil
dca
0
0
Exnmple
then
cos z/ cos ba
=
)'
Applying It's
+
.'
ab < x/2
9.1) that if
;'
cB'lds.
.j-
cos z/ cos ba is a solution.
=
r(z)
(0,r).
1t(#())
u(z)
(inExample
We will see later is a local martingale
'
.
0 then there is no solution.
=
shows that
eollztiox
We
'f
i . ..
(
.
'
'''
g
:
.
sup E. (exptpcwl) s2 '
g)
laroofpiic
.t,
>
zrn
() so that ee,
>
-/)
n')
1 s : sup 9 So it follows by induction that for
''
( > (
Espnxrn
=
>
Pvrn
integers k
all
d V 1z'(p)S : Fhere
k 0 we have
K
-
=1
(x)
4
s J7 k=1
1 4:-:
=
we
cxl
1
J7 ak-,
. .: : .
% n') .
=
4
k=,
Careful readers fnay have noticed that we left r.t However, by the Blument 'h a l 0-1 1aw either Prg computation is valid, or Prrx = 0) 1 in which
(
..
1 .
1
-
za
r
=
tb(z)
Lt c* : s> Ic(z)l.By (6.3a), we can pick rc o mall tltat itrr I.& Bz I > r) and r a, then l'xexplc*tr'r) < 2 for all z.
Since 1/z is cnvex,
:+
r
(6.3b) Lemma.
inft
:
Let 25 K a.
11-D(z, 2J) C G and y E D(z,
Th e remson for Olr interest in this is that it for y e D(z, J). If Dy, r) C G, and r 1D(:)
=
K ro then th
Splexplcg'rl'tz'l./ (Tr)))
= .;'plexptc*Trll
o(z)
:'
'
xdwyt
*'
'E
Co
k ckl imptis
2 (:x)
and using
(++)
jopfy,vj
s2
.
c.d
a simple covering
,
w
k 0
we
ttltzl
1/2 and
integrating from 0
dz
have dz
(++)and (+)it follows that
j
(6.3b) and
rd-l
s(.,aJ)
Co k 2-1 (2J) a oy.n)
C
k
1t)tzl
--
D(z, 2J) D Dy, J), tp(z)
by
1
-1
.(g)
dz
0(,,,)
o k 2-1 (2J) d 2
memsure,
l/.zalexptc*garl)
tp(z) k 2
tp(z).k :-z co (2s)a
jopLu,vj 1stzlrtdzl
to be a probability indepehdehtk
are
ztptzlvtdzl
k
where again Co = d/cd. Combining ijlies
the f>c4
Jehsen's inequality implies
d
wij
z and using
=
aDztr)
Combining the last two displays, multiplying to 26 we get the lower bound
K .G(exp(*Tr)to(#(Tr)))
where z' is surface memsure on oDy, r) normalized since the exit time, Tr, and exit location, #(Tr),
dy
'
), then
Varkovproferty
strog
o(a)r(dz)
and
k f.(ex#(-c*X)+(#(Tr)))
j
f'alexpl-c*Trlz
Rearranging
shows
dz k 2-1
.E'.(ekp(c(s)u)(#(Tr)j
(2J)d
wybK 2d+2o(z)
Proof
:
s
-
:::i
= fxlexpt-c *Tr))
2
=
=
rd-l
1). ltearranging we have
=
D(p,O
z)
s(,,,)
(z : 1zI 'tz'(z)
.
0 out of the expected value. 0) 0 ln which cmse our = 1. E1 cmse Erexporu)
=
of
cd
,.,(
where Co Vld is > constant that depends only pp d. Repeatirk the flrst argument in the prof with y gives that cr. k -c*Tr
have
. . '..
4
-a
is the surface area
the last inequality by
j
-1
.
Equatiim
=
Eexptpalftl 1)-/ 4 rn .
(72
(+)
DO
Ez expldlw)
and
%a
1)-/)
-
1 sup Patrn > n') < w 4 = eo K 4/3 by mssumption,
Since e' is incremsing, and
(
D(g, J) c: G, multiplying integrating from 0 to J gives If J
> y)
1)y); rn
-
ms i!l the
The Schrdinger
.
-1
8d wyj Co
argument
:ic
:-(d+2) wy)
lead to
D
Chapter 4
160
(6.3c) Theorem.
Partial Dirential Let G be
Equatiops
a connected
1o(z) < x Proof w < (x) an open a subset with zrp so wy) onc
Sectfon 4.6
open set. If w
for
all z 6
/
(:x)
then
(A3)
G
(6.3b),we
E1
< x.
WC
MYOCCCCI to tMc SRiQRCRYS
I'CSl;It 5Ve ,
5V1t
to
lf there is a solution 'p(z)
%
of
(6.1)that
ErfBpj
explcrl)
toof (6.2)implies that Ma Since f, c, and u are bounded, letting s gence theorem gives ,tz(/,)
=
'u(z) =
th: lt
Strengthen
161
x.
see
--,
CeSX l s ion.
/
(6.3) Theorem.
that if 1t)(z) < x, 23 K a, and D(z, 2J) C G, then tlJ(z) < XJ is on D(z? J). From this result, it follows that Gz = (z : considered subset of G. To argue now tht Gz is also closed (when mq of G) we observe that if 25 < a, Dy, 8) (EEG, and we have z,z E Go large, y 6 Dtza, J) and Dtzrj, 2J) C G, y e G then for n suciently From
w
Te Schrldfnger Equaton
ErlBvt
exptcr);
r
is bounded, it must be
exptc,)
1r
is a local martingale on (0,r). A t and using the bounded conver-
< t) + EsuBtj
explc);
r > t)
!
Since J is bounded and u?tzl Sxexptcr) < x, the dominated convergence thepremimplies that as t x, the first term converges to EofBpj exptzl). =
-+
(6.3d) Theorem. Let G be 1G1< x. If / i2$ then
a connected
inite Lebsgue measure,
opep set with
To show that the second term 0 we begin with the observation that since conditional expectation and the Markov property (rt,> t) E Ft the dennition of -+
,
't
1D(z)
qup
,
< txl
imply
'
'
E
'.
fpr Proof Let ff c G be compact so th>t 1f? ff l < y the const>nt in (6.3a) 0 c For each z e K we can pick a ta so that 25. K rc and D(z, 2&) C G. The open sets D(z, &) coker ff so there is a qnite subcover Di, &), 1 K % f Clearly,
ErltBt)
-
=
exptcr); r > t)
.
=
.
'tt7(zf)
suo ls-sz (6.3b)implies
wyj
K 2d+2o(z) for M
If y 6 H = G Markov property
-
=
< cx:h
p 6 Dzi,
sup wyj
vK
ff then Ey texplc*rJzll ,
.;),
.
K 2 by
r) + Ev
a1l y E
'r(p)k
(6.2a)
.
so using
rn (exptcolzpllra); 1t?(p) Ey (exptcrx); f'f'p Bpn ff 2 + K (explcrx); e ) i 2 + 2M =
=
Now we claim that for
so
< cxl
BvH
.
the
.
. .(
.
'
.
strong
l'
.
=
(A1) G is a bounded (A2)
f
and c are
connected
bounded and
open set. continuous.
t)
G expt-c*l.!%tr
S 1) k e > 0
r > fz.tl'ztftllexptctl;
t)
:-l.E'.(lu(#t)l exptcr); s e'-'lleullxfwtexptcrl;
S
x, by the dominated convezgence theorem since as and < x) 1. Going back to the first equation and the proof is complete. shown 1ttzl -+
u)(z)
.l%tr
=
'p(z)
(6.3d)established,
r > > f)
The flrst inequality is trivial. The last two follow easily frpm (A1). See the first display in the proof f (1.2)in Clzapter 3. Replacing wBtj by 6,
: Ik-4
we are now mpre than ready to prove our uniqueof the results that follow we will now simplify result. the statements To ness list the mssumptions that w will malte for the rest of the section.
With
.
EzkztBt) explcrllF); EruBt ) exptclzstlt) ;r
=
r > t) v > t) =
-+
0
Es exptcr)
< tx,
in the proof, we have Cl
This completes our consideration of uniqueness. The next stage in our always furtunately, is it has been. Recall that here and in program, ms emsy as what follows we are assuming (A1)-(A2). (4.4) Theorezn.
lf v
e C2,
then it satisfies
(6.1a)in G.
Capter
162 Proof
d Patial
DiRrential
The Markov property
('r >
implies that on
f.(exp(cr)J(lr)l.'&)
=
=
The Schtdinger
Section 4.6
Equations
to the previous Emse.We begin with the identity established for a11 t and Brownian paths LJ and, hence, holds when t
s),
explollbxlexplcrlytlzl)
r
exp(c,).??(.s,)
cBsjds
exp
The left-hand side is a local martingale on (0, so the right-hand side is also. If 6 C2 then repeating the calculation in the proof of (6.2)shows that for
=
0
T
1+
0
Equation ther,
163
which hold
r(tJ)
=
T
ctrldr
cB,) exp
ds
,N
'r),
Mgltiplying by JBp)
'tl
and taking expected
values gives
,
s e ( r) ,
,
'?( s-lexpto)
/' ()as
.p(.o)
..
.
+
-
-
r(z)
cs)
0 + a local martingale
(srlexptcrlfr
Conditionzg ..
.
.
:
' '
L
on
'?()
.
The left-hand side is a local martingale on (0,r), so the integral on the righthand sid is also. Howeker, the integral is continuous and locally of bounded variation, so by (3.3)in Chapter 2 it must be H 0. Since E ()2 and c is 1Ar continuous, it follows that + cv H 0, for if it were # 0 at some point then ? EI we would have a contradictlon.
(.nr) +
Erl
=
j (c(l,)
ctsrlfrj
=
a
EnlBvt
> 1?1(z),
J(lz)1(,
1)
> 1)
0
-+
E1
This brings us finally to the problem of determining when is smooth enough to be a solution. We use the same trick used in Section 4.3 to reduce 'v
C. Applications
to Brownian
c is Hlder
continuous,
then
Motio
In the nxt three sections we will use the p.d.e. results proved in the last three to derive some formulas for Brownian motion. The first two sections are closely related but th third can be read independently.
Caper
164
Patial
d
Dirential
.
In this section, 4.4 to find the
we will use results for exit distributions for D
(7.1) Theorem.
If
Summing the lmst expression on
for the Ball
4.7. Exit Distributions
.
...
.
.
..
.
' .
.
.
.
:
.
.
.
.'
l1u(z)
:
.
-
r(z)
.
memsure.
If we replace the 1 by
then
n solves
(7.2a)In D,
wE
C2 and Lv
(7.2b) At each point of 0D, The llrst, somewhat
(7.3) Lemnla.
-
e op
z
1)
is
=
painful, step in doing this is to show
If y 6 OD then Lky
'' :
rhj,
0 in D.
=
-
-
Iz
-
'
'
kd
:
E
(z
-
the 1g12,
.
.
(z
y4
-
-
collapses
expression
(-Iz
pl d+a y4
-
.
-
pl2 + 21z12
-
2z
-
.
lpl2 lzl2)
y +
-
.
2z y #
IzI2
=
.
-
=
0
.
.
.
.
' .
' .
lpl2. ' '
.
opep
set D,
'
' '
.
'tl
.
fz).
'p(z)
and
continuous
(12
-
of tlt t- So bringingthe is a linear combtpatlon (tnsl e the v ) iliflkrrptiationupder the integral (andleaving it to the reader to justify tltts uqing (1.7))gives .
0.
=
=
p12 =
since Iz
D
z 6
(4.1):
the Dirichlet problem
lt-#tz)
-
J(z)
+ 2d
-
to be a probability
Jdo ,(z)T(#)r(d#)
y
.
,f+z
-
-
clmss theorem shows that if of the monotne Proof An application (+)holds for J e C* it is valid for bounded mesurable f. In view of (4.3),we can prove (+) for 1 E C* by showing that if V(z) = (1 Iz12)/lz pIf@nd =
.
-
l:!) 12 dT(p)zr(dg) an Iz pI
is surface measure on OD normalized
+ 4:
-
-
.:
';r
Iz
1zI2 z 14 !/1 -
pIa
-
dz + (1 IzI2) Iz p1a+a jz gjdx.a -2dIz d+2p12 + 4d(Izl2 z y4 + (1 lzl2) 2d = Iz pI Iz pIdA2 Iz p1d+2
then
-
where
-2d
=
165
gives
.
=
1
=
.
the Dirichlet problem proved in Section (z : lzI< 1). Our main result is
and measurable,
f is bounded E=1B.4
'
for the Ball
Exit Distributions
Section 4.7
Equations
y sqtisfies
(7.2a). To
(7.2b),the
check
flrst step is to
1 1z 12 ag't#pl
show
,
-
Proof
f(z)
To warm up for this, we observe Dlz
-
pIF
=
Di
(
yp/2
zj
p.f)2y
-
p1z
=
p1P-2(zf
-
yi)
-
=
an Iz -
pl
H
1
apprpah but we prefer to use a soft ncmpzpputational This is just 1, I is invariant under rotations, instead. We begin by observing that f(0) with and LI 0 in D. (For the lmst conclusion apply the result for f H 1.) = : inftt If I > rJ. To conclude f > 1 now, let z E D with l1 r < 1, and 1et r Applying (1.2)of Chapter 3 with G D(0, r) now shows fcalculusy''
:'::z
i
'p
=
so taking p
'-.d
=
Fe have
Di tx (z) Diferentiating
=
=
(-2z) Iz
1
.
-
:1 a
-d(zf
(1 IzI) Iz
+
-(d
+ 2) gives
=
yi4
(-2) Iz 1pl d + 2 (-2zf) Iz pId+a :w)2 dtd + 2)(z d a lzl) (1 di4 + pjd-l-z pl jz Iz -
=
.
.
=
:1 a+a
-d(zf
Diikvz)
yi)
'
-
and using our fact with p
again
-
a
-
.
f(0)
=
EoIB.)
=
f(z)
where the second equality follows from invariance under rotations. To show that r(z) ly) as z p E 0D, we observe that if -+
-+
-
-
-
-
-
-
-
1 Iz12
0
-
kz z)
=
--,
Iz -
zl d
Ip -
zl d
as z
-->
y
z
# y,
Chapter # Partlal Diential
166
4.8. Occupatlon
From the lmsi calculation it is clear that if J > 0, then the convergence is niform for z e Bz6j = OD Dy, J). Thus, if we 1et B3(J) = OD #n(J), then -
since .J(z)
=
Ir(z) T(p)I
jy
=
r(z)
=
-
+
sup
'
'
.
whre .
lJ(z) J(p)I
pl
-
Axlog
-
. .
j t= r(#2 .
'
E
.
'
.
.
.
..
yI
p la.(j
-
d d
1 2
= =
cj
k 3
1)/2gW/2. .
.
y
.
l
z
-
-
'
.
.
. .
.
.
.'
. .
.
-
zE.8t(J)
-+
small
it &is. This where
Exercise 7.1. The point of this exercise is to apply the remspning of this sectipp 1et to r inftf : B3 / HI where # ((z,p) : z e Rd-l y > 0). Ff 0 E
.
.. . atE
.
.
1 lzl2Gz, jz zja -
.
y.
tert
-
zepsinjE
.,
meurabl
r z-/-,(s,)a /cs(z,,),(,)o
Cl
.
tuzded and
lt j i
(.1) Therein.
The derivation given abpve is a little unsatisfying, since it starts Fith the ad thn verifls it bt it is simpler thafl aid' With lflvir's ! ( . 100-103 in Port ad . . trsfoimations It Stone pages (seewhy (1978)). ls rh'. . th merit of explaining 1-p(z) is the probability density of exiti: g: ki is a nonnegative harmonic that has :(0) 1 and kyzj 0 when z e OD and z # y. z ansWr
lz
-
:z7
Cd lz
J(p) tzlzlxldzlj j
ztzl'mtdzl
is,
Gtz , yj
The Erst term 0 as z y, while the sup in the scond is shows that (7.2b) holds and completes the proof of (7.1). -+
'E
.
In the lmst section we considered how Bt leaves D (z : lzl< 1). In this section will investigte how it spends its time before it leaves. Let r inf (t : Bt / we of Chapter 3. That y) potential and be the kernel deined ih let Gz, (2.7) DJ .('
now we observe
ztzlltzjxtda)
f 2lIJIlxfo(J)
.
167
=
fy)
-+
Ball
q
for the Ball
.LL:i ..
.
0 and
-+
1. To prove that -
Times
-
/so(,) a(2)''r(d2)
Occupation, Times for
Secton 4.8
Equations
Gn4, p)
.
Gn, p)
=
=
p)x(dz)
-
.. .
.function
=
-+
roof
-+
=
Rd-1E
=
ptz,p) where Cd is chosen so that
1
zz
,u(z,,) J
(Iz
j' do :,:(0, 1) =
where
Cdy :1a 'F y ald/a
-
/
=
(b) Show flz, p)
=
(c) Show that if zu
=
J -->
do
k, pn
(d) Conclude that Ez,yjfBpj
H -+
0 =
We thin'k of Gnlz. ''''''''''
.
.
.
0)
1.
thenutza,
pn)
J do
ylfo,
ptz,
-+
'k
J, 0).
''
.
'G
.''#
''*
0).
=
.
.
.
.
=
m v. uosa;,
G(z, y)
z
.j
az
-
-
-
z + 1 G(1, yj
-
pl +
z+
z
pl + 1
k y and
(1 (j
z)(
-
.,;(j
Geometrically, if we 'Ilx p then z that it is linear on (0,p) and on GD(#, #) 1 #2. =
-
.
Jx
cases
u)
.
.
0 in H. (.2)
..
.
.
.
-Iz :z.c = -Iz
(1.7)to conclude Lu
(7.1).
with
at z spends at y expected amount of time
Considering tlle two
0 in H and use plz,
. .
&otz, yj
is bounded and contzuous.
(a) Show that Llta
Section 4.5
as the exoected amount of time a Brownian motion exiting G. To be precise, lf before c GD tn the starting spends exiting G is in X before Bt (z,yj dy. Our task in this section is to compute Go(w, p). In d = 1, lvhere D s (s1, 1), (8.1) tells us that '
1 and 1et
deltkz,yt-f,
(+)in
Combine
1
z
z
-
: 1
t7(-1, yj
!/) +
-
: -
(1
1
-
= -1
1
g
u
y z
gz
H
If 0
< y
1 from (a)iand recalling # 3, we lzave
Summing over i gives 1
'q'f
'j''p Multiplying by 2
we see
J''(r)
(b)
=
+
tl
gt
1
--
,
(lzI)+ zjzl T (171)
=
sltis:es
f
that
1
/(1-)
.
g-1
J'(r)
r
'
'
2p.f(r)
-
() for
=
.f'(1+)
r < 1
.
'
'
'
'
'' ' '' '
'
'
'
'
(ik) n is C
.
and hence
-(/(1)
'
'
'
to
F(r) is continuous
at r
1.
=
Exnmple
=
mlPtzz + m + 1)
m=0
arcslne
=
=
-
co (ar)2m c(c) I'---2 + 1)! (2r?z m=0 check
Pzlh
f'rj
=
Ca) -
Jr
r
J ?? (' .)
=
C(c)a r
2 ? .f''(?.)+ -J (r) r
s i,,l,tc,.) =
27 this matches
=
=
Clclc r
a
(1951)derivation
ra
sinhlcr)
Remark. The not depend on ahead does not value for which
dr
-
7-(1
its
'7r
-+
the otlzr
see
arcsine
#1y,x)(z)
-a
=
-
1 -Ar + 2
can now forget about we have
2 arcsintu)
-
-
(9.3) Lemma. Let c(z) C3, bolmded, aud satisqes
# 0.
coshtcr) =
ca
.f(r)
+
2C(c) Jr :,
sinhtar)
(4.2)in
law,
with a,#
Chapter 1. To
k 0. Suppos
'p
is
-1
cv
=
Then
origins
'p(z)
dt
=
e-Gff r
e-pll't
0
Poof Our ssumptions about imply that + c(z)'p(z)) dz in the sense of distribution. So two results from Chapter 2, the Meyer-rfanaka formula (11.4)and (11.7) imply 't?''(z)
sinhtcr)
of
reader should note that by scaling the distribution of S/t does t, so the fraction of time i (0,) that a Brownin gmbler is converge to 1/2 in probability as f x. Indeed r = 1/2 is the r) is smallest. the probability density 1/x r(1
For the last equality prove the irst we start with
for a11 z
=
r)
-
-2(1
'v
2C(a) -
27 and recallipg 1/2 thr solutior
cr
'
the result in
l(s 6 (0,t) : B k 0) Iklf 0 K 0 K 1 then
=
1 a g' 0
% 0t ) =
c(a) sinhlcr)
the solution we want, we that it works. Diferentiating
tRc)coshtcr)
=
sinhtc)
I::l
law. Let Ih
Proof
is one of the happy families pf Bessel flmctions. Letting c 1), we see that when d 3 and hene v r(z) = (z 1)r(z hms a simple form
and simply
c
9.3. Our third and llnal topic is Kac's
(9.2) Llvy's
Stwell
tfound''
Since
sinhtc)
C(c)c-1
-
.
=
Having
1
=
=
Facts (ii)-(iv)give us the information we need to solve for /' at lemst i p'rinciple: (b)is a second order ordinary differential equaticm, so if we specify J(0) = C and J/(0) 0, the latter from (i),there(x7).is a unique solution fc on (0, 1). Given Jc(1), (ii)gives us the solution on (1, We then pick C so that (iv) holds. known'' fact that the only soT0 Carry OU.t O1lr Plan, We begin With the ltltions Wlich tO2#) stay bounded near 0 have the form crl-td/zlf (ayalmy(r (b) where x (a/g)v+2m S(Z)
-
C'(c)c-1
-
-,1)
Solving gives C(c) = 1/ coshtc). Exercise 9.1 when z 0.
-
,
C(.(z)coshtc)
=
=
and the reaij we note that by applyink (b) together, 'p tie equatipns (a) with ad uking of (t.6) (Which rfers prf > the 0(0, 1 t q (iii) ) iu (5 Ca)) * tn cntlde '
Laplace Transforms, Arcsine Tzaw 173
=
1
$
j
.j
r(#t)
jo 'p/(.8,)
.?/(.1)
-
=
dBs
-
.
j (1
+
c(#,)r(#a))
ds
Section 4.9
Chapter 4 Partz'al Differen tial Equations
174
in c(f,) ds and using the integration by parts formula, (10.1) Letting q = exptct), of bounded which = locally and is = with Xt Chajter 2, X q variation, w have
JJ
It is a little tedious to
these
solve
'
,
W-#-#-W
W=
Jo
explcalz,/ll,l
'p(.s(,)
z?(f)exptct)
=
-
t ./r-
explc-lfl
(&
+
dB'
gives a solution.
c(s,)s(s,)Jds
What
+
jv '??(/,)
(9.4)
'D(0)
exptct) + fn'exptca) ds is a local martingale. is bounded. As f 0, lh x, explq) K e-' K marting>le and boppded cpnvergence theorems gives
boupded sincey isusing the 0, so
a
p(z)
EZMZ
=
=
ECM.
(9.3) tells us that
'pt)(z)
=
=
Cezs/*
1l(z) .
J
'tl*
-/'p
=
To flhd
1)
.
+ 1 we write
y'pl
=
'/S0
=
1
,?
-r1
2 1 -K0 u 2
'p
=
1?0
+
2(.+p)
Be= zjw+
-z
+
xl-t
where
rl(z)
This identity
O
.. 1/7
and
+
1
.+p
z >
'v
buded, ()
x < ()
&
C
1 1 = B+ a a + # B/(= 2(cf + p) -
so
up by
we have
-z.2/a
1
1
= a + p W'a = From
2/y
e
=
n
(9.3),(9.4),and
the
-
2
.
-
1 2
.
tfz
z. z--2
g
=
:-(E'+#)'
*
1 =
yy
(x,
j
e-tzt
t
,
t
-
Z'
0
s
dt ds
-#a
-
0
s(t
-
s4
ds dt
of Laplace transforms it follows that
uniqueness
again
CK' e-G(-')
W
o
Ez expt-p.Jftl
.t
p)
+
Fzbini's theorem imply
and invoking the uniqueness
and B we note that we want v to b Jl
171
+ 1
Picking the signs to keep
de -w
(x)
dt
= 'y
j.
with
z>0
To solve the equat ion and note that
so
to fid a bounded C1 function
we wnt
.-@f
d
-
.
.
(X)
) ds
exptc,
Er
=
ata
To go backwards from here to the answer written in changing variables z = 2y, dz = (1/2) 2.tjt dt to get
0
0 using and Fubini's theorem leads to the c(z) nosv Plugging in the dehnition of El formula given above.
we have
warm
1
=
p
+
-.+
-+
'
(9.2),we
asl-lTn
1
W+
=
explc,ldc,
vtz)
:. '
that
interested in is
we are
t
.'
check
to
easy
s=W-W-+-#
+ 7):/J
j
=
but it is
eqations
'p(#t)
'
Mt so and q
Laplace Transfoms, Arcsine fgaw 175
t =
t
-
'm n we
-7'
e s(t
-
s)
ds dt
have the desired result.
C1
jjd
' . .
5 Stochqstic '
' ..
.
'' :
-
.
.
.
.
DiFereptial
'
. ..
.
.
.
.
E .' .
.
(
..
' .
.. .. .. .
..
..
..' .
.E
.
.
.
. ..
Equations ''''''*.
'
'
.
.
.
.
-.
.. . .
.
..
'
.
..
(
..j( '
. .
.
. .
5.1. Exampls ln
Ethis
chapter
(i SDE's) that
we will be interested in splking stochastic E informlly ks are writtn
ilations
diseretial
'
dXs
(X,) ds + c(.X',) dBs
=
or more formally, as an integral
equation:
..'
j
j
(*)
Xt E
'.
: .
.
.. .
' .
' .
' .
unt.dm
1/(.X'a)dq +
Ab +
=
.
.
.
E .'
E
.: .
..
' ' .
l
:
'
;. ..
i
'' '
. . .
.
' ' . . . ..
... beginnipg of, Sectiop 5:2. We will give a prcise forpulatiop of @)at . the In @),Bt is an m dimensional Brownian motion, c is a x z?z matrix, and to make the matrix operation wcyk- out zight we will think of X, and B as column vectors. That is, they are matrices of size d x 1, d x 1, and r?z x 1 respectively. 'Writing out the coordinates @)says that for 1 K i % d .
. .
.
.
. .
.
.
.
.
.
.
.
m
Xgi = ,
xc +
biA) 0
ds + q
'E
J ajy
0
njxj)
dB a
the equation Note that the number m of Brownian motions used to cdrive'' generatibtto additial d. This ds not less than or more be may dilculty, is useful in some sttuations (seeExample 1.5), and will help us distinguishbetween cT which is d x d aud T which is m x m. Here C'T denotes the transpose. of the matrix c. To explain what we have in mind when we write down @)and why we want to solve it, we will now consider some examples. ln some of the later examples Tltls is cstomarily deflned to be process. we will use the term dxusion ttstrong with continuous paths.'' larkov this section, llowever, in process a 'any
'ase
'
Captez 5 Stochatic Difirential Equations
178
Section 5.1
the term will be used to mean tta family of solutions of the SDE, one for each starting point.'' (4.6)will show that under suitable conditions the farrly of solutions desnes a difusion process. rownlnn
Expnentlal Exnmpl where .X is a real number motion. Using lt's formula .1.
Xg
=
a'Y +
j
pX, ds +
a standard
j
'f
cXs dS, +
1
j
j
/2*,
. . .
:
.
.
zi
c() .
.
'
integrals,
't
(8.7)i
Chapter
j
Jkk(x,)ok(A',)
ds
0
k
aitA)
=
i j
z z
+
Substituting dXk J
=
Xia dXja +
,
0
k(x,)
ds +
0
J7nj ,.
i
.
..
zibiz)
we
get
+ zd/tz)
gives
j
t (.E'(.;v't.zq, i ) -dt
.
.
.j
Ex,
-
iexjj
j
,
.
odyytz)
yst)
justifying the pame lnflniteylmal covarlnnce. vaz-lanc and c(z) c(z) When d i= 1, we call c(z) the lnfnlteslmal stnndard the infniteslmal dviation. In Example 1.1, l(z) is o'z proportional to the stock price z. The ininitesimal drift (z) = (/I + (>2/2))z is also proportional to Note, however, that in addition to the drift p built in to the exponential there is a drift c4l2 that comes from the Brownian motion. More precisely it comes from the fact that e= is convex and hence exptcft) is subinartingale. a
)
Di lzl
j
Xj, dXi +
+ aijz)
f(-Y,) ds and digerentiating again =
.
.
.
+ zbjz)
f=0
Subtracting the lmst two equations
:
j =
zibiz)
=
.
....
.
(atla)jjotl
yt
'.'.p,
X3iXj
.
1.2k Th Bessel Ptoces.es. Eknmpl Let W'i be a d-dizensional Brownian motion with d :$. 1, and Xg = I#J$1.Diferentiatint gives
ds
0
by parts formula
Usipg the intgration
(x,) ds
.
.
then the formula for the covariance of two stochmstic 2, implies =
aij
0
'
J'0
.
't
=
'
.
(aY, Xjt
,
=
J(z)#T(z)
=
(x ) da
gives
g
.
i
.
+ E
txa)
drift. is alled th infniteslmal Because of the lmst equation, To giv the meanihg 6t the the cocient w hote that if
.X',J
.
d EXixil j
=
.
0
diferentiating
.
-+
=
0
179
values
Xsibj (.A-a ) ds + E
.
=
i d f(z) W.,;'.;L1=()
E
''-
Using Exji
t
i
+ (.
expectd
J z z + E
=
'f
&(z)
=
)
If aij is also continuous,
ds
= Exponential cz. so Xt solvej @)with (z) (p + (c2/2))zand Brownian motion is often used ms a simple model for stock prices because it stays nonnegative and Quctuations in the price are proportion>l to the price of the stock. Tp eyplpil the lwst phrme, p retprn to the geperal equation (A.) apd qupIf .X$q z then stopping at t ATa for slzitable P ose thatb and tr are contmuous. (x) we stopping times Ta 1 x, taking expected values in @)and letting n have t zi Exji ds + E () So if is continuous =
i
ftlja'i
Lt Xg Xt ekjtpt-l-ls) one dimensional Brownian
Mtlo.
Bt is
and
using the mssociative law, and taking
Examples
t.xiXjj t
(X,)dBi,
,
j. .
!
gzi
Dlzl
jzj
k.
=
jzj
-
z,)
jzj?
g ,
Zlzl . . .'
E
.
.'
'
..
y uu
=
. .
'
.
'
' ..
So using It's formula Xg
-
A'o
=
S
t I/vf '
#F, (, I I
dea +
t
1 -
2
()
d ) ds 1#F,l -
.
j.
jzj
Section 5.1
DiRrential Equations
Chapter 5 Stochutic
180
side, since Bt is a We will use Bt to denote the flrst term on the Tight-hand i.e., Bt is a Browniau motion by Lvy's theorem, local martingale and (#) (4.1) in Chapter 3. Changing notation now we have =
,
d 1 ds 2.Xa -
(1.2)
Xt
@)holds
so
X0
-
Bt +
=
(z) (d 1)/2lzIand
with
=
-
0
o'(z) .
Exnmple 1.3. The Ornstein Uhlenbeck sionalBrownian motion and consider
(1.3)
dX3
.
.
.
E
'
. .
:
'
Eetht be
.
:
.
.
.
.
. ..
' ' .
.
difnen-
a one
.
..
Xt
=
dBt
#
ea'
.X%+
dXg
(1.7)
d To
.
'tl)
=
e-q'''p
U1
=
i'
Xt dt
j
dt
1
-
.
n2
take -.y
Jtz,
'
-p/2
g)
=
z
Note that c is but we need the drift a unit vector tangent to the bz, y4, which is perpendicular to the circle and points inward, to keep ( X) from iying ofl'. always
circle
,
:
t
=
1
-
Xt df
=
-z/2
(z,p)
,
-.-XdBt
=
@)we
this. in the form
write
#fa
0 check To this informally, we note that diferentiating the flrst term in the proddt and diferentiating the second gives # dBt To check this uct gives -aXL formally, we use lt's formula with fu,
. ..
.
Exnmple 1.5; Brownlnn Motion on the Circle. Let Bt be a one dimen= sional Brownian motion, 1et Xg = cos Bt and X lin 1, Bt Since XI+ (.;% X) always stays on the unit circle. It's formuela implies that
-a
(1.4)
.
j
.
.
of the velocity of a Brownian particle which is which describes one component experiences slowed by friction (i.e., an acceleration timesits velocity aLI. where solution explicitly. This is one cse we can flnd the c-Gf
j
,
-ce.iLit +
=
Here V( is an Ornstein-uhlenbeck velocity process, and 5$ is an observation of the position process subject to observation error described by a d#Ft, where W' is a Brownian motion independent of B. The fundamental problem here is: how does one best estimate the present velocity from the observation (K : s K )? This is a problem of important practical signifcance but not One that we will treat here. See for exmple Roger and Williams (1987) 327-329, Wksendal Chapter VI. or p. (1992), .
Process.
181
,
1. E
Examples
Exnmple
...'
'
.
1.6.
Branding
Feller's
'
.
.
Disuslon.
This Xj
k
0
:
and satisfes
:
(1.8)
.
conclude
to
dXt
pXt dt + c
=
Xf dBt
To explain where this equation comes from, we recall that in a byanhing d idetically process, each individual in generation m hms a idepedent distibuted number of ofspring in generation m + 1. Consider a sequence of branching processes (Z;j r?z k t) in which th prbability of k children is p2 ,
(1.5) Theorpm. -Gf
ze
When A
and variance
0.2
=
Jf()
e-2Gr
the distribution dv. z,
of
X is norm>l
with
pan
Proof To se that the integral has a normal distribution with mean 0 and the e-atz indicated vari>nce we use Exercise 6.7 in Chapter 2. Adding natuk El n ow we have the desired result. =
Exnmple
(1.6)
1.4.
The Knlman-Bucy dl d/i
=
=
Filter.
Consider
(A1) the mean of pz is 1 + (A2) the
variance
of ,2 is
(A3) for any 6 > 0,
d#f
(#a/n)with pn
d
Ek>,s
J2a
with 2p2
Since the individuals in generation tributed number of children E (Z1nlZl
tdl'li
Pkdt + -cI4 dt +
,
and suppose
,2
-.+
p
-+
>
0
0.
-+
0 have an independent and identically disnz
=
var(z1nlg;
=
)
nz
nz
=
)
=
nz
1+
.
.
%2
p n
-
182
Section 5.2 It's
Chapter 5 Stochastic DiFerential Equations
So if we let X1
=
of
l
( j XlN/s
var
''=
X''( 0)
2)
-
zj
zJa2
=
lI, pages 176-191 for E.
.
'
5.2. It's
n
-1
=
183
'
z7a
'
E
(0)
XH
z)
others.
some the
-1'
(X1N/a
E
(1981),vol.
arise from genetics. See Karlin and Taylor
then hngtjln
Approach
Approach
.
n
(' :....*.
o?
anb
variance the The lmst two equations say that the inflnitesimal mean rescaled process X;t are z#s and zok2 These quantities obviously converge to those of th proces Xt in (1.8).In Exgmple 8.2 of Chapter 8, we. will shw that under (A1)-(A3) the processes (X1' t k 0J converge weakly to (.X t '0J.,
by giving some esqential defnitions and iptroducing a will kylin th eed fr sme of the fifnaliti. Shat W will coterekmpl uikiks rutt mai prctd exitence bsint the d to ou. nit' tor then entaikld SDE, which wws proved by K. It long before the subjt i brile of the general theory of processes. the complications To flnlly beome precisv, a solution to olr SDE @)iq a triple (.X, Bt where Xf and t are continuous processes adapted t a flltratin so that 7
.
setion ..
.
''
. .
'
. .. .
,
,
We begin this
.....
''''''
. .
.
.
.
.
'
. ' . .
. .
.
.
.
.,&')
,
.-&'
Exnmple
Dlffusion.
Wright-Fisher
1.7.
This Xt E
(0,lj
and satisfies
(i) Bt is a Bqowpiyn mption with respect to &, i.e., for e>ch s amd ff-j.a Bt of and is normally dtstribvted with men 0 nd variarp: s. is indpent -
dXg = (-a.X
+
7(1
,
u'V(1 aX) dBt
.;V))dt +
-
.&'
-
The motivation for this model comes from genetics, but to avoid the detail of that subject we will suppose instead we have an urn with n balks iu it that with may be labelled .A or c. To build up the urn at time m + 1 we sample aln with probability replacement from the urn at time r?z. However, we ignore the draw and place an c ilt, and with probability #/n we ignore the draw and tb mut>tions. ink 1I genetics terrns the lmst two events correspond plac ap pf when the the number of /'s in the urn at time m. Now Let Z;b on a given trial is .A's in the urn at time is z, the probability of drawing an ..
'
-
,fraction
.
-be
.
.
.4
. ... ( .
ps Sine
we are
sampling
z
=
.
( ) 1
& + (1
z)
-
.
g. . . . . .
('
j'
)'
@) ..
.
E
E
,
,
.
'' ' E .
' .. .
We will integral
(
.
A
,
''
' .
:
.
'
..
E'.'
..
#
' ' '' .
' . ..
' ' ' . .
:
' '' .. . . .
' ' . (.. .. . . . .
.
l
. .'
.
tr
0 ..
.
.
'
'
'' :
.
.
=
=
nz ) = npn nz) = nyatl
' ' 'i .
:
'i' . .
.
and
(.X'I'/s z var
j
=
(Xy/aj Xn(0)
=
z)
=
=
Exnmple 1et
2.1.
and
t-(>z+ #41 -
pa
'. . . . ..
i
.
,
' '.
(
.
.
'
.
.
.
,..
, ' . . .
'
.
.
(..
' ' 'i ... . .(' . : .
' '' . .
'
' ..
.
. .
.
'
g
' ' ' : . . .' ' . ': : . . .: . .. . . .. :
'
i .
' .
.' .
r ' .
and ldcall# bouzded so the
'
.
g
E'
.
'
'
(
,
. ,
.
XtB
.
,is
..=
p,,(1
j ' .' .
'
.
it will hold for h It is emsy to see that if the SDE holds for some adapted #f). When Xg is the flltration generatd by Xg to FF the filtration = generated by the Brownian motion, then we can take h an d X g is called solqtlon. could satisfy (..) witholtt strong dicult imagine how It Xj to a B eem' qdapted pl b th is famws dtle H tp h tp , T>np> showing ut re a being that this can happen. i y E
y.
-y;
z))
.
vyj
1
E..
.
y yj
,
y
q
,
0
=
:
:
sgntNa
=
0
'
) d 0,
J(z)
d
3
.
The baslc
estlmate
which
(2.3) Lemma. Let r be a stopping B (4td + 16d2)ff 2 Then =
is the key to the proof is: time
with
respect to
&,
let T
:)
and
1et
.
-sgnl)
EL
-1.
'
( j 't sgn(-N,)d#, .
=
-
j
z
sgntI#,l dB, + 2
!
j
jo
1fyv.=o) d,
E
0
1(e.=p) dBs
Turning to positive results:
0
jo
IF,
-
h l2ds
*
.
2s
(c
So the left-hnd
side of
s
(2.3)is
.
t
E
A
7012 + BE
-
The inei uality is trivial if the right-hand side is infihite each term on the right is fnite. To begin we recall that that suppose 2a2 + 22 Using this twice 2a2 + 2( + c)2 2c2 + 42 + 4ci + + c)
'
=
Z 12 S 2E IFn
E
=
2
-
Proof
To prove that the second integral is (andhence the rightzhand side is -W)) recall that #Fk is a Brownian motiou. Example 3.1 in Chaptrr 2 impls that (s : W'a= 0J hms measure 0, and hence f
If
rcssystpv'az
1(11:.::z:)ds
=
E 1(sK t I'F, = :
0JI1:z
K 2.E'1F:
1:1
-
(a)
A
j
12+ 4E nssuyspz
j
+ 4E cssuyspz
:Ar
tK )
-
tA r
bzs
''
2
c(X,) 2
) ds
-
J(Z,) dB'
so we
n
(c+ )2%
5 Stochastic Dirential
Chapte
186
To bound the third term in k lmplies that (
(a),we
(K)
E
:
:
biz't
ds
S
((K)
0
()Sj KT
ontinuity
with the Lischitz ' 4E .
.
.
'
l 'r. . ..... .
(c)
': .
:
.
the
0
)'
' E
(K)
J7
.
fr 1
y To bound the
second
(y(F,)
.
rrpr
,,
j
s
(a),let
term in
'
..
i
Let
.l .. '' . .
=
X
=
70n
IAC
SIIP
-
0 K, K 11
''-
12
XH-
,!
(2.3)implies
that for n k 1
T
-
(
X j tjs
Aa(s) ds
%'B
0
.
1
1
,
,
E
i
1.X-,1 xa0j2
.
. . ..
:
bzs) ds
.-t
YO
artd observe that since
142(t)
sup 0 K ST
A i'
)-l
S 4T E (
E! .:
of approxlmatlons.
sequence
Zr&()
2
4E
s
..
.
of
187
lds
.
give
assmption
tar
sup
y
m
jj '
Convergence
ar
f(Z,))2 ds
-
d
l'??()I2
0KaK
.q
t
,
1t
and that the rightllkand
side
sup
0K,K1
Io.lzlsl
hms flnite xpected value
we
have
tAT
Mt
(c.(#-,) Jf(,))
=
0
an
d let
for the
i maxtmal ra = inttt : lMtl k n) A r. variance of a stochmstic integral imply that
X
y
z
(11)
SIIP
OKKTAJ',X
z
i ; 45 (W/ara
y
)
-
k
1(f) S Ct +
inequality
'he
E
dBs
.
-
>nd
hy
jgtu)
4s 0
ttk formula
Combining this with
)j2ds
(2.5)
y
yk (a
-
(2.4)and Ar,(t)
f2)
using inductior we have 2@N+1
tn
< Bn-c
+q
V
(n+
1)!
:'
Letting
n
then uqipg
x,
-+
.
'
E
l
LE sup
j
.
nszsc-
Ar
(b) >pd E
E
(d)
d
nssc-
s
16s
'
..
'
'
4f a litnit. Cieb#sev's .
. .
emsily get the existehc
'
'
%22ns(T)
2rn
side is summable,
ipquality
so
the Borel-cautelli gives
.&n-11
c.lzallz
l(n('i&)
0
(2.5)we
2
Tar
J
:
.
P
(tn(K) -
=1
'
Flom S lk OWS that
have
f Ar
J'l sup =1
w
2
c(g,))
-
d
K 4f
onthpity
( . .' . . .
.
(c()
0
Lipschitr
IK Zs 12ds -
P
ds
Since
Ea 2-n
uniformly on
2-n i o.
it follows that with probability
=
1,
0
A'C
-->
a limit
XT
DiFerential Equations
Chaptez 5 Stochatic
188
We begin by showing that
The llmlt is a solutln. occurs uniformly in L2
The last contradiction implies that the desired result follows.
also
convergence
.
al1 0
(2.6)Lemma. For E
Km
< n
-
It's
Approach
E)cP,
E
> 0 is arbitrary,
vt) K (X +
2
k=m+1
,
,
,
=
.
a k (r)1/a
J7
/
so IZI
,
a
but
189
To prove pathwise uniqueness now, let (X Bt Si) and Zt Bt FI) be two solutions of @) Replacing the hi by Ft h v h2 we can suppose h .,V. To prepre for developments in the next section, we will use a more general formulation than we need now.
x,
X,n l2
1x,m
sup
OKaST
S
Section 5.2
=
g'
Ilg112 EZ24l2. If n < x, then it follows from monotonicity expectedvalue and the tripngle ipequality fpr the qorm 11llathat
Proof
Let
=
azld Zt be continuos (2 8) Lemma. Let to Ft with processes adapted and : inftt r(J4 y = Zz 1et > Suppose J. that Bt is z IMlor lZI and Zt satify @)on (0,r(JMj.Then Brownian motion w.r.t. Sg and that X Zg on (0, 'i'i
of
=
.
.
n
p
jo
(2.3)implies
and Z3 are solutions,
zA,r(a)
w() S
BE
j j, llr
XT 12
s
0
-+
so (2.7)holds
./
1F,
E
E
.Er
JJe
..
ds for .
E. .
(
.
..
. .
with .A = 0 and it follows that p '.. . :
.
''''Temporal t''''''''''''
coeciets
InhomoMenelty. .
.. . .
i .
Ip. some t dpend n tin, :.
.
d
.
!
.
#(s) d
+
s
j' e(s) ,
'
'
.
.
.
j vs,
ds
0.
H '
.
t
E
tdf
applic>tions
it is important to
. ''''''' . ' .
allow
' .
the
that is, t6 copsider 't
(p
.,4
s' p
,
y
t
.
d
-4
.
sp
E
The last result implirs th>t twp solutipns must agree until the flrst time thi implies that the two t lt'e ba 11o f ittius R us one them leaves exit at must the same time.) Since solutions are by deflnition continsolutions X,d we in us t hav' '(J4 1 x as R 1 cxj and the uous functions from (0,x) desired result follows.
yl1( k 0
,L
.
% 12ds
-
zrli
-
of
. .
.
X
j
are continuous, w(t)k #(t)J Since # = sihc impossible #('r), but this is =
:
. .
.
(2.3)to get
in
'
and
.
4(z) z +
E
12ds -
vt) % .A + Xcpl. () % ..
4(t) =
.' . . '
Since
.
(z4+ eleft then
=
X*J
0s,sg'
Spppqs
.
.
-
J
The key to the proqf is
Grppwall'j Inequallty. Tlin is continus.
Zt
and
A'J
:::q
-
(lststr
.
.
Prof
g
'
'
.
.
:
,
that XT is a splution, we let $
Uniqueness. ....
:
. .
+: &.' cxn and using Fatou's lemma proves the result for n
-co by (2.6),so Xt
d
.
.
, .
=m+1
v=-=-
=m+1
K BT E
7) (2 . .r
:
g
,
'
'i
=
-
.
'
?
n
T1 ''''!
'
'r4.)
11 57 tu, x-krllllaa,(z')1'2 Il--su-p--lxrk
11 cssu,spylxam xsnI s z -
=
ds
it follows that if
w(t)
(*8
Xz
=
an
+
0
bs, .X,) ds +
In many cmses it is straightforward example, by repeating the argument can show
9
s, X,) dB.
to generalize proofs from abpve
with
@)to (*.). For
some minor modi:cations
one
190
Section 5.3
Chapter 5 Stochutic Df/l-erenfal Equations
(2.9) Theorem. If for a1l fort : r, l(qltf,0)IK Jf, Ic'J(f, z)
-
z, g, and
=
< cxn there
T Ikt
If(f'0)I
tanabe inequality implies aj diagonal
ap
tfyi/d Setting c = 1/p and tn that explodes at time 1/c
Rearranging gives .P(T,z
=
=
-
n
%Ts
=
w(z)k E
193
XL Suppose a' = 0k > 0 so t is strictly increasing. Let Ta inftt : Xg n). While Xt 6 (n 1, n) its velocity bxj ) k nl so T,z Ta-z S n-J. Letting n-J Tx = limTa and summing we hav Tx % EQI < x if 6 > 1. proof since orfly We like the last depends on the msymptotic behavior it xplicitly, which hms the of the oecints. One, can alsp solve the equatipn ot explosion giving exact the time. Let 5$ 1 + kt. tfi'k = %?dt so advnte ctl-r = then yk if we guess (1 -
limnx Ta To complete the proof and X will be a solution of @)for t < Tx + (x7 a.. If Artl z then using the optional to show that T1 now it suces stopping theorem at time t h Ta which is valid since the local supermartingale is bounded before that time, we have (e-A(AT-)w(x,azk))
Extennions
Ja-l
lzl)J
with J > 1
Ja
p-2(u)du
,
=
n
of solutions.
That is
approximations
of
Equations
5 Stochastic Dfsrezlta.l
Capter
194
Let 0 % /rz4u)K 2p-2(u)/n be an d
function
a continuous
#n(u) d
For I in have
(a) we
1
=
#n integrates
of
=
yu(-z),
to 2 ove:
c-l)
(cs,
(d)
JJ#a(u) du, for z
=
= 0 for IzIK an ; 1 for ca < 1zI< ca-l 1 for cn-l s lzI
>
0,
t
,(xa2))
-
.
' . .
yo
dBs +
-(at) j ' w'-(a,)da, -
+
j
jo
l'la I s
martingale
A(t) is a local
so
)j w:(ak)
pn
lzl E (ar,,
-
(a) we
lf2(f )I S
j
.j, :
.
'' .
e lc
-1
.
(z)dz
=
whrr tx7
..
'
. .
for
.
f
is
any
'
.'
6
>
.'
.
.
'
. E'
x(#(f)) with g(0) Proof Let g) be the solution of p?(t) e. It follows gt) proof of show from the that J(i) S that g is small when 6 is small To (2.7) note that q is strictly ipcrewsipg and 1et lt be its inverse. Since g(()) z, we ' '(()) H'1/x(p((z)) 1/x() nd hence if c < (z) 1/ g h =
=
.(xa2))a
ds
'c
??
:::':
:p-2(ja,j) ,?z
A
g:
'
()
-
#(c)
gi
1/x(z) dz
=
r
u
ln words the right-hand side gives the ainouht time it takes g to climb from Taking c = t' and b = 6 > 0, tlte assumed divergence of the integral c to ipplies th>t the amoupt of tipe to climb to & approaches (x, as .7 0 and the E::l desird result follows. .
there are stopping times Tm 1 tx
t
a slight improvement of Gronwall's inequality.
.
+ f2(t)
=
with
-.+
0
note that #?(z) = n and is 0 otherwise. so usmg (i)
in
ds
,.t:
=
,
''hsjfcxs)
E Alf A Tm) .12
d(a),
1 lzI)we have
/, (.sIa,l)
j'olx(J(s)) ds
(3.4)tepma. Suppose Jt) increasing for z > 0, an has .
tvfll.ks -w--(a-)t(-'t.l)
(b)
To inish the proof now we prve
'
and using the
(xn
-
> A(t) + .I()
with cn-l)
(-Y,i)) ds
-
-
0
To deal
ttat)
ds
-
-+
= .X2 z driven t)
=
dB' /a(A,)f&(-V)&(.X'.7)J
+/+ 1/'
ds (-Y.7)I
-
-+
:
=
j
.E'I(-Y))
(x) and using Fatpu's lemma, then letting Letting m n monotone convergence theorem (recall pn (z) k 0 and yu(z)
,
folrnula gives
.
of x we
(a)-(d)we have
-
so It's
195
so
'
.
j
concavity
the
' xx(la,l) s x(sIa,I) /' /s
Combining
and it follows that patz) 1 Izlas p 1 x. SuPp ose now that 71t and X2 Ay tskospttitions wlth X1() by the same Brownia inotion, ad 1et At = X1 Xt2.
a,
(ii)and
so using
1,
' g
dz *uzt
w's(z)
and
Iy$(z)Iis
(c(x))
I#a(z)IS
ds
dp
wa(z)
=
observe
.E'I.n()I s
E
xi
=
.
Extensions
Ja-l
Since the upper bound in the defnition uch a function e sts. jae (
t/u(z)
(cw,ca-1) .
Ga
By deqnition
in
with support
Section 5.3
4a(lzl) K 2p-2(Izl)/nwhe .E
..j.
P z (Iy
s
j; vgs
?z
c. Exnmples ..
' .(
' ' .
.
.
.
.
the comWe will now give examples to show that for bounded coecients bination of (2.2)and (3.3)is fairly sharp. In the llrst cmse the counterexamples will have to wait until Section 5.6.
Dilrential
Chapter 5 Stochatic
196
Exnmple that
(z)
Consider
3.4.
Equations
(z) lzl;A 1. (2.2)and (3.3)imply
0 and
=
=
k 1/2
holds for
pathwise uniqueness
Secton 5.4
in d = 1 in d > 1
Jk 1
Example 6.1 will show pathwise
Weak Solutions
197
the Browniau motion the solution at the same time. In signal processing applications one must deal with the noisy signal that one is given, so strong solutions are required. However, if the. aim is to construt diflksion processes then there is nothing wrong with a weak solution. Now if we do not insist on staying on the original space (f),F, P) then we the map LJ (X @),Bt (u?)) use to move from the original space (Q, F) can to (C x C, C x C) where we can use the fltration generated by the cordiaates to1(s), 2(s), s K t. Thus a weak solution is completely speciqed by giving the joint distribution (.X Bt ). Reoecting our interest in constructing the process Xj and sneaking in some notation we will need in a minute, we say that there is uniqueness in distrlbution if whenever (X, B) and X', are solutions of SDE(, tr) with Ab i.e., PX e a1.) PLX' E x1.) XL z then X and X' have the same distribution, tr) for all E C. Here a solution of SbE(, is something that satises @)in the sense deqned in Section 5.2. .and
-+
faits if
uniquenes
(s
1/2
0 and Jv have created a secoud solption starting at 0. Once construt inflnitely many. Let % 0 d let we hve two olutions we can .' .
'
'.
..
'
' '
..
..
Xa tk a weakening of LlpsThe main reson for interest in (t ) ls th ..- . .. ... .. . . . act als condition continuity att inyrovemnt, is chitz .of However, . . . . ..the . . . . on counterexamples. Show and further sharpens the line between Georems (a) holds when C > 0 and in t.hat
xerciso
.2.
,r
-
..
.
.
.,
.
.
.
(ii) (3.3)
s(z)
=
(b)cflsider
4,(p) where
t
C'1OZt1/')
fO'
c(e-2+ z) oi
'
z
with p > 0 atd g f) = expt-l/p) pp(log(1/p))(p+1)/p. =
S k
d-2
show
''
'' .
.
.
'
.
(4.1) Theorem. tribution.
that g'tt)
i:
(#(tj)
.(,
,
.
If pathwise uniqueness
holds then there is uniqueness
suppose(x1,s1) aud (xi, sc) are two solutions wit x x z. proof remarked above the solutions are spciqed by giving the joint distributiqns (A =
,
#)
.
) Since CC c) is a complete
seprabl
,
conditionaldistributions Qi, in Durrett (1995)) so that
.)
Pxi
deflned for .,1.1#i)
e
=
fhetri space, we can flrid regular : C and e c (seeSection 4.1 .4
Qf(f,.)
Let Po be the memsure on C, C) for the standard a measure k on (C3, C3) by g'(.c x
.1
x .Aa)
=
jgo
ciii
(.d
Brownian motion
.
Intuitively, a strong solution corresponds to solking the SDE fo: a given Brow. weak solution we are allowed to constrttct producing nian motion, while in a
in dis-
Remark. Pathwzse uniqueness qlso implies that every solutiop i.s a stropg lution. and the conlpsion However ij pot that proof is more complicated so important for our purposes so we refer the reader to Revuz and Yor (199t), p. 340-341.
(Xi
c-2
5.4. Weak Solutlons .
Exnmple 4.1. Since sgntzlz 1, any solution to the quation in xafn/le 2.1 is a local martingale with (X) H f. So Lvy's theorem, (4.1)in Chapter ( 3, implies that Xt is a Brownian rriotion. The last result asserts that uniqueness in distribution holds, but (2.1)sllows that pathwse uniqueness fails. The shows that the pther next result, also due to Yamada and Watanabe (1971), implication is trne. =
'''''''''.
d.&@c)Q1(tJc,
.1)Qa(tJc,.A2)
and
define
If we let y/@n,
2)
1,
and a diagonal matrix A with diagonal entries c UVKU. a is invertible if and only if a > 0.
then it is clear that
tJf(t)
=
.!.
(yl
-
,
,
y0)
(x2 Blj
aud
(y7jy0)
-
,
kE
a
.
.
a
.
k 0
so
.
=
:.::z
cvT(z)
jz 1
2
.y
#=
x
=
=
= Take C or .A1 = C respectively in the deqnition. Thus, we have two solutions of the equation driven by the same Brownian motion. Pathwise uniquenerss implies F 1 '.F2 w ith p robability one. (i)follows since
X-1
2
and recall from Section 5.1 thaj
(:1#12 and (4.3) Theorem. If oTazjo k llc(z)al1c(p)ll S Clz c1/2(y)jj (C72al/2)jz pj for z, y. then Ilc1/2(z) s
cf(A-,) ds
=
We say that X is a solution to the solves MP(, c), if for each f
-
ar tcal martingales. :
bixs,
Xgixji
and
ds
:
pk'
. :
t -
0
0
(
.. .
bl
(
'
j
.
b
m foi
..
nd
or
,
(xa)ds .
.
f
.
.
..
.'
'
0
' y'
'
To
' . .
.
.
To be precise in the desnition above we need to specify a flltyation .J for the nlf iptersted lpcal rrtaxtlnkales. ln posing the martiugale ptoblem we a the undetlyig geherality of without that loss the proces Xt so we n assume = and generted by Xt the with flltratitm is the Xt@) tlf one spie is C, C) o'1$ and foi above bmsic data the b are In t he formulation a t fflarobtaiping with of problem tr fropz c. the problem. This presents us tingale pypiti of intoduing solve by thi problem we begi Tt? sqme !: jizlce i j j y aij i: at tht symmtric, if l, is, is z a;i (# X )t (aY X )f, a '
tn
'
'
/
.
see why we need a
lzl. In this
.
.
..
.
:
E
.. .
.
.
.
.
:.
.
'
.
4: gt ar ipsc j ( r
case
J(z)
e c2 to
=
'
2
clpl
pj for all z, y.
-
-
. . . .
.afctzlpl
then llc1/2(z) c1/2(p)j1 < d(2C')1/2jz
. . . . . .
.
c./(A,) ds
=
g'
Ip
mp.x lKtKd
j
. .
.
to
.
suppose more smoothness
Suppose aij E (72 ad
!
equiklent E
When a can degenerate we need Contilmtms Sqlare yoot. (4.4) Theorem.
(
. . ... . ..
.
.
:
E
it
The second conditioh i, of coure, (.X', Xiit
-
-
martlngal and j,
simplyX
Xt
'
p1 for all z,p
-
-
(.X'..;'Jf
that
E
Uwv-h, UVhU, so if we want c = co'r we can take tr (4.2 tells us a entries matrix with where A is the diagonal (4.2)tells us how to flnd usual algorithms of The for fnding U are such that root the square a given c. J(z) c(z) will measurable resulting apply them then the if we square xoot to a c(z) sophistiation smooth with measurable. start takes also. be lt to more and construct a smooth l(z). As in the cmse of (4.2),we will content ourselves to just state the result. For detailed proofs of (4.3)and (4.4)see Stroock and Varadhan (1979), Section 5.2.
.42
Let c(z)
k
t
199
=
CX1 l1)
.,
Weak Solutions
Section 5.4
198 Capter 5 Stochastic DiEerential Equations
alz
upschitz
get a
1
lzl/2 is Lipschitz continuous
d = 1 and a(z) = only if k 2. and >t 0 if
consider
'.
,
Returning to probability theory, our flrst step is to make the connection between solutions to the martingale problem and solutions of the SDE.
.
=
E
E
,
o
zia . z g .
i
J' ;
So c is nonneqative useful information.
z i aj k c.J, k z J
-
.
-
-
zi tzk k
--
,
k
i l 2-l
defnite
and linear algebra
(4.2) Lemma. For any symmetric orthogonal matrix U (i.e.,its an
,
square
enlarge-
.
.
,
,
(4.5.)Theorem. Let X be a solution to MP(, c) and c a measurable root of c. Then there is a Brownian motion Bt possibly defned on anJ). ment of the original probability space, so that (X, B4 solves SDE(,
E
> 0
E
Proof lf and let
o.
is invertible at
X this is easy. Let 'Ftf
each
-
i
provides t!q with the following
(a)
Bf
=
)'-) 0 J
From the last two deflnitions and the nonnegative deflnite matzix c, we can llnd vectors of length 1) rows are perpendicular
t
oj-
t
1
0
xi:
mssociative
=
J$
'n
-
=
jt() i (x ) ds
law it is immediate that '1
,) c(X dB ,
-
(x,) dni
'f
(b)
=
X,
-
A-a
bvn4ds
-
0
,
.5j
are local martingales with (.&, Bijt = in Chapter 3 implies that B is a Brownian motion. To deal with the genral case in one dimension, we let
so @)holds. The
z'
jt so
=
1
(todeflne thi.
may
-
f =
'
we
,
o o.x
,
)
J dpl/
dy + *
'
of qo(4.5) shows that there is a 1-1 correspondence between distribttipns ..,,.. . . problmj Whiclk b# definitipn lutions of the SDE and solutions of the martigale ar distributiohs on (C, C). Thus there is uniqueness i distribution for th SD2 solutton. if and tmly it the maa-tlngau problezfz hs a unique our fal topic is an important remson to be interested in uniqueness. . .
..
'
.r
'
Our result for one dimension implies that the Z,i may be realized s stochstic integrals with respect tp independent Brownian motiops. Tracing b>ck through the dennitions gives thedesire representation. Fo) mre details se Ikvcta and W>tanabe (1981), p. 89-91, Revuz and Yof (1991), p. 190-191, or Karatzl ind (ZI Shreve (1991), p. 315-317. ;'
=
B
201
.;,'
I., let I/'/a be an independept Brownian motion need fo eplarge the probability space) and let
let Ja
Exercise 4.1
#0
1 if J(X,) 0 otherwise
(
=
Wak Solutions
Section 5.4
200 Chapter 5 stochaticDiEerential Equations
'
() t
. Bt The two integrals are well deqned since thelr . vrinc. processe ae . K martingale with is Chapter implies that Bt 8 in 3 loal so a is a (4.1) (#) moti. Biwhin To extract the SDE now, we observe that the associative 1W ad th fact H 0 imply that Jsxjj .
. .
.
''' ..
:
''
. .
,,k.
.
:
.
'
.
.
. .
.
' ''
. .
are locally bounded functions and MP(, c) Markov property holds.
(4.6) Theorem. Suppose a and has a unique solution. Then the
strong
=
E
't
.1
(r(.X',) dB, 0
f, dF,
=
= In view of (b)the proof tltis Fe note that
>;
be
will
t
X
jo
tfyk
=
E
r.t
L JF,
-
when
complte
2
Js
'o
-
,z,2
jo
=
zt
(zf ,
zJ)
't
kt
H
0. To do
jo
ax,
) ds
=
Urxs)
=
1 if
=
orthogonal
matrices
dF,
=
o
01. 1.&)
con-
Exl'jv
=
.
() To begin to turn the lmqt paragraph into a proof, fix a starting poit will follow thq stand>4d complicated simplify rather argument, this To we za. practice (seeStroock and Varadhan (1979), p. 145-146, Karatzs ad Shrk and (1991), p. 321-322, or Rogers Williams (1987), p. 162-163) of giving the details only for the case in which the coecients are bounded and hence
.#'tii
'. '' '
'''''
=
x
't
and
0
0
xi-x.i
aij
-
0
we t
l
ds
=
0
have
htnixs) ds
. .
.
..
(a'c) da
are martingales. We have added an extra index in the frst cmse so we can refer to al1 the martingales at once by saying ttthe Mi be a yegular condiConsider a bounded stopping time T and let Q(Y, FT. given 0) distribption (.X+f want fo# We to claifit that t k tional ''
otherwise,
.r
(x,) ds
-
.)
j
.
t =
.
0 =
E (F
Mfo
'
0
.7)
.
(I(y)a
,z,2
Ui,kxsjak,txsthljx'a,
=
0
hj
JJJa dF,
we show
To h>ndle higher dimensions we 1et J(z) be the structed in (4.2)and 1et
lntroducing Kronecker's
(t
t
E
a'
=+
0
.
Propf For each z E Rd let Pr be the probabiltty peasure on (C, C) that gives 4h distribution of th uniquewsolytion. starting from z. If you talk fmst distributioh of the proof is easy. If T is a stoppmg time then the conditional (.X+,, s k 0) given Fr is a solution of the martinjale problem starting from Xr and so by uniquness it mudt be Pxv). Thus, lf we 1et 01, be the rndom R we shift deined ln Section 1.3, then for any bounded neasurable F : C property the Markov strong ave
,
(4.8) Lemma. For .P.0 a.e.
f.J,
the
all
i.i M.l.
-
o/.l r
=uij
t
o
ov
Capter
202
5 Stochastic DiRrential
Equations
Section 5.5
Change
of
Meuure
203
.).
are martingales under Q@, To prove thi we consider rational s < t and B stbppig theorem implie that if .A 6 Fr thh .
(((.6 i
Ezz
-
Mii 1s) ) ,
'
e
Since (F): < .Mt, (3.7)in Chapter 3 implies that invoking(12.3)in Chapter 2 we have deqned Q.
'
Ai
01' 1x)
j
o
''
that tlie ptitmal
and note
.'&
=
i' l
0
=
J
tef
1
-
'
'
.
.
(
'.
'
'
.
.
'
..
. .
:
.
'
...
.'
. .
'
E
.
..
: . .
.
.
.q .
.
(1
o
g. .
So by the fovmula fp: the
ch ange
o f Measure
)t
since
'
cls, z)
j .zl.t
Jcp(#,)ds, let
-
cts,
)
uj
5$ .which exists
(z) (s,z)
and
let
o
x,)
.
d
t -
aa
-
t
cfls,
=
0 t
=
uner q.
ij t
csacs,
X4) ds
=
0
0
Zc-lts,
z) by our assumption that I c l % M. Letting tions of Q nd P to & we deine our Itadon-Nikodym
ts,
.
d% dPt
=
a
=
exp
yk
-
1 .j(J')t
d (a, i ),
A'jl ds
% ML
Pt be th restrir deiivttv t be
Qt ad
.
'.
'
'
'
'
'
.
.
=
0
'
ce, jtsj Xa) ds
f
.x-
j ) (s , X a ) d s
=
()
0
t
E
=
X1
-
0
1
xJ),
pj (x,) +
bjs,
.;Ll ds
but the other half is
easy.
We note
j
aijxs)
=
and (12.6)in Chapter 2 implies that the
.;L) d(X, X),
Xslcjs,
1
J (s,X,) ds
(.x
since
(F)t
,.#)a
(
T
is a local martingale/o. This is half of the desired conlusion that under P
.
cts,
=
a
:
d(-Y'
(c a)j (sk.Xal ds
t
Tt
E
vn
integrals
Using Gixsanov's foxmula now, it follows that
,
y
0
0
=
ts,
s
=
.;@)
Let
of stochastic
'
z). Ij. follows that
c-1(z)(s,
=
'heorem.
Piof
0
=
(5.1) Suppose Xt is a solption of MPI#,), which, for ioncfeteness withut of and loss generality, we suppose is deflned on the ianonical prbability td. Suppose c-1(z) space (C, C,P) with & the fltration generated by lTa-lts, z)l z) and memsurable and exists for-all z, that is has % M. We probability e4uivalent to P, so that under Q, measure Q locall# can defne a Xt is a solution to MP(# + c). u'kj
*
()
cea cts,xa)
=
t
E
i cea ct s , Xs ) dX-,
=
't
(a,.X'
Our next step is to use Girsanov's formula from Section 2.12 to solve martingale problems or zore precisely to change the drift in a' existin: soltiok T accordate Example 5.2 below we must consider the cmse in which tv a'ddd t: t drift b depends on time ms well ms space.
.
covyxiauce
'' .
'
5.5.
j
ce, dFa
=
.q.,
.
aud
)
.#,
PTOOf.
is a martingale,
To apply Girsanov's formula ((12.4) from Chapter 2), we have to compute a-1 and dte X),. It's formula th asspiative law imply that 0 a '
Letting B ruh through a counta le collection that i closed under intersectiop it fotlows that .J'., the ekyected vale of tM!fl r.- .#.f,U)t..s zd genertds whlt inflies 0, ul4enss Vsing now gives (4.t) and tofnflt.kl.. tr (4.8). i
tn
ds
cov>riance of Xi and Xi is the same '
,
n
Exerclsp 5.1. To check yom understanding of the formulms, consider the cape in Fhich b dpeq nqt depend on s, start with the Q cozstruted abpye, change to a measure R to remove the added drift and shop that dRtldot = l/m so R = P. ,
' .
'
.
Exnmple 5.1. Suppose Xt is a solution of MP(0, f ), i.e., a standard Brownian motion arzd consider the special situation in which (z) = VJ(z). In this cmse
204 Capter 5 Stochastic Dirential recallingthe deflnition UX
) we
have
,1
.
K
j
=
(5.1)and
of
X in the proof
of
'
. TUCXS)
X'
.
=
Section 5.5
Equations
V(Xz)
l' ..
. '.
U'(X)
-
dPt
exp
l .
jl
k
.
de ldp, 2
.
AV(X,) ds
(g(m) )/' -
hux-,
+
n
In the special case of the Ornstein-uhlenbeck is a positive real number, we have J(z) expressionabpve pimpliqes to
proess,
-a1l2
=
d% dfl = In the trivial and Luzj
=
t 2
I
-al-'
exp
dPt =
It is comforting sa tisfy
to note that
Qt(.;% =
p)
should
=
be since under
(5.1) deals with
da t
(z)
-
p
=
1
.
p Xz
exp
.
p A .
-
.
when
A
=
exp
Jz y .
-
t:e
the
and
-2da,
=
-
-
p z .
Q, Xt has the
-
same
:r
=
d
we
.
have Uz)
p
=
+
between solutions to MP(,
c) and
MP(0,
c).
.z
Remark. Combining (5.3)with (3.1)and (4.1)w see that if in addition to the cT where o' is locally Lipschitz, then MP(, c) conditions in (5.3)we have c upique solution. using hms a By (3.3)instead of (3.1)in the prvious sentehce result in one dimension. we can gt a better
=
=
y4
:::i
jjz/2j)
is
Proof Let Z and a be as in the proof of (5.1)and 1et Ts inft : IARl > n). From the proof of (5.1)it follows that cttt A Ta) is a mrtingale. So if we 1et cvt A T,z) then under hn be the restriction of P to hhn and let dolldln c) solution of MP(, a up to til'ne Ta. Qn, X3 is(txdftic Th growth odition implies that solutiops of MP(, c) do not ekp 1o d We will po* show that ih the abqence f ekplsios aj is martingle. First, to show at is integrble, we nte that Fatu's lemza imlies
;yy
ms with
1-1
Bt
-#
'
#@.
.
uniqueness. colresp'
Eat
%liminf f'tatt
A Tr,))
n'-x
1
=
, ,
where E denotes expected value with respect to P. Next we note that
t
.
'E
1
1. Suppose P and 15 are not locally equivalent measures. equivalent to 1l, Q1and Q2are not locally equivalent locally is Qi be equal.
,
(a)
ndep
,
CASE
Izl2)
.z4(1
-
.
Let P and P2 be solutions of MP(7 c). By chage of measue * c producesolutions Q yd Qz of MP(# + c). We claim that if .F # Pk theh Q # Q2.To prove thls we consider tw cmses: Proof
K
=
distribution
nder the hypotheses of (5.1)ther (5 2) Theorem. between solutions of MP(#, (I) and MP(# + b, tz).
cf(z)).
z''L.
Then there is a 1-1 correpondence
at
our next result
of solutions,
+
i =1
2 1p1
//
'eorem.
jzda
:azjxs
I#tI2 Ptvj
-:
-
(5.1)
Suppose k is a solution of MP(, c) constructed pn (C, ). c-1(z) exists, Tc-1(z) Suppose that is locally bounded and memsurable, and the quadratic groFth copdition frorp (3.2)hplds, That is,
z, the one dimensional density functions
(2JrI) -a/2 exptr-lp
existence
where
-2az,
=
(5.1).
(zztzl
VV(z)
=
o
lt will
(5.3)
p
j
= as it
I+
drift, i.e.,
cmse of constant 0 so
dQf
2
+ al-'
ds)
AJ(z)
q # q,.
The nekt can considerably relax the boupdedness condition in cover many examples. lf it does nt cver yotlrs y te tat tlte min of ingredients of the proof are that the conditions hold locally, and we c) of solutions MP( 1 do ot explod. knw that l'estl
Ivv(x,)la (z)
and
Ohe
,
u(xa)
-
205
2. Suppose .P1 and PL are locally equivalent. Izl the definition of tn, (F)f and and X are independent of Jl by (12.6) (12.7)in Chapter 2, so doldp
Nikodym derivative: -
Measure
CAss
to
Thus? we can get rid of the stochmstic integral in the deqnition of the Radon-
dQt
of
=
.' -
formula
applying lt's
Change
Then since
!
(b)
f'lm
-
aavkl
=
ftla
-
a(Ta)l;Ta
S E (a(Ta); Ta S
The absence of explosions implies that
%t)
@)+ Eo
as n
-+
fltyl; Ta
-
(:x)
and cannot
E (a(Tn); Tn K
)
=
Qn(Tn K @)
-+
0
>
t)
This and the fact that fta;
(d) sn e.g.,
x.
-+
a(t A Ta) '
.
..
Tk
t)
>
'
1
(b),(c),>nd (d)imply
(5.6)in Chapter 4 of Duirett
and from this that a', 0 dameas that f (5.1)ad
t:vtt
.
)
Ta K
A Tn)
(5.2).
1
-+
Let Bt be a Brownian motion startig = 0, and let :7 > 0. Then
.
El
(
'
...
.
Our lmst chore in this sectitm is t complete the proof 1 by showing
(5 4) Theorem.
It folloFs (see
.
The rest of the propf is the
a martingale.
:
'
'
(2.10)i
f
at
Chapter contin-
0, 1et g be a
In the last section we lqarned that we could altr the b in the SDE by changinj the measure. However, we could not alter o' with this technique because the quadratic variation is not alected by absolutely pntinuous chapge of measure. In this section we will introduce a new technique, change of time, that will allow To prepare for devrloprprnts in Section 6.1 us tp altr the difiksipp coeient. will allow torthe possibility thatour processes are deqned on a iandom tize we interval.
#(t)1 < e
-
and
i,
.'
. .'
.
. .
'
. . . . . '
.
. . .
.'
''
.
' .'
. '. .
are local martingales
E/2
=
d11 =
exp
dBs
,
j.
() :
1,l '
.
=
'
'
5
'
.
.
'
-#(t)
'
-
,
.
.
e(c) / -
ddop: .
lc dn
c
j
2
( j dQt dy
=
-
exp
()
gxs
= g
) ds
tt x
.
('
sup I.Dt 0KtK1
:.
.
of Time
5.6. Change
(61) Theorem.
uous function with g(0)
P
and hence has exthat a = g'sj is a (a
'
'
(a Iaas )
JJ
-
.
cu in L
-+
that (1995))
% s K t is
..
.
that
A zk) c:h E
at
.
f(a(Ta);
-
(JJ
Change of Tlrzie 20;
dsj 24 dB. since exp (1/2) I2,I? is a martingale pected value 1. For this stqp it is important to remember non-random vector.
implies
is ar martingale . =
Sction 5.6
Equations
5 Stochastic DiRrential
Capter
206
' . ..
E
ds l
. .
is a local martingale on (0,) to be a solution of MP(/#, algj. side
Chapter 5 Stochastic Dierential
210
('
Then MP(0, c) is well posed, is no explosion.
Equations
i.e., uniqueness
in distribution holds and there
Proof We start with Xt = Bt a Brownian motion, which take g(z) = (z)-1. Silce gBtj % Gl for K Ts inftt gBj ds < (:x) for any t. On the other hand 0 ,
=
:
Jf
' ..
''
* :(f,)
./-
ds k
fR-1l(t: lftl S 1Jl
lfl
(64) The-emk
Conider
*...
.
.
.
.
(i) 0 < Ea %ay) % CR (ii) (z) is locally (iii) z (z)+ c(z)
soltitio .
of
P(0, ltj
of the SDE.
E
To begin to carry out our plan, suppose that Xt is a solution of MP(, for t < f. If J e C2 It's fimula implies thatfor t < ( 1
(1.1) ' .
. .
(
J(Xi) .zziJ(.X)
+
j9
J'(.X',) dka
= local
# pj
mart. +
j
.1
2
ja y''(.;L)d(.X')a
z,J(x,)
ts
(I)
Captez
212
Section 6.1
One Dimensional Df/fu:ons
6
where
T(z)
(1.2) .
1
,,
: c(z)T (z)+
=
,
bzjl
nd c(z) = (8 - (z). From thi we see that fXt4 is a local martinlale on 0 and noticing (1D) implles c(z) if and only if J(z) > 0. Setting L1 gives a Erst order dilerential equation for f'
(0,() >
=
fj' Solving this equation
I
J
sexp
tht
J(z)
.,4.
+
j'a
S
dz4
(./* o (/'
.
dz)
c(z)
n
Any of thes functios call be called the natural 0 and B = 1 to get .
(1.4)
-
scale.
(6.1)in
e (0,#)
z
.
w(z))< x 0 for all z e (0,#)
qep)
Tc)
nslate the system by when a1l of the 0's are replaced by y's. holds that in the integrals see (2.1)
.'
'
.
To flnd such a g it is convenient to look at things on the natural scale, i.e., let K w(.;V). Let f = limxt. w(z) and 1et r limwjp w(z). We will ;nd a C2 function Jx) so that is decreasing on (', 0), f is mcremsing on (0,r) apd
=
=
(2.4).Using
decresing
on
(', 0)
0 (2.6a)folloWs emsily by induction from the de:nition in (2.6a) the deflnition, (2.6b)follows. (2.6c)is an immediate ' .. ''
consequenceof (2.6b). ' Or next stp
=
e-'y(yk)/.('F)
'
y.
lA(z)l
1)
et
k1
n
x
(2.3). move(2.5),we begin
E !E
dyj' ' j,
(1.8) in Chapter 4 implies that if
i-e-j J' satisfes To
0
cxl we have
-+
(e-Tr
>
.'
1 and deflne for
H
=
1''%t
(e-fa
fL) k
h
.'
.
iqta < (&)
-
let
lnz)
(2.5)
z).
we
0
=
-
(2.4)
cxn
=
.
shows that comp ete.
ms n
-.+
xt applying the optional stopping theorem at time e -%f X) % fbuj for t S a) we conclude that
1 Jy)
whichis
0
-+
.J$t7-t. < :&) #.-!(p)(o < ksfalse, i.e., (a1)implies (a2),
'ra
=
-+
n
T!
< y < r,
tat conclude
tetting(
0=
0 as
'-+
It A Za 1et bn 1 r with l > y, and 1et gk Let 0 optional stopping applying A t, time at theorem the cx)j
(a)
of
217
E1
E
is to use induction to show
(2.7) Lemma. A(z) ; (A(z))n/n!,(2.5)holds,
1+ l
and
S f K explll)
antt Proof Using (2.6c) (2.6a)the second and third conclusions are immediate of frst The inequality A (z) i (J1(z))n/n! is obvious if the ozw. consquences
218
Chapter 6
Sectlon 6.3
One Dimensional Diffusions
n = 1. Using (i) the deflnition f and h, (iv)the result for n
Jn(z)
of
Jn in (2.4),(ii)(2.6c),(iii)the
1, and
-
(v)doing
little
a
deqnition we have
calculus,
of
To evaluate the first integral v'yj
,,dy fv dz :ys-,(a)
j jo S * *A-,(p) jv# /' = j d,Ja-,(,).fI(p) 'r (J,(p))n-1J1(p) s jo (,,
=
/j(r)
dz
dy
z)
-
1)!
*
z) ,
-(1
+
dz
+
exp
=
1?)1+J
(y,(z))n
ts
p)1+J
j
exp
*)1-88
-1-
dv
n!
d-
d'tf exp
s
1+
1l)1+d
+
-
+ 1
+
so
)
1+
-
yggkq
y.qo
To the proof now we only have to prove (2.2a)and (2.2b).In of view it sces t proye these rrsults with f repl>cd by hi Changing (2.7) = = variable z w(r),dz p z('p)dv then y = tp(u), dy = p z('u)j u *e have
--p
1/7(p)
=
'u)1+:
-(1
1+ ,
X
+ 1
..4.
1 so my)
=
*
1
-
219
0
> -(1
- /- (41+ )/-tl ( - 1' j* (t1 s ) -
=
#
that if y
we note
so w(x) < x. To use (2.1)now, we note that c(p) thre is no explosion if and only if
2
,
j
exp
=
Recurrence and Aanslence
z1+4
-
complete
n(z) j-dy /' dz p (#(z))l4(,/( (?dy 4(p) dv 2 =
= cx) if
c
j, jo w,(s).(s) j - j,
=.
Letting z
1r
=
))
,
(,
where in the lmst step we hav changed variables z n + 1, y of the l's. The next lemma estimtes the last expression and
q'rj
u
d,?,yl,,l
(2.8) Lemma.
2
(s
X(r)
=
E
=
j :
p d'D
w?(s)u(s)jv
I'p
j
y.p
2
(x)
y
ts
-E
A similar argument shows that the second and the proof is complete. factor of 2) f) To see what
(2.1)says
concrete
expmssion
case
=
1,
=
=
-x
in
i
)
y
y
x
g
t?
y
(2.1)is (exceptfor
a C)
1zl)J/2
where 6 > no explosion.
-
z
:
0. When We will now use when J > 1. Whn occurs integral in (2k1) is x.
+ 1 to get rid that it is
0 < &K 1
variables
z
#
l+
6
1+;
-
z
1+J -+
(1+ J) =
zy-: y+
1+4
1+ 8
w' dw
z
-
=
y-
ms y
x
-+
have
we
v-vzy-n
x
J
K zy-ny
1
dz exp
a
w
-
,
dw
g
+ zy-nj' z ms y p the dominated convergence theorem. followsfrom
Since
6.3. Recurrence K 1 the
exp
y
dz exp a. JVY'V
dz
#
Changing
(
'
jj
We considr
Exnmple Let (z) (1+ coecients are Lipschitz continuous, so there is (2.1) to reprove that result and show that explsion and the secnd y < 0, p'yj k 1, so e(-x) 2.1.
c(z)
in a
s
(p(g) /(s))
mtsl
Proof
d'?z/(1z)
'p
shows
If &> 0,
.,(s).(v)
and using Fubini's theorem
=
.-.+
-+
x,
the desired result El
and Translence
The naturalvscale, which was used in Section 6.1 to construct one dimensional diflksions, can also be used to study their recurrence and transience. Let Xt be a solution of MP(5, a) and suppose
/(z) Let Tp = inftt
scale
and 1et p be the natural
&2(z),
=
0 : Xj
>
j
=
z
j
exp
!,) and 1et
=
(3.4) Corollary.
with c'(z) > 0 for a1l z
continuous
(1D) b and c' are Let c(z)
Section 6.3
One Dimensional Disions
Chapter 6
220
.r
v
a
vy
(z) v
z
show
T. A T. The flrst detail is to
=
''' '''
(3.1) Lemm.
If
z < b then
c
aq we
fjhhhp) lmplies =
..
k
C, r
where
X
-G1 so p'yj
With (3.1)established did for Brownian motion. optionalstopping theorem
1 then
C(1 + zl-r
=
* 0
ln the borderline cmse r
pLb) w(a)
exp
=
1-r
1 the
=
C
-
((1+p)1-r
-
depends
outcome
on
lj
the
dy
< x
value
of C.
-
''' ..'#''
Lettinx (J)(x) = lims-.x w() and w(-x) since e is strictty incremsing) we haye ''''.''''
'F
.
'
''i.
(3.3) Theorem. PaIT. .&(T
. .. . . .
.
<
l mart.
=
-
.
and (ii)to give a formula for the Green's function GD (z,:v). As in the discussion of (8.1)of Chapter 4, we think of Gstz, g) ms the expected amount of time spent at y before the process exits D when it starts at z. The rijorous ztieaning of the lasf sentence is given by (i). The analysis here is similar to that in Section 4.5 but instead of the Laplacian, we are concerned wit
zJ(z)
T(X) T(Fn)
'
satisfy (1D) and are periodic with period (p)for al1 y. Find a necessary and sucient
Funtions
6.4. Green's . .. .
=
E
and note that 1 k 0 in (w(c),w()) with J&(z)
Functions
's
Green
E
>
=
EnMZ
=
E.M.
#(Sf) dt
Ec
=
0
deriving the solution (4.2).The next two pages will be devoted toand verify the answer who equat.ion content Readers in to are guess (4.2). to the skip to (4.4)now. To solve the equation it is convenient to introduce Solvlng
can
,
m(z)
=
1
,(z)a(z)
224
Chapter 6
where
w(z)is the
(4.3)
1 2m(z)
natural
(
.d..
1
p'z)
d
scale
and note
zd;
.c(z)
tfz
+
(fz2
2
Breaking the first
that
dz.f
( ) -w'z(z)
c(z)
ld oytz) dz
p'z)
2
-2(z)/c(z).
tet
(8)
-g
-
' :
.
an d integrate to conclude that for some constant
1 sy)
.
:
(a)
In order to have
#(w(z) w(c)) =
/'
-
j .... .. . ,,(z)/E E !
.
:
.
.
2ja
-
' ..
Plugging the formula for
r
we have
dysyt
j
dgsv
dzmtzlplzl
Fe have E
.
(Pt
.
/(c)
-
. . ..
w(z)) (1w() w(c) (w()
=
.
=
-
dyqv
.
E
m
have
we
b
2,u(z) /.
q
'
dysyq
j
tr
dzmtzl,tzl
p
2(1
-
-
j
u(z))
z.
c
j
to
z, and recalling
that
r(z)
=
2(1
u(z))
-
(d)
2u(z)
+ dzmlzlptz)
dzmtilft)
writing
w(z) w(c) < /( ) w(c) s r (zj zq;
j
y.b
/
'
j
dysv
and
r
dzmtzl,tzl cancel with the
=
tfzmtalptal
dy s(p)
-
; dzmtzlplz)
/.
dysyj
Using Fubini's theorem now gives
s(z) 2(1
(e) we
j 'r
dz?.,,(z),(z)
2u(z)
denne G(z, z) to be
jz j
dysv
.
''fs
ilp
+ If
u(z))
-
/.
dzmtzlglz)
e(z) w(tz) 2 :() /(c) (w() w(a))m(z) vb, w(z) (P(Z) P(t')) rr(Z) 2 ) vaj e
15,
dysy)
-
=
-
.
-
when
z
k
he.
z
sz
-
(4.4)
-
'
-
(f)
z
-
then we hake
(b)
/
,,(z))
-
r
-
=
(1
r!/
Multiplying the lst identity by 2 ja dz m(z)#(z),
.'
/- /-
# into (a)>nd
u(z)
a
b
'
. . ..
=
dus''b
va)
j' dy s(p) j' dz,n(z),(z)
Using this in (c)we can break ofl- part of the second term flrst one to end up vith
j.!/
dysy)
b
2
vbt
f/yj
=
0 we must have
-
have
/.
a
-
'p()
jy
zyp (smtal,tzl
=#-
r =
,
p
Multiplying by sy) on eah side, integrating y from 17(c)= (1and s = p' we have 'p(z)
q
syj
Ralling
r
J'= %-. LJ :
d.v
(z, ) we
z) and one over
,
.-xam/zov,
=
-
dz
,
+ 2u(z)
'a1
,
1 dv s(z) dz
225
an integral over
=
=
,
ja6dy into
s(z) 2(u(z) =1)
=
sine the deqnition of the natural scale implies w&(z)/#( AT) .Pptr > k 1)M) -
-
c) k
K (1
kM4
-
(1995)nqw,
we get an upper
/-
-
t'tlrrttzel'z
6.5.
tE
a. So
&'qc,&)stays bounded
(?n
a11 EI
undry
Behavior
In Section 6.1 when w constzycted a diflzsion on p. half lipe pr > bmpded intervalj and we said that we gave up when the process reached the budary I!I it had exploded. in Fhich Fe cap this section, we will identify situations extend tlke life of the process. To explain how w might contizpe F begip Fith
as
-x
0
(w(z) w(a))m(z)dz < -
E
a
El
cxl
The. two conditions in (4.7)are equivalent to f%(To Th) < x. To complete the proof npw we will show
=
::
=
=
Jf ,!5(T.
T)
A T) < x.
j'
jt
PaIT. < :J) > 0 implies .&(Ta < cx)) > 0. Pick M large enoug so Proof that #OIT. K 1') k c > 0 and that .J%(T S M) k E > 0. By considering the flrst time the process starting from 0 hits z and using the strong Markov ( prfrt#, it follws that if 0 d z < then '''''
-
.
.
.
P 0 (1b S
)
=
f'o(#.(T
K t = Tz' );T S
).l3n(:r',, s t) s
.z'wtn
s A similar argument
bound
.
-.+
.
3 now, we have
Z'
-
The frst integral always stays bounded ms a apd a if arpd only if w(a)> c
(4.9) Lemma.
229
lt follows by induction that for a1l integers k k 1 ;
-x.
(
2 e(z)
kM)
>
the proof of
=
0 if and only if
>
E
.
AT: and repeating
g'(0)
-
.
n
.&('r
(3.2)implies
Ezra,sj
=
Boundary Behavior
/
.
.
f)
.
.
.
.
.
''' .
To use the lmst recipe in general we start with Xg a solution of MP(, solves MP(0, pxtl and let p be its natural scale. ) where y j
h(#)
.
'
K t)
=
s i)
k
Combining the last two results
< z
0
for all ce
ct'q.u, < .:J') proof of ih the s (4.9)we frst get tr 6 (
s(z)
0 then .E'r(-e,
lk
E
M(O))
I < cxl < (x) = x = x
dz
of m to bring out the analoa
To see that this means that the process repeating the proof of (4.9)shows .
m(z)
(2.1),we have
.E'nq-e,e)
(5.2)Lemma.
z
jvjo
,
antiderivative
;'
/.'
m(z) dz
dz
This leads to four possible combinations,
and
=
Boundary Behavior
boundary.
m(z)(e(=)
-
that
we note
O
w(z))dz
=
j* 1
1
2px
dz
=
cxh
Section 6.5 Boundary Behavir
232 Chaptet 6 One Dimensional Dihsions To calculate J we note that J x. The combination I
into or
out of x,
and J
x
1/(&2z) so M(x) = (:x) and says that the process cannot get
m(z)
p k 0,
when =
=
(x7
=
k
so
x is
a natural
boundary.
Exercise 5.1. Show that 0 is an absorbing boundary when (a)p 0, (b)p < 0.
boundary and
t:xn
is a natural
Exnmp e 5.3.
The flrst conclusion is reascmable since in d k 2 we can start Brownian motion at 0 but then it des not hit 0 at positive times. The second conclusioh ca be thought of as saying that in dimensions d < 2 Brownian motion will hit 0. Exerclse 5.2. Show that 0 is n absorbing bounda if y K =1. We leave it to the reader to ponder the meaning of Brownian motion in dimension d K 0. Exnmple
=
Consider dXt
on is the lndex of the Bessel process. To explain the restriction and to prepare for the results we will derive, note that Example 1.2 in y on Chapter 5 shows that the radial part of d diiensional Brownian motion is a Bes:l prpcess with 7 p (d 1). The natural scale is Here 7
Power noise.
5.4.
process.
espe
-1
233
(0,c=).
1/(v
The natural
'(z)c(z))
=
z-2J
scale is
AJ
=
(p(z)
dBt the
z and
=
speed
mrmsure is m(z)
: '
so
>
1
I
=
0
z 1-2: d z
< o:) = cxn
=
if < 1 if 6 > 1 -.
-
(z)
exp
=
=
1 *'
-
1
-%
d#
#
1
v/z dz
dy
ln z
=
(2 1-v
'
=
y)
-
if y
jj.
=
vp
From the last computation we see that if k 1 then w(0) < < 1 we observe that the speed memsure To handle y .f
=
-x
1 1
1
Combining the lpst two and I
=
m(z)
=
So taking q = 1 in the defnition of f 1
f
=
.C1-'y
o 1
-
To compute J we observe that for any y 1
J Combining the lmst two
=
z'Y
7
Z v+1
dz
natural absorbing
-v
we see
entrance boundary regular boundary
regular
dz
=
z=+1/(y + 1) and
< cxl
that 0 is an if if
2' 1
j
.-.
2:
dz
g
.z
(x)
th.t the boundary poiht 0 is
we see
if if if
0, Pllj (c) Plh p.m) = 1 when J k.1.
H'J
and < x)
Exercise 5.4. Show that the boundary at and entrance if 6 > 1.
=
x
1.
.5-2:1
=
()
(b)Ellj
(s.>1) ds
a
y
#
The frst detail is to show that X3 will eventually
la:I:S'-
so hlaj thorem
t:y
''
and nondegenerate
k jsinhtgll
:
..
locally bounded
i
Proof
+
k a coshtaxa )
'
optinuous
ee*# ,
2
a i ..j- coshtax,lcfftxa)
arguing
'
.
Dlmenslons
:::1
(i) b is measurable
ey -j-
our choice of a implies
ln this section we will use the tehnique of comparing a multidimensional difconitions fusion Kith a one dimensional one to obtain sucient for (a) no explosion, (b)for recurrence and transience, and (c)for diflksions to hit points limr-..x Sr Throughout this inf (t : I-'LI rl and vb or not. Let Sr will section we suppose that Xt is a solution to MP(, a) on (0,Sx), and that s>tisfy the coecients =
k 1 and
=
..
.
ds
=1
Recall the deqnition given in Ex-
by flrst considering Hint: simplify calculations that the value of a is not ifnportant. '
d
1 . 2 i tr coshtax,lcfftx,) + I'''-I (?
mple 1.7 in Chapter 5. Show that the boundzy point 0 i absorbing if if regular entrance if
)(A%) ds + local mart.
-:
cxnif and only if J > 1.
-->
sinhtaIj
=1
-
J(#,)1(s.sM)
EM
formula we have
d
t
ltvl
235
:
l-YtI
=
r).
1zlK r.
Yalz)
hemark? actng cosn so tlkt
exit from any ball.
C
Then there is a constant
(ii)imply that &
0
we can pik
an tr
lrje
s
0.
Chapter 6 Onq Dimensional DiEusions
236 for z
-w(z),
k 0 and w(-z) =
Using
(4.8)now
b = r,
with
the spqed measure
so
rrl(z) c
1
=
e (z) z
w(r) w(0)
d%
c
=
0
=
w(=r) w(r) w(-r)
=
So
that
noting
an
2w(0)
-
2e(r) w(-r)
Our plan is to compare St
-apjal
,
-r,
=
Section 6.6 Applications to mgherDimensions
introduce the natural
we
.1
((),x)
on
nd speed measure
mtzl for Zt:
=zgy) oj j (-
exp
=
bLztjdNt
w(z)
scale
s(z)
-
-
-
1.412with
=
/z(Zt) dt +
=
237
,a(,)
w(z) ji=s(,) dp m(&)1/(s(z)p2(z)) =
we have E 0 (-r,r) 'r
r 1 ezpz e-zpz dz gp o 0 1 (-n2#IzI e2#r) :-2#lzI dz + gp
(d2#r
=
)
-
+
=
In view of Eellez''s test, (2.1),the next Tesult says that if Z3 finite time then X3 des not explde. .'
-r
equal
The two integrals are
r
j.
E 0'r(-r,r)
=
so
.
.
.m
-
.
u.
(6.2) Theorem.
,
lj dz
2V(r-Z)
-fj
'
.
t a2J(r-z) - 2#2
,
.
-
-
z
p
r
1
=
2p2
'
(j
(elpr *
-
,
-
-
* dzm(z)(w(x)
./r,
r
p
1 and (z) c: =#Qrt(z), th To compare with (6.1)now, note that when c(z) remark after the proof says we can pick a = 27 + 2 so the inequality in (6.1)is ::
E 0q-r,r)
S
coshttz; + 2)r) gp + :
,
,
With the little detail of the fniteness f Ersr difl-usion and a one dimension>l duce a compyrison between a multidimensional which is d We begi by ihtrodciik thi yiis t Khsminskii (1960). on, in of deqnitions that may look strange but are tailor made for coxrijutatitm and of proof (d), (e), (6.2). (a)of the ' . .
..
'
a(z) v(r)
=
sup
=
zzTctzlz
t
a(z)
/z(r) =
'
'.
tetting
.(z))
-
x
-
:
.
=
limryx %, then
.
t-j
r( gn a )p y n
pl# the deqriition to
Lxg)d a (p izjxxjayj ,
x
(j) dt
1.
0
e -'.'(.R,,)2Arji(A%)
,
,
tekz?
.):..' (yjgyr'jd
-e-'#(A,)
=
d
and
.-,
have
we
E1
2 ms lzI T formulate a resllt that allows de(z) to Dll function' if and oly if deflnition: J(t) is s'id to b
l
?/
=
while
ivhr tr (c(z)) i:z E cf#(z) is the trace of c. We call detz) the effpctive dlm' sih at k beuse (6.3) nd (6k4)will sltw tht there will be rettrrence 2 in a neighborhood of infnity. r transiezce if the dimension is < 2 or >approach (x) need
p'(IzI2)2z .lJ#(Izl2) J''(IzI2)4zjz.j # .Dfptlzl ) g (IzI2 )4z7+ g (Izl2 )i .Dp(lzl2)
in
1
gives
cal.culus
is a local supermartingale
239
:(z)
=
we can rewrite
Using the deflnition 9 0 and p npw
(a)
Combining (d) and (e) shows that e-fg(.&) Rt k 1 and completes the proof.
it follows that
Since e-fg(Zf) is a local martingale
to Higher Dimensions
Secton 6.6 Applications
One Dimensional DiRsions
Chapter 6
=
E(s)
(1gt)-F =
is
a Dini
function if
p/log s.
Suppose detz) k 2(1 + e(lzI))for lzIk R and J(t) is a Dini thn is transient. To be precise, if IzI$ R then .&(Ss < x) < 1. Xt fukction,
(6.3) Theorem.
-
(d)
I
+
e0
f-#(l,)
'
+ #(-%),'(A,)
# a(*a)J''(,)l u
To bound the integral when Rz k 1 we use (i)the deqnttio of v and (ii) the equation in (a),the fact that g k 0, and the deqnition of p -#(.Ra) +
(e)
t
pt-Ya)
-:(./,)
2
Using
This time
#(r )
above.
little
Since
Let
's'rj
-w'(r)
we
=
lzlk
the conclusion to
p. C1
let
we
j
=
now extends
243
1 :(j)
r ex p
Ja )
-
dt
t
-s
ds
#'(r) k 0 and
have
r#''(r)
e(r))#'(r)
(1
+
-
0
=
1.X12and using (6.5)shows #Rtl is a local supermartingale while Letting Rt Rt e (0,p2). If 0 < r < s K p ujing the optional stopping theorem at time v = Sv h Sa we see that =
which has
i)(tiiI2) k tz,.iRrb y
=
#(r2)1%(,$'r < &) + #(s2)(1= Prsr
Rearranginz gives
-
and hence satisqes
r/'(r)
1Ah1 Letting Rt 2 and Rt e (0,p2). If 0 < r = Sr A Sa we see that =
0 (1 e'(r))#(r) us tzk g (6 5) Aws pR,4 is a local supermartingale +
using the opsional
Kp
w(IzI2) k ErvRr)
while
ssopping
k vrzlprsr
tu = inf (f >
IzI
'rh+1)
< x)
=
0 for 0
Let 0. so llA Jll n we lu e llla ;II 0. < E/3 for t < (). Using the triangle inequality and the contraction Illrlia ln 11 property (1.2)now, it follows that for < tp Proof
,
--,
-
-
.
-!
=
-
+ IIQA All + Illa TII IIZJ TIIS 117l/ 71/nII E/3 E/3 + < e K llJ AII+ -
-
-
-
=
and cqp-
1
-
!
Prof Let f e 'p(T) and ga last deflnition, we note that
=
JJT,.Jds.
-zll
1111s
fl
c
T&f h
J
-
AJ Its domain O(.A) is the set of which there is a g so that
Before delving into properties that cannot be generators.
T.Af ds
=
Remark. Here we are diFerentiating and intejiting functions that take values in a Banach space, i.e., that map s 6 (0,x) into L*. Most of the proofs from calculus apply if you replace the absolpte value by the norm. Readers who want help justifying ur cofnputations can cnsult Dynkin (1965), Volme 1, pages 19-22, or Ethier and Kurtz (1986), pages 8=10.
is deflned by
of a semigroup
generator
%f
1im
=
At0
f for
as
the limit exists, that is, the
which
c
-+
0 since
f fpr
1
-
-
g
0
-.
as pz -+
J e p(T).
nga
.
Proof
1 e /(.)
Since zp is a maximum,
J(zc) k fyj 1Jl/'(za)
)
Now c+
c
%+s1 ds
=
0
=
TaTds
ga + tl
:'
i'
E
ln,-s -
'' .
,-
-
v-g
-
111 ds
-
0
/-+jjz'-y nyjj
y)j yt
s
.
.
for all y then
.J(zn) %0.
;(zo) K 0 so d.f(zc)
K 0.
ds
-
-
+
.
1
j
ds IITJT T11 -
-+
0
This shows that ga e 'p(.A) and Aga Taf f. Since the first calculation gala and DT) shows 0 the flrst claim follows. converges to f as c e =
-
for the
()
. (( .
(
of generators, we pause to identify some operators
and
the motivation
explain
J-llz)z- ttlds-
'
If
To
so we have
0
.
(1.4) Theorem.
Xxf
=
-
Rzirk. While T(T) is a reasonable choice it is certainly not the only ohe. A common alternative is deflne the domain to be Cft, the continous flmctions which converge to 0 at (y) equipped with the sup norm. When the semi-group We will not follow :Il maps Cz into itself, it is called a Feller semi-group. this approach since this assumption does not hold, fpr example, in d = 1 whrn (x) is an entrance boundary. The lnllniteslmal
f e 'p(.A) then Af e p(T),
t
Ttf
ll7l; T,JII llT,(TlsaT T)Ilf II:l-,r,T T11 =
If
.
.
-
of actual generators, we have
(1.5) Theorem. #(.) is dense in p(T). hd C D(X) 1)1 . d . WXf XXf :,
=
To prove th last claim in the theorem we note ttlat the semigroup traction properties, (1.1)and (1.2),imply that if s < t
Returning to properties
-
-.+
Section 7.1
Chapter 7 Dl/lbsloms as Afarov Pzocesses
248
141h6
The fact that Af e 'D(T) follows from (1.3)which implies Tsf p(T) and O(T) is closed. To prove that If e we note that the tion property and the fact that f 6 'p(.A) imply -
'p4d)
jjn71t-11
Ev (;(Xt-a)
jjm Ln-xyjjs jj s jjny -
J
()
-+
-
-
mxyjj s
jj jjm-, LrslyIIT'T 1 T xyj
Our flnal topic is an ipport>pt play a minoi role here. For each bounded measurable 1 by
.
p(z)
s
() fince J e O btatned by integrating the one
.-y
To make the conlwction
+
-
awd
(
jjm-stxy zxyljj
llxy nzyll
0
.->
-
e
p(r).
The
it.
abve
snal equatin
.
'
. .
Clearly
in
(
.
'
.
'
.
.
.
probleps
(1.6) Theorem.
J
.u'/e is a Pp martingale
with
gxg)
respect
fxz )
-
j
-
we
ad
To by Xt
memsurable
or each
is integrable
are bounded,
..r.
g)ytzl dt
e-ty(.m)
Ec
=
.
6
for
dt
1-1 onto ,1)(.A). tts inverse is
(
.A).
-
of U and changing
we have
*
e-tm+sy
..
that Uf
conclude
will
fact is that
subtle
nlldt
-
(:o
1
-
ds
to the flltration Fj generated
for some developments but operator 0 dehne the resolvent
concept >
J)//l. Plugging in the definition =
1 =
memsure la
Alxs)
Tr.J(p) dr
j
0
0 h 1 c = h
.u1
and prool-sincey A >
nf-s
=
Juzl
If
e-t
U maps /(T)
Proof Let Alf vriables s = t + li
will now prove 6 ,27(.A)then for any probability
p-, -
*
J
:z:
ll&'lllS lIJII/. A more
(1.7) Theorem.
'
(t.5j is
EI
between generators and martingale
249
drl
.J(Xr)
J0
'
..
A.f
ygyvj
-
avjj
-
S
+
-
Genertors
wkichis 0 by (1.5)
'
'n-sl
t-a
-
=
jjr;
j
-
= rJl-aT(!8 T(p)
SIl.4; and that the rkht drivativ of If The last equality shows that A%! is %Af. To see that the left derivative exists and has the sarri value w note t at jy
1Xz4
-
1
=
and
But for any g, it follows from the defnition of Tr and Fubini's theorem that
contra-
./)
mxyjj
-
Semigrpups
'2)(.)
-4
Taf ds
e and
Auf
0
f;
=
-a
f
-
T, J ds
e
-
we note that
.
sinceusl is '
r'
and compare the last two displays to
conclude
that S 11/11) (using1171J11
'
( .E'
( .. '
zpul 1.,--) M1 + s. -
By the Markov propezty the = Ex.
(.f(xt) j'
second
(J(.;V-,
lx-t
-
-
j
y)IIs c,y j
.z-,)
-
Ilxsgy (uy -
-
#
side
term on the right-hand
) fv'%) -
dr1
./(.;)
-
f-,
yyfyvj
ts;
+
is
J
()
1
(1 e-')1IJll ds + -t
.j
jn
c-,jj/jj
j
I
j
()
-
..
IIT'T Jll ds
0
-+
-
(
n-,jjyjjas
,
h
0. The formula for Auxf impltes th>t as and its inverse is -+
/
ds+
,
-
Ajlf
=
J.
Thus Ux is 1-1
Chapter 7 Dihsions
250
To prove that U is onto, 1et f
z (J
-
A14
=
*
d
j*
e-' w
0
e
= e-f
-/-
Integrating by parts
Ildt
Section 7.2 Examples
Afarkov Pzocesses
as
'p(.)
yjty
e-f
-
j*
d
e-z
7lJ(z)
pg dt
w
-
l
,
AJj
=
f
;
q
Markov processes >pd
-lte-
compute
-
1+
. .
(c-z
t)
-t
-+;0j
jj7lltz) J(z)
-+
1)
.
.
jpz',
.
dvgv
.p(z
,
jpz,
dyj is a
-'.
dvgy)
-
t as
-
ykj)
..-
.
1 , an d
=+
c
-
Let (S, Sj te a memurale txnmple 2.1. Pure Juinp Varkov proesses. x) be a transitlon iernel. That is, lnd let Q(z,.A) : S x c (0, space t$', each .A e (i) for x Q(z,.A) is memsurable (ii) for each z e S, .A Q(z,.) is a inite measure
o)y(p)+ o(t2)
pz,
jlytz)
1+
-
+ -f
of
jpz,
dylgly)
.
=
Sitce t -1 (c it follows that
j
e-
it follows that
and the proof is El
7.2. Examples ln this section we will consider two f>milies their generators.
c-AJ(z) +
=
-
0
-
E=1Xtj
7lT(z) lzj t
vje-tlldt
+ c
=
Doing a little arithmetic
G)
x
Wt;
Combining the last two displays shows complete.
=
Agj dt
pgdt-
(e-'mJ)
=
X%defnes our pute jump Mazkov process. To compute its generator we note that Ng suptn : Ts K ) has a Poksson distribution with mean t so if f is bounded and measurable
(1.5)implies
and note
251
o(t)
+
uausition mobability
ytzlljj
-+
c
0. Recalling the deqnition of Pat, dy) it follows that
-->
-+
(2.1)Theozem.
'p(.A)
'p(T)
=
=
L*
and
-+
(iii)Q(z,(z))
=
0
Al)
Intuitively Q(z,.A) gives the rate at Fhich our Markov charins makes juppp frpm since invksitle. z into and we exclude jumps from z to z with they are To be able to construct a process Xt a minimum of fuss, we will = supgE Q(z,S) < x. Deflne a transition probability P by supposethat
=
JQ(z,
dylly)
-
J(z))
..
.P(z,.) .P(z,
1 yQ(z,.)
=
1 lzJ) y( =
-
if z
(
.
Q(z,5'))
.
.
,
.
.
.
u
.
,
event.
Exnmple 2.2. Dlfrusion processes. Suppose we have a family of memsures Px on (C, C) so that under Pc the coordinate maps Xt () = f.J() give the unique solution to the MP(, c) starting from z, and St is the flltration generated by the XL Lt Cxa be the C j: functions with compact support. .
be i-i-d. eyponential with parameter that is, Pti > t) = e-'. Let tl t2, = .jza j; e a ar kov chain and let Tn Let Let Ta = tl + for 0. 1 + s n k with transition probability P(z, dyj (seee.g., Section 5.1 of burrett (1995) for defntion), and let a Xt = ya for t G gTa Ta +1 ) ,
We a:e now xeady for the main
(2.2) Theozrm. Suppose a and b are continuous Then Cx2 c @(.) and for a1l f Cx2 e Alz)
=
'j'
1
aitztlhjft
+
and MP(, c) is well ppged.
biztDilzt
253 Proof
Chapte 7 Diffusions as Afarov Processes If
J e C2 then It's T(-Y)
J(a')
-
formula implies
Let #(p)
Proof
t
J7 i
=
ds + locat zilart.
Dilunjbix''t
(.;%)
Ej(!& an)2.
=
1
+ ij7
Diilxstaitxst
(.X)
-
ds
+
' .
.' ..
.
the local martingale,
of times that reduce expcted yalue gives
:E
.
J(z)
-
stopping
the coecients and the bounded copvergence
zslxtt -
.
y(z)
't
=
4
y
+ J7 i,J
Ez
=E
step
are
bounded
-
is to bound the movement
t
j,
-+
For
-
zfltxa)
+
-
B T
+
W
g
ck(X,)Jds K ztrB
.4)
+
r? it follows that
=
Pzsc
.4)
K t) K ztvB +
result.
E1 -
thE
,
z
/
ff
,
-
J(z)
-
uytzlj
,
jtjxytx,) s tsa
fzj
0
=
-
and
Af
lz)
=
0. So
xytz)
t
.
using
suj.
TQ
%%
(2:3)again
E=1Xt4 t
ds
i C
sup Prs
gives that for
sup
DEOK
comes from' using the strong
wlierethe second equation time Tx. (2.4)shows that
0 we can pick E (0,1) so that if lz pl < &then I.4J(z) .A/*(p)1< e7. If we let C = supz l..J(z)Iand use (2.3) it follows that for z E ff
ds
tl
J(z)
J-l i
=
Loilx-tbix-,
jo'zyt-v-lds
=
'f
E=
=
Since /1 k 0 and ltXs.)
now we have to show that
conclusion
jjEzfxt) The first
'
5-.7./,, J .j.
To prove the
ds
then the integrands theorem implies
1
(2.3)
fat-xasr)
.
are continuous
E .
j
..-+
ds
Dit.f-nlaijvn)
;
f e G%2and
xilbi (-Y,) ds + local mart.
Letting Ta be a sequence of tims that reduce the local martingale, stopping at t h Sr A Ta taking expected value, then letting n x and invoking the bounded convergence theorem we have
tAT,X
.
If
-
:
0
+ 1J7.&. 0 i
2(.X'>
0
,
a/(.X,)(-Y,)
Er
=
1
have
we
.
tATa
Ezlunsn
-z
formula
253
'1
J7
=
0
'
Using It's
-
0
'
lf we 1et Tn be a sequence at time t A Tn and taking
Section 7.2 Examples
Pys
f
Afarkov Pzocesses
as
Section 7.3 Diferentiating
the desired result follows.
arbitrary,
For general b and a it is a hopelessly dicult prpblem to compute ,27(.A). To make this point we will now use (1.7)to compute the exact domains of the semi-group and of the generator for one dirpensional Brownian motion. Let C be the functions f on R that are boupded >nd uniformly continuous. Let Cu be the functions 1 on R so that 1, #, .fM 6 Cu .
(2.5) Theorem. Afzj
For Brownian motion in d for al1 J e '2:)(.A).
1, D(T)
=
Cu,
=
Cu2 and
'p(.)
=
,
r(z)/2
=
=
sense
X
l:1T(z) TtJ(p)1K 11T11Ipttz,z)
-
J
my,
The right-hand side only depends on lz pl and converges to 0 as lz !/1 0 so lf E Cu If J E O(T) then ll1JlJ J11 0. We claim that this imylies < E/3 then pzck f e Cu-. To check this, pick E > 0, let t > 0 so that IIZJ pl < then l7lJ(z) Tl.f(p)l < EI3.Using the trangle inequality so that if Iz now it follows that if lz pl < J then
e-x-evy)
'Zi'l /
dy
2J(z)
-
J-x
lt is emsy to use the dominated convergence theorez to justify diferentiating the integrals. From the formulas it is easy to see that g, g/, f e Cu so # '6 /2u and gtgqyj .:y(z)
.:,(z)
#/
gtz)
=
-
(z)/2.
J(z) by the
(1.7),it
proof of
follows that .A#(z)
=
.
To complete the proof now deine a function f E Cu by
we need
/(z) lt
=
-
Uf
to show 1
tz)
=
=
u
j
D.
,1)4.)
%2.Let
E
Q
and
(y;. .
satisfles
1 'i'y (z) ,?
-.+
-
-
dy +
Jl
The function y
z)l dz
255
!
pT
e-u-rsvi
pflz, z) for the Brownian Proof If t > 0 and f is bounded then writin transition probability and using the triangle inequality we have -
Probabilities
again gives
Vi-'
Since .A:(z) is larger than Cu2and consists of distribution and lies in Cu
For Brownian motion in d > 1, T(-)
Remark. of the functiops so that Lf exists in the (seeRevuz and Yor (1991), p. 226). .
g''j
Aanslon
ptz)
-
0
=
-+
-
.
wyyy
.f11
-
-
-
A11solutions of this diflrential equation Since / Ux1 is boundett it follows that
.ry
have the form + Be Uvf and the proof is complete. -e
=
-
.
(:1
-
,
IJ(z) J(p)l K IT(z) TtT(z)l+ I7lT(z) TlJ(p)I+ ITlT(p) -
-
To prove that -p(.A)
implies
Ufj
=
ct =
me
Our frst tmsk is to show that if we have
1 E Cu then
jn
gzt (2) -,/a =
e-tp-xlzi-.ytpl
dy +
(2)-,/2
(r = r
e-u-rtvby)
jr
=
e-@-p)ziIy(g)
Uxlkj
dy
r -X
e-(=-OYiQ'J(!4dy
a and
that if
b are
continuous
and
MP(,
c)
is well
d
forJ e cx2or ,
z)
'p(t,
changing notation
=
T1J(z), we have
d'v -dt z.p(j,z)
(2.1)
=
-X
'
-
we saw
-jX.f(z) = ZXT(z)
and noting that the terms from diferen-
dy
In the last section posed then
Probabilities
dy
Uvjf E C,. Letting g(z)
T
Differentiating once with respect to tiating the limits cancel, we have
'(z)
j
c-Ir-#IVWy(p)
7.3. Transition
< e
(3.10)in chapter3
begin by observing that
(2)-1/2
.f(p)1
-
-
where .A acts in the z variable. In the previouj discussion we have gone from the process to the p.d.. We will now turn around and go the other way. Consider
(a) e C'1.2 and (b) u is continuous 'u
(3.2)
'uf
=
on
Lu in
(0,x)
(0,x)
x Rd
x Rd and
,u(0,
z)
=
J(z)
Captez
256 Here
we
7 Dihsions
have returned L
=
as
Afarov Processes
Section 7.3
to our old notation 1
so (a) tells us that that u(t s, Xsj implies -+
J7cfJ(z)ZkJ(z) + J7(z)D.f(z)
'*'
ij
-
j7
To explain the dual notation note that (3.1)says that r(t, 6 ,i7(.A)and satisqes the equality, while (a)of (3.2)asks for u E C1l2. As for the boundary condition (b) hote that J e z)4.) c 'p(T) implies that 11Tt1 JlIc.7.. 0 ms $ 0.
,utt -
fxjj
s, .Xa) is a bounded martingale ms s t, so the martingale -+
utj,
.)
-+
-
.
To prove the existence of solutions
(3.2)we
to
Section 4.4.
(3.3)Theorem. Suppose a and b are bounded is, there are constqnts 0 < J, C < (x7 so that
n'ct Suppose also that
IcfJ(z)
fOr
a1l z 1 y
-
cj.(,)Is cIz
a is uniformly
p1,
-
Itz)
elliptic,
-
tplls'elz
that is, there is
p1,
lpI2
k)
Then there is a function ntz, p) > 0 jointly continuous in t > 0, z, y, and C2 which atisfles acting of dpldt function Lp L z on the z variable) (with as a so-that continuous bounded >nd if is aud J
theorem
o
Izl
=
Js
pf (z,p) dy
n(z, y4 is the trnnsition
density for our disusion process. Lt D J0. To obtain result for unbounded coecients, we will consider
ke the connection
ptlz,
=
the SDE
with
It's ut
-
Erfhh
=
z)l K 11J11. l'tzlt,
solution
s, X,)
-
A)
'
=
du dt
--(t
to
(3.5)we
turn to Dynkin
Voly II, (1065),
and suppose that (HC) and (UE) (3.6) Theorem. Let D = (z : 1z1K ptRt, y) > 0 jointly continuous in t > 0, hold in D. Then there is a function C2 as a function of z which satis:es dpRldt Lp (withL cting on D, y z, e the z variable) and so that if )' is bounded and continuous =
of MP(,
c).
If u is a
utt,z)
)
formula implies 1z40,
of solutions
.0
we prove
a nonexplosive
'utt, z) Proof
implies
principle
(3.4) Theorem. Suppose XL is bounded solution of (3.2)then
:)T(:) dy
To prove exiqtepce pages 230-231.
/p1'(z,
p)T(p)dy
=
satisqes(3.5). -
r,.Xr) dr
the' connection
To mak
with the SDE we pzove
4
+J-1 i
0
ofzlt-vrlftxr)dr + Ditnxlaitxr)
-u
0
dv
of MP(, c) and let r
(3.7) Theorem. Suppose Xt is any solution D). If is any solution of (3.5)then /
XL
.1
1
+ )-l ij
local mart.
'tz
stj,
z)
=
>;.(y(a'
);
'?-
>
t)
=
inftt
:
258
Chapber 7 Dihsions
X implies
Proqf The fact that u is ontinuous in (0,t) x It's formula implies that for 0 % s < t A r u(t
-
s, aYal
-
u(0,
A%)
Section 7.4 Brris
Marov Processes
'as
'
=
du
--a
0
+
)C 0 1
.&)
-
dv
r,
.;
+ y''l -2
(t
it is bounded there.
DutxrlftA%l
dr + local mart.
-+
-+
l'.(J(.A$); r
=
>
.,4
E S, z
-+
for fzxed z 6 S, .A
-
u(i,z)
for fxed
dv
-
-
the next, we will mssume that the reader is familiar with the bmsic theory of Markoy chains on a countable state space as explained for example in the irst five sections of Chapter 5 of Durrett (1995). This section is a close relative of Section 5.6 there. We will formulate the results here for a transltlon probablllty deqned s). Intuitively, .P(Ak+1 e dl-'k = z) = ptz, d). on a memsurable space S, Formally, it is a function p : S x J' (0,1) that satisfles '
so (a)tells us that u(@ s, Xs) is a bounded local martingale on implies that 1It 0 as s 1 r and u(t ).X,) s, A%) s, .X'al ('r > t). Using (2.7)from Chapter 2 now, we have
(0,tA r). as s
t)
259
-+
#
Dituxrlaqxr)
Chains
-.+
(b)
In our
applications
-->
ptz, .A) is memsurable ptz, .A) is a proba j; ility memsure
to diflksions
t on
n
we will
ptz,.A)
=
take S
Jx
pztz,
=
Rd,
and
1et
f) dy
where pttz, y4 is the transition probability introduced in the previous section. By taking t = 1 Fe will be able to investigate the asymptotic behavior of Xn (x7 through the integers but this and the Markov property will allow us n as real numbers. to get zesults for t thmugh the x We say that a Markov chain Ak with transition probability p is a Hrrls chnln if we can flnd sets .A,B E &, a function q with (z,p) k 6 > 0 for z G y C B an d a p robabiltty measure p concentrated on B so that: (i) If inftn k 0 : Xn 6 .4J,then Pz < x) > 0 for a1l z a eS =+
Combining
(3.6)and (3.7)we
-+
see that
.4,
Emlxj Lettig R 1 (x) and ptt, y) ;NR(z1 #) R is increasing, -+
r > f) =
we
lima-xpRt
=
Jo
pllz,
p)#(p) dy
(z,y), which
have
,
(3.7)implies
(3.8) Theorez. Suppose that the ma4tingale problem for MP(, c) is well and satisfy (HC) and (UE) hold locally. That is, they posed that c and hold in (z : 1z1 K JO for any R < x. Thn for each t > 0 there a lower semicontinuous function p(t, z, y) > 0 so that if f is bounded and continuos then pflz, yj dy Exfxtj .
'is
=
Remark. The energetic reader can probably show that pt (z,y) is continuous. However, lower semicontinuity implies that pttz, p) is bounded away from 0 on compact sets and this will be enough for our results in Sectiop 7.5.
7.4. Harrls
Chalns
trx Jc
=
exists since
.
ln this section we will give a quick treatment of the theory of Harris chains to prepare for applications to difusions in the next section. In this section and
(ii) If z e and C B then ptz, Cj k qlz, #)p(dp) c In the difl-sis we consider we can take X = B to be a ball With radius r, p to be a constant c times Lebesgue measure restricted to B, and q, y) mtz, #)/c. See (5.2)below. It is interesting to note that the new thory still contains most of the o1d one as a special cmse. .4
=f
Eknmple 4.1. Countable State Sace. Supppse Xu is a Mrkok chal on state space S. In order for Xrt to be ountabl a > Hryis hai!z it is eqessary with Pzt-''a and sucient that there be a state u fi some rl >- 0) % 0 for a1l z 6 S. 'tz
=
pick r so that pty, r) > 0. If we let Proof To prove suciency, and B then hold. To prove ncessity, 1et ('u) be ('p) (i) (ii) /7((u)) > 0 and note that for a11 z =
.
Pkn
=
u
for
some
n
k 0) k .E'.((vYzz)p((u)))
The developments in this section make the theory of Markov chains on a
>
,4
=
(zIJand
a point
with
0
l::l
are bmsed on two simple idems. To countable state space work, all we need
(i)
E(' 115,:* .1:::::,,* 1111::* .1:::;1!!-* 11:::::.* .1::1111:.* j'.1::1111)* :,,' 11::::211::.* .1:!5115;;2.'* 11r::!:;' .1:::111!'* 11(:::211,:.* dl;:!!iii;l?'d 11!:::21.::7* ilts:'d :,,' 11:::::1,* .1211,::* :jj' ,,,' :,,' ;::;i!:::,' 41:::2)11::.* ::::::::::::::' klp!i,lk--d ::;:::::::::::' ;::;1!:::-* ;:::i!:::!' :,i' ;:;;1!:2:y* ;:::i!:::?' ;:;;i!:::,' !,,' dl!:::qlll;;?.'d .11::::((::;7.,* :,,' ;:::1!:::,* ::::::::::::::' ;::;i!:::t' !y' 41::1111(.* ;::;1!:::,* ils)lk.,d :bii!p;d 1I!::::Ii:!'' ..jI!::;,'' 'E11:':;IL.' .,,,' ,;;i!!::;Ii' ilt::!qllli;?''d (::;1.:::-* j'y':111:':111.* :llEsllk.k:,i!,;d (J1rq')I;q'r!1i.' (iii,d il::l)d ..-ii!!EEEE::)I'' 'llkzird ;I!EiI1;-;:Ei!;;' 112E52::* 'lIi:E::1,' (:11::--* ,111:;.* :i1!F::I,:' 11I:Eii!!.' ;1pE5i1k,,' 11::4)* jyd '1EEiE:'' 411:)* ,,.;i!!E(((:;;I'' .,.ii!!55((E;;I'' :1I(r7'' ,,-;i!!E((E:;1I'' (:11::-.* dllsE'd 111:4* (11(:7-* (iii.d ()1(::-* j't')'k';:!!5!1t..' jyyd .j;jjiijii;' ((.yjj' -',,' .j' 4512:.r:11:.::1k.* ,-,' -11:;,* .-,' ,;i!jiiii;7' .:1!:,;,.* .4,* .,j,' .,,,' .-.!:111::* ,--' -.41!:::;1.* .-.' y.' ,jj,,,' -,r::,,-,::j,,' ,;jj,,,,-' -j4!!::;p'' ..-1144)* .-,' .;j!ji!ii;7' ,-,.jj!'''''' ....'.........' ,...t-.111!1...* ---' ,....-..:1!1.,.* .''..'.'.'....'
:,11::-* (::11...* .:i!:,;j'' )'.1Ik!!!!IIj@;'' $' jiii,d .j' j'y' rjll::-d -;!!i!!I;7' liid (:!IIIEEE)-.,' '1IE::::,h' 11!::::((E:27'-* .1!::::E2E:;71'* (()1(..* -1,;:115E;12-* iii.d (411:.2,1;.* (' y' .14:::E2(2:;?1,* .-;i!iiiii;7' ::1IEi7'' qlllEd )' (lli::llk.d ':IIii''' 11!:::E(:(:271,* tltd .;i!jiiii;7' ((::1i' t':lllq':ll:r':llk.d ((:11i' (2::11* ((:),' ((()1h..' ((:)1i' (4:)1h* ((::)p' ((::)i' ..'IlE()' .'j'(tf r'-1!::::1h:!'' .,,,' -112!1,11-.* jjy:::,,d -j,::,k-' ,'.i!!!::;1''
.' Chapter 7 DiEusions
260
.112:;.
!11::k.
dl;::::p,l
-jj1!::;;,.
' dlEEEEEEllii
.;jjiiiii;7'
;:;;i!:::t. :II
11::(. :,,. -;:;!!::r'
l!llEq' ':1IiE7''
11::::)1.::. :.I
.;1!!E11--'' 11!::::: 11:::::,,1 !1E::::E
4t
:
:II
also follows easily 'rhesecondthenequality then this is the same n d r:ll:r':ll:r'll-.'1It:k. ()1iy'r1h,. .111:k,
'112:k-
:jiisl. 11EEjj:,. )ll:!-. .!II,j:1Ik. (II:':1Ik. IIIEEE:::
',!i.;''i.;p-'
41:::::.
':jlii'''lll::'.
-
.
. EE ( .. . y.( j jg:: yy y j yy E ErqEil,,jli,rjjiijljrllylyyk,jjyjyji.,y,,g :
:.
.
g
-
qj k g;. ( j j .y jj y r
:
. .
:
: :
.
.
.j . .. jj gjjj g Ei EE. Ei. i l g q E(r q !.. !y .!.jjyjjyyqyyyiy.yy!y(yyyyyyyjyyyg.jyy. E E j;.iEji. j 'y @ !j j .j .. . .. q:. q y . yjqy y g j ..y y j y ( . .q. yyjE-yyjjyjj. jj,jj;yy y g j.yjy y j.y...g.j.....yyy..y.y..y 'E. .'jEjy ; j ' E y yyj .Ejy E.q . E.! i . ! i . ( q . .E..j...(.ri....;).q.qiy..)yj yyjjj.,(jyj.....-jyjy.yqj ..Er y.. j . ;yy!j(!yqjjyj..;;yjyy. y.jj.,yj..,,.j, y.. (yy ( .. y j.jj.jjy.y..jy.yyyg..y ' .. ..yy..q. gjj;yyjgjy y j . .'. . Ej..iyjj;jyjy j j. j y.jgy.j. . . jjjyjjjyg . . ... y. j.q!qjjqgyyyyyjjjy yyijgyjjjyyjj..j'jj.jy.;j.ygy.y.jy.y. ( ; ;yrr.iqi,rit,ykj,yyjygyl.yk.,y.jyj,,.,gy,yjj . . yg .(r..jyjyyj)yyyqyjjkjy jjjg j jy..j-jyyyj.jy. y . j y y y j y y j yyy j yyyjj y ,l,E,E,iyq.ljqgyjyyyy,jjyj,jjjjyyjgj,jy,yyy,yj . . yjj yjjj jy yjyj ygyyy. y.. yyy.jyyjEy jgjyjy . .,'.'.:.,,(,..;y.).i(,.,p.,i..,.,jpii,,.,y;,,k.,..ykyy,,,.
y . . y y, . )' j)j y. jy ... ))ikyy)(.-(y .j . -
-
-. . .- -
-
E-
. . -
-
-
-
. -- -
.
:
.
.
.
.
y g. - -.
.
-
--
-
. -
- -
-.
.
.
.
E -(-'.
-
.
.
.. E. -
j(j. j.
:. : y . :y . - . - -g --y -; - . . y---. - E .
-
-. - . .. . E
.
,
,
'
E (;E .. . . !
!
-y..
y
.
.-.
E..
.
:
E. y yy
. y
-
Pv'z
making
6
'p.
-
-y
y yy y
jg
..
-
E!
.
. y .. -
.
.
.
-
-y
(
-
.
y
.y
.)
-
- -
-
,
-
.
-
E .. - .
-- -
-.E
. . -.E . . E-.E j E..EE. .. yy. yy . -: y: . ... E E.E E -yy. . : y -. ..g . gk gy y g j . g g g.j.j j .g g . ig g. -y y y y j. g -i -
.
E
.
-
.
- . .
'g . .g
-
.: , -. g . .. . .-
.
q
-
.
,
.: - q ! .y. . E . . . . ..y .. -
.
.
-
-
-
E.
.
.
.
-
y
y
-.
'
'
.
y...
-
EE
-
y. y
. ,
.
.
12
,
..
.
g
j j .
-.
-
.
-
(0: l
-
-
-
-
'Ey,Erlqy,yyjyp.l.qijyjijjpjjjj,yyiyjyyqjj,yr,yjj,jjyy,yjyjy,qy.,ly,qjy
. . ..
j jjy,
y. ;-.
-
.
.
-
--
-
.
.
-
- - --
-
-
- - -
. .
- .
.
.
y j: ( ) rjy j jj j j rr . y l (!y.(y . .. rjyjy . j .jj. .j.j)j.y jyjy . . j.jyyjyjj.;jjyj'y jj j j yy.y . .jj ;g.jyjyjjjyjjyy .. yjjij:jyjqi(jj(jyy j ) j yy jj yy y.jy;yjy.yj . j.j
j
.
p' is the transition probability Lebesgue meaure on B4 but
.j.jg y.;yyj,yyyj. y.yj gyj j. yyyyyjyjyjgyjy y..E.jjlyjyjjjjijlyjjygljjjjyyjjgk,,ijyjyyEyjyq,jjjyjyjyyjyjjyjyyyjEyyyyj . . y q( (y.,Eiyylty,ylljyiyyy,jjjij.jq.yj/,ji,..,yyjjyy,yjjy,ljjggyglyyyjlklgl,jyyj,,,y,, ....jj.j)(y ...
. . . . - ...
,jj,.,,,:.,,,-*.
.
(4.2),and
in
. E
y - . .
j ,.
y
. y y. ' i. i . E
-
.
.
-
as
vk
-
-y -. - . .
i
y. .
'
.
::.z
.
-
..,jj:::,:,
=
.
. .
.
Xn and R are constructed roof bserve tltaf if and S) 1 then a'Y Xa is obtained from Xo according to
'y
-
.. E.
E
.
.,j,,,
(,%S4 then .G/(#s) on
E. : . EE E i. EE : E E.(yr,.)E;ty)jEij,.j(,;gy.;)'i)y'(!jrj,gyj : yy
. .
-
E
.
.,,jr,,,,
eS
by . ... yj..) jyi(. a transition j j '.'rqy!(lik;yjjryjt(yj;y)y.jyy:((y.r(y.j..;.(.jj. ;r...,,,.Ir.r(,;y,.,,r!;j.y,j)I,qjrq(,yyyyyyyyj . (. ry.. . (y j y c1 , r .. '...E'.((gry..j).( (yjj:jgijqg.j',yjgE..y.y.j...y..j . yjy j.. . . Before deyloping the theory we give one example to explain why some of E) (yj gq;j j! ly yy;jj (yjyy .jq.jjyyyy ,...,..j,.y,.,rj.(..,,,,,.j,.,y,.,,....,.y.y.,,,.,.,..,,,.y.. . )y;;.y(ry .;yy.jyg j..j.j.. h t t a temnts to come will be messy. e s i . i !g)jjyjjjrjyjjj jy jg y.. ...yyj.j.y.jjyjgjjgj;yjjjjgy j.jy .yy.yjjgj.jgjjjjgygg j y..jy y j g g j . ..y.E. .(yj.yyjyjyjyy iy'Eyjr.;;yy(.jjjIg)jlj!.yyy .. i). Exnmple 4.2. Perverted Brownian motion. y. '.:jyj. i)j(j;y For z that is not an integer gj.jj .;;;.(j..j.jyqyj.yyy.jj.j E. E ( k :jy q ; ;.:g ) . ..:l j . j j q(,.y.j.y. . k 2, let ptz, be the transition probability of a one dipensional Brownian .....;. y.qyyy.;yy...;yyj,yjjyj)yjj.gjjy.jyyyj...jyjyy.y.yyj.yy.yy y;.q. ;!ipyq,y motion. When z k 2 is an integer 1et yyjyrjj. ,y.jy.yyy(.y. i yy .r.. ,..yj.j.. '..'j'r.;)(.i.!(;;g)::)yyr)y!;;.rj;yry yjj..j r yjgjy j .y yjy.yy j . y y . j . j . y .;'('.;.y;y!jy!yj;yy(yjj;j(jqj.j.y.g!y.j.y.j.y... ' ' . j( F(21, + 1l) = 1 z-2 . . ... (.j( ; j, y y . . ... .jj .jy(jjjj.jjj .j.j. j. jj gy.-y. . i .jy (. j j y y j.jjy j.g ;y jjjjyjyjyjr;jyjy.yjjjjyj.y jyjjjy .y ..,y,,..,.y.y,y..jj.,yyj.j.y.y,y,j.j;,yjyyyyyjyyy.jyg.ygyy.y,.,,..jy ptz, Cj = z-2IC r) 1) if z + 1 C yg j.jjgyy.gy ........',.!.l..;q.,.,.,j,!jr.yI,yj(!yjqy.y.,,,;.y,j,..yjj;yr,y....(........, yy ( ( j(;j.;jjy j,y j gyj E . Ey y i iy y
'
..'.. j .g
.
=
. - ..
.
.
-
.
-
.
Eplxnt
:
-
:
-
.
'
mobabilitymemsure
. -.
z
1 dp
.
- -.- -
.
for
-
.
E-
. .
=
g
--
-
-ty
.
.
:
..
-
If p is a
/(a)
J(z)
:
,
.
.
(4.3) Lemma.
=
,
,
.
'
-
J(z)
:
-
. .
'
-- .. -- .
.
-
.
.
--. !
yj .. j ' j. ... .gyjyy . .. j . .. yyyy. y.y y
il:EE!!EEE!i rillq'''ll:::::,,-
.
-
,j.jj.yy,j.
. -
.
.. .
-
-
).
.
.
' : . .
.
-
y
-
-
:
.
.
.
.
:
.
.
(4.2) shows that there is an intimate relationship between the asymptpttc behavior of Xn ad of Xn To quantify this we need a deflnition. lf f is a bounded measprable function on S, 1et j' = ?g/', i.e.,
qq ,rqyqylyqiyiljjiEjiyjyljyjljl'yjyjy,yjylj.jEyyyrjy.yyyijy : ( jy y. y
(Illi'''lll::::l,11::EE!EEE!E
.
.
... r..).!jj y.j y yjj. j.jjj jj.jjg
i !EE E (EikgjyqrrylEgrqjiryjjjgjjyqyy,yjyy ;. . y - . y y.. . E.' ( ( ( . g. . q : y,jEyyqjlj.jjEpjilyjj,yljy.yryyyy,, .( y: yy yyy
- ' . ..
.
j
:
tnstead
#at.Xk
of a Harris chain
jjrjjgygjjyjyggjjjrjjjyjjyjjj,jyjyygjjgyjyjggyrjjjjjjyjjyyyyjyjyyyjyjg,jj g g = .. . g. jyyyy gy . : . : .EE.EE.i(.: Ei-ig! : y E!r E . g qjjj gy . !((.i.(E.!(y g ;y ..yrq -y y.y- gy .j k.g . . ' (. ' j'g(Eg!....j.jggE;.gg.!.j.;Ej.j:jy.(jjq.y.gy.rjjq y . y y j . .. . y gi.j ;y;... jg.y E .. - .(q .E y. yr( y-yy - .y .gE.E.g ;.y jjy.j((..q.- il.jlj..r.l j ( j. j y-y gyq;yjgjgyj :..tyt jjjy j . yjg .ygtsin -. E i. Ei E -. y(. (,;jy- -!y j y y y y- y.yg. .(....,,q...ygyyjy,jgjyyjj,,j,jjtyjjjjjyy.y.yygj.j g . y j y yaaaayygyyayyyyyyyy.yyyayygyyyyyyyy.yyyyyyyyyyyyayayyyyyyyjyyyyyyyyyayyyyayyyyyayy g
.
.
.
.
..
.
n
+ 2 for
/
(takeW
all n) >
=
B
=
(0,1),
p
z
0
-
:
:
-
.
.j
..
.
. .
,
:
.
.
.
(2:1,1 ,. dennition.
'
.
.
.
.
.
.
.
.
.t
-y
-
. .
-
t'y jj.j.jjgyj..jj
-
from the ms one trnsition
.1121,.
.11@:;.
j.
y
.
11::f1112.
(:11(::111.. ilii!ll il!ii!lk,. Tllr:':ll.. (:11::7.
y
-
y
.
=
-
:
. y
-
-
,
=
!::::::::::::;.
.
. .
.... gy.. jj(. j.j jgg j . . . (tg'(!
Before giving the proof we would like to remind the reader that ma- . .,,y. .:y, on the zezt, t.e., . zn . .oue nrqv cuww sures , multiply the transition probability . accorclh: . want t show pr/ to anu tneh yp. lf we frst make a transition S O C one according to #j this amollllts to OI1e tllaflsititm accordill Eiil. ;11!1!1k-. :12!511--. IIE!E!;. :1!E5,1k-.:1EiE5;. '1It:k, (::!,p?,''' 'lliF:'llir''.liis!r. ...ki!!EE(:(;:1''
:
-
.
-
E.
.
.;j!r::;p,.
11::(( 11::4) ::;iIr:2:'
- .
y
,;jjqiiji;'
il::iill;.
!
.
.
.
. :
.EIIii''' 11::E!j4E44 jl:))i (jii). ..-1Iq(' tll)ji.. 14::() 11::4) -:1!:-;,-((:1,1 (q:1p.
Proof
-
-
:
.
.
:j,
.:10:,;,1
:. :
.
11::4* :::i1!2::-
11::4( ':11:':11..
,;jijiji;:
':i!:-;pI
.
.
.
''
--,-
-,-
:
-
-
,1:::::)1-
:211r:::111::::11k.. :111:2::11r::2111.. 414:(). (q:),1
.
-
-
fen
2111,. .
:.E :
-
Here and in what follows, we will reserve J1 and B for the special sets that elements . will of S. We ocur in the deflnition and use C and D for generic simplify notatiou by writing #(z, c,) instead o >(z, f.)), ?z(a) O o 11::(( 41::::)1.::. -ij1!L,-iI:. 'dI@:;. ((:1,1 Our first step is to prove three technical lemmas that will help us cafry out the proofs below. Deflne a transition probability )) by
.
.
:
-
-
,
.
=
tj::ll. ((:111
11::(
..
:
.
,.
-,41t:::;,-
:::::::::::::::
.
.
.1(::::(2E:;?.-
,:jjji@2i;7'
:::::::::::::::
:
- :.
.
-
-
:II
::::::2:::::::.
-
.. . -.. : : y .. : . .. : : .
.
:
;:;i1!::)'
.;;@4r:;;,-
::::::::::::::E
.
.
':lliir''dl:::::),, 11::().1(:::EE((::71il:EEE!i!!i (2:),1 .. ((11i.' 11::44 !1(:q(
.;;i!ii2ii;'
-.
-
.
-. .. : . . :y
=
!II
.. -.
:
. : .
,'
..!!iiEi;r'' 11::(( '2111((( 11::EEEEEEE1
-
:
--i-i-hi-hit. -I,-i- -.---t-h------p-i-ti(ii),,, -u---iu---t-z--p--ttbeing which corresponds such point below) in B manufacture to a a (called rlll;;. occmsionally :1!111;.'1Ii::::,,. out needed plan notation this iiii. is '1IL,(:I1,. The to carry ''1i';.'ik;,'' obnoxious but we hope these words of wisdom will help the reader stay focused . . on the simple ideas that underlie these developments. with chain cha will construct Markov Xa Given a Harris a on susj, we (S, B U (a) : transition probability # on (,V,Sj where ,V S U (aJ and J' B e d). Thinking of a ms corresponding to being on B with distribution p, we .. . deflne the modifed transition probability as follows:
i-
y( . lj$jj4jy14j)j)jjy4j ..k... y .. Section 7.4 Harris Chains 261 l'( y (. ( y . .y. . (4.2) Lemma. Let 'F,zbe an inhomogeneous Markov chain with p2k z? and l nk-j.l zp #. Then Xa = 'Far,is a Markov chain with tr>nsition probability # and Xn = F2s+z is a Markov chain with transition probability p. (, .
-
Afarkov Processes
us
' :.E' : .-
! (' 'EE.EE(E@ ..'...(gj':;jjij). (' .; E' ;yqyry . yy ry y,g .jy y ( ' .. Eg.Ey j j j(jj.jg y jjyg.yyy. j.jq..jy..j. . .((E. .EEj;.(jjjy . y .; yjjyj.j;jg !y.qy y ; jy j....y..yg.jjjy.....y.j.yy. (y.;'..g. yy;ijj)y qiiEljlyyl'.yil,'lyElliyyjyjjjj,E,yjjgyljyq.Ejyryq y j ... .. .... yy(jjjjj.qy...jy.. j..jy.. ..y. i (E(Ej) :. ....(.(E-.y;..E. jyiE:;;ErIyyj.;i.(.y..jyqy;yy..Egy.jy..y... yy . j y y ) ',......,..,,..,...,,,.r,.,,,,,,;,,,,,,,,,,, jj . .j r j . ..j jjjyrjjt. .y j.y.j.:.y . i y.. E; E E j E ( ( ( .,'E'E'Fi:El(:',k.iE,i;jij)krijyjjjyy,,y(,,! ...( .yj.yy(jjyjy;jj)yyjjy. jjjj..(y.(.g ( ( ;( .r,!','iii,;;':;l)l'qi,k,(y.jj.:,j',!.(.,((. . ... j ).( jg(yjijjyj jyj.y.y.y.j .. yyj(.jy yy yEEEy('Ej;l((y,yr.rE).y),yjjEyjq,jy yjjgyy y y.yjyjjyy.jjy j.jyy.j....y.... yy .y.y.y.. jj'j . g y q j . y .. y y EE,'llqijlylEyrirElkyjyjy,,jyjyrylyyj.,,ljyyjlr,yyyyjy jyjy j gy . .....gjr.gy;gyjyyjj.. gy.y..yyg
.
-
-
.
In words, if # acts to p since n
according
.
-
I can sympathize with the reader who thinks that such crazy chains will not arise xpplicationsj'' but it seems easier (andbetter) to adapt the theory to include them than to modify the ssumptions to exclude them.
jyyyyy y;.;.. gyy yyi.y..jl,..irjr'jiiyljyjy,jpyjjjy,,y,ryriyylyjjyyyry,jjj,,yyyj,jy,y,g.,yyyylyjj,yjy,,,y .jyjjjkjjjjyj yyyj y.y .
..... t'-.
-
.
-
E
.
g --; . .
-
-
-
'','.r')'i''.::)''.!.;.'i..ii;,,;y(yjyqjg)yj..,yj
jjj jj j ! jg j(jjjykjjjy.jy r..r)j!yyjjjj('. 11:::::::::11. jj.yy..yjyj.jyyyjj.. .. . j)y j .. ; .jj g . .. ..;.j)jjjjjy j j . y j g ..yjy y. .jj .yyj.jy..j.y j y:jy..yjyy jjryssyy. j..(;j.;.j;kjj(yjk . j jj.jy .yj j...j . j jjj.jj kjj jj j y . y y)y yy . .. .i.i ;..jjy j . j y.. y E .- j. q. ;-j - g ; ;q y E.! (;- ,- (;y - jyy y j gy ( y - gj y . y y yy yjyjy; j yjy y ;yy .yy.g. y (ii . y y g.y.yy. ...( :: kgiy.(i((gi jj: y.j ; y y y kyk; j(yyyy . . y y y y.j y. . :
::
:
:
.
.
,
,
,
-
.
.
a. Recurrence
and
translepc
g
-
-
,
, .
.
-
-
.
.
-
.....,qy....,jq;.,.-p,.,t..ji,i..(;(j.)..jj.y,j.!(gr,,,.,,,).yyjgj,..j,..j...;).g.,y.
R
=
We begin with the dichotomy between recurrence and transience. Let a). inftn k 1 : Xn lf .&(S < x) = 1 then we call the chain =
. .... rk.)qy(rj'y (y) '. . rjj yt(-. ( j yy ...r....y.....;r2.q,j!..j.j;.j.y;.!jyjj!j,.jyyy.j.jj.,yjjy.j;.jj.jj,.jjjyj,.,.j,y,...yjyyq( rtyyjjjyjjjjyyj.jyjj.rylyyyyj,,lryyj.yl.j....y.
' ' (. y(Ey y jyjjEy E ..E.j' jjjE y; l .jgjrjjg::!.j7.jj. lp:gyEy ... Ej' ;jg j. yg.y..yj.( (,;,rqjE.!7r)yty)j!I)j,;.;yjgkj.rjjjj;,;(jyyqq.y..j.y,j,jkjjj.y.(j.qjq:. j y!gk);))'. .......;.(.q..E...:,j(j.y,,.j.yj.j.jjjjjy.j.yjy.j.yj,.j,j.yyjrjyjyy,j.yj..y,.y;,.yj.y.y. !
:
:
:
-
:
:
:
,
-
:
.
y
.
.
. : -- -
y y,
g
-
.ti;.E
rjjk (. yyy .: ....t..tE..j.jt) ..,.yq.jj...,,.Ek.j.j,);..y.jj.,!tj!j.jy.q.jrjjyy(yy,,.,.jg.j.jy,y,y.,.y,.,r.,j..,.j.-,y,....
.( ;; il1l!li l E : . :i ..
.
.
i @ E iE: .
. ....
262
Captez 7 Dihsions
as Afarov Processes
Secton 7.4 Hazjs Cllains
otherwise we call it trnndent. Let Sz R and for k k 2, let recurrent, Th strong Rk = inftn > Sk-l : Xa GJbe the time of the th return to = #tz45 < xl < x) Markov property implies Pllk 1 so #.lxk a i.o.) in the recurrent case and is 0 in the transient case. From this the next result follows emsily. =
=
.
=
=
(4.4) Theorem. Let (C) Ea 2-n#n(a, C). Inx)the recuzrent cmse if (C) #.IXS 1. < then 6 C i.o.) = 1. For a.e. z, =
263
0 then the deqnition of implies .&(Xa E C4 = 0, so 0 for a11 n, and pC) = 0. We next check that # is Izet Gk,&= (z : #(z, a) k J. Let Tc 0 and 1et Ta = inftm k Tn-1 + k : Xm e Gktsj. The deqnition of Gk,&implies Proof
If
(C)
#.IXk e C, R J-finite.
=
> n)
=
=
.plTs
nlT'a-l
0 for some n then =
-
jpfpy,
)
=
a i.o.)
K
x)
#n(a, dz).Pa)(S
1. If we take .A B (0) this is a Hxrri chain by Example 4.1. Since p''40, 0) 1 it is recurrent. =
)(2 Pa(Xn6 dy, A
=
=
(a).
> n)#(p, a)
n=0
Ea(a = n=0
=
=
jp-k e
.
n=0
CAss
1
u azg
> = E a(-1k+,6 c, R
.,z
=
.&(Xm
S
=
=
n + 1)
1
=
=
#(a)
=
b. Stationary
where in the last two
C
measures
(4.5) Theorem.
Let R
=
inftn
k 1 : Xa
R-1
#(c) s =
deines a (4.4). p
J-flnite =
pr is
stationary a
J-flnite
a).
In the recurrent
cmse,
=
5-7
1(x-Ec)
n=0
=
=
e y'.7at-1k
c,R
> n)
n=0
measure for #, with # 0). By assumption Uask = S. If :(D) > p(a)p(D) for some D, then LD n&) > p(a)#(D n&) for some n and it follows that p(&) > #(a), a contradiction. (:l (4.5) and (4.7)show that a recurrent Harris chain has p, J-finite stationary measure that is unique up to constant multiples. The next result, which goes in the other direction, will be usful in the proof of the convergence theorem and in checking recurrence of concrete examples.
(t
.
8) Theorem.
If there is
is recrrent. Proof
Let #
which
'm#,
=
(i)of
a stationary
k
probability
a stationary
distribution then the
distributipn by
(4.6),and
chain
note that
the deflnition of a Harris chain implies #(al $ 0. Suppose that the part chainis transient, i.e., Pll < x) q < 1. If Rk is the time of the 1th return =
C)
=
p(&)#(&, C4 +
s-(0)
dylpy, C)
then PRk
< x)
=
qk so if z
#
Cr
Using the last identity to replace ptdpl
on
the right-hand
side we
Js-(.) X''d#)X#' + /s-f.)
p(&)
'7(&)#(a'*)#(#'
C) +
p(a)#(a,
=
have
Cr
E=
1(# a
n=0
C)
t:e (0
P.lRk
=
=.)
-E e
=
S
a=0
#(t)
1
1
-
is a stationary
x
1(#a=.) n=0
''
=
=
cxl
q
distribu-
.1::11111.* 41225:1:* :11:;;* .15515,:* !:::::::::::::' :,,' ytd ::;:::::::;:::' ::;:1.:::,* !,,' qjd ::::::::::::::' :,i' :::;1!:::,* :,,' :.,' i::;.!::,?d yyjjyjsy.d i:::'',' dll::!:::d :11:;;* 11Eir2' ((IIirr11L.' 111::)* ;::;1!::)-* ::1,..:;1,.':4,:.* 114:)* 114::4* .j' j'p' ::11:.:,1k.* ;11!E,11-,* :41r::-* '1Ik.j:lI.' :lljii!lliiisii,d '!It,(:11.' ::11..* ::::4!:* (:11..* g' (' .1Eii5!,' CIIES!Ik,-:E!!!I;X '1Ii.:11k.' (iii,d E:ll:::llt,d ;1E!5!1;' ;ll:r'd jl'jtd ::...:1111:2.'* :1Ir!!'' 11EEE!,t' .1EEii!':' :Il(:!'-' :)11r:::111:::111..* 11IEE,:,' pj.y' yyd E:I(,.' j';11!E511--* 1j' y' $' )' j'(('.::;1.:::,* (' t';::;1!:::)* jjd j'y'(jIl;::,,' jjf
y' )' t't' (211:.:)1k.* dl!!::E:El:;?.'d .1!::::E2222?.-* dll!:E::EE:;?.,d dl!!:E:llE:;?.rd (111:::11..* ,;;jiijii;r'''':'l'' (:11(2q* ill:iilll.d 11(E2E11,* IIE!E!,;X iii.d 11(::E:1.* fiit.d y' .;;idriii;;' jjd t' ((:)1h* ((:)1p' ill:E)d
.(' .-.' ,;;!iiiii7' .-'' ..-;i!!::iii!!:' '-'' '!:;':.:'''d .d ':iiiiii;k'' '''''''''' .' -(' ;' 111::::2* 411555:2* llrrr:rlir!'d :,,' :::::::::::' g' :,,' ('y' ,'y' 111:::::* kll::i!!,d
.' '.!r::t!!l
.1::1111:.
(4.9) Lemma.
There is
.
.
(5.4)of
For a proof see
(4.10) Tieorem. :iii.
''lIi:::),I
(iii,. '1It:;,
an mp < x
''''
E
ty) >
0 for m
f
111::::)14! ill::::ll,l
In view
of
k
.
(4.8)
) ... y. ..
.
-'.
a
#
.
.
. .
11:::::1,.
''!!',;'''
.. .
g'#
=
% (-)11
')
:5ii,.:jiil;-
Let 52
-
s
.
=
m=1
-...
=
114::::),111(i2::
dl::ql 11::(( -:;:i!:::,'
i:iiili (:11::-.
j. j. )' jyj j i.
y
q!
j .. .jj...jjjjjyyyy . .. i ! .i .' ' E.E'i'.i.E(1.y'yj.).;y..;grIryjjjjj!jqr!jy.;yy;i .j.g.yg; i.. ' ; ,r r; ..'y..'.j......;yyg...ryyyyy(y(yy.y(jqjgry;jyy;yjj.jgy.gj;y. ..(....jqjryq.y-y..y.q.jg. .. ....(.'..i...y.y.'yyE'y.jy;..y'.yyyy!-yyjy.yyqy.yyy;.y.yy.yy. yyjyyyyyy.y.yy '' qjryj. j)j jy)j (.;(gy ..'.@'i.E(.)i('gi.r,;r!))),j)j)',',,.!,.., ..-.-. --y-.y.j.--jjy. .yg.tyjt, -.;- , ;; -.), ; ! y yyk(. yjy.r-.-..r .y;;j y .
E
j
-.
-
-. E - Ey E .: . . !..y - .y! y --.EE.i;-- . - y . . -. - -- . E ! . . . y- y . y y Eyy.. .. yy-. . y -y yy -j y . EEE . Eyy y .
-
.
.
.
.
(C1 x C%)
'j
-. -y
- -
';
.
-
-
*(C);'i(C)
=
.
'
-
-
:
:
.
.
.
-
-
.
j.j -
-
-
. -
-
.
--
-
deflnes a stationary probability distribution for #, and the desired conclusion tollows from (4.8). (Xs '.Q)and ' To ptov the convergence theorem now we will write Zconsider Zn with initial distribution & x #. That is Xu t:v with probability 1 and hs the stationary distribution #. Let T inftn : Z-rt (a,a)J R(4.8), (4.7),and (4.5)imply i x # n) y y. . jtyjjyyqj yj j.. ; ! ; . j;rj( jl .....r(y(jyi...!j.j!yy)jyj(y;jjy.y);yj,.jjyy.g.yjy.jyyyy;r;(yyy;yj..qj lnterchanglg the roles of X and F we have j y .y . ....'..j.q..yEqj.)(!-j(Ej.r.jyqjy;;i.yjljj)y'. yyjjy ..yyj.jy . .jljjjyjy ..jyj.gj.y .. ;. j'i y p !...'... j ( ( ( j , j q. ; .y ) . y j k y ) j. . j . j .. j.jyj. -.yjj .'...qrqjg...y.r.jjyyijy(jq;i..jyjjyjy!jjjyjygy)yryE.yyq,jjy!(yj.jjy. Pku .ryyjj t'y y!rjj . J3tt e j .. . . j)j yjj,jjy jy e c) + PCT > n) . . .jjy)y j y y . .(i''';;,','E,i.!'7'i:i.'i!)II;!jjjj(Ejg.j .. y. yjj.y. q ))! . q jyjyjjyj y q yj(r;,jy y y y y .j . jryjjyyyj.j.y jjyjyjytgj . yjyjjjji,ltjijjjyyyjj,jyjjjjj,jjr,yj .. and it follows that ;(y j ) j r g j j j. . j .jig y . yyjy j . yjj.j.jj y y j y l ( . ; y ( . . ..'..q'.. y(j.j.y(j(jy.j!!jjjyjy!rjy,y.. ;jyyyyj.j .y y y);jjyj)j((y.j,yyj; jj.jy.yjj.jy.yg .yj .. .'...('.EE.y.yy;.y..y.ji,pjjyg;ij;y'j ;k yy. yy jjjyy;yyjy-jjy (j.y jyykjiyjj.jjrjyjyyj. .. j j!g y j y j'y y y j gjj y ,y .........j.;.y;y(;.jjyjyEy'; y. .. yjy yyj;y , yy g y. jy yjyg yjyjy ;j . y p g .!.q.y)(jqj!;E-kl'y y yyjjy.y yy y g y . y ,j;( jij1. y j j , g;gjyyyk. . . .....(...!)Ej)iij'('('5 .iiyi j!j..y... .yj!yy.j;;).yy , ;rk .y!)jj-gryi j;.j!!j.,yjy.y.;j .
-
(2:11.
::)),'.:::: (:11:,. :::2!,:
is recurrrnt. To do this, ditributio fr #, ahd th two
=
.
,
'-1i,;'Ii';,''
.;;@iiiii;'
chal
'#
y
. .
..
0
:. .
.
..
..
0
..jy,.y.tjy.y
E .y @. . .... - -.gi q.q EE.y ... y y . ..s,gyss.yjyjsjsyjjgyjyyyysygyyyjg.jysgygjjysjy,sgyy
-
>
a) >
:
.
..
(111.. (211:':11-. ,t (41144::),! S x S. Deflne a transition probability
:jisl;. .1:::::)1-
..
(a,a)
:
:
,
:
.11.1k.
'!!i',p,'.
a)#N+M(a, k #f'(z1,
y g g g gjgy yyg ,E'.lp.Ey,,,l,ijlEyiiyr,jylig,qjjjjkyig,jqggyyylyy,j.y.y,,yqj,yy,, . yg yy.,.g y
: - .
.,,;i!!::E7;1I'',:;,I::,.
-!%(.?
a)
#A-(z1, a)#f'+3T
y..,.jy..yy.j,y.ygy.jy,jj.jyyjyjjj.y..yy
.
t:v
k
,
, ..,.
then
dlisir,.
II#''(z,
.
y
y
... .
.
'#
a)
>A-+T'+M(za,
'Cl!E;')..;'yj;jiyj(j!jEiyEyy!!qjE.y;((,.yyr.y
.
''''''''''''''i!'-'.
:11'':1/':)1,.
''!i.F''ik'-' 11iEiE!t: ljiii!r.
y
:
.
11::4)
((:1)1
.
: . : g
za), (a, a)) > 0. and hence jN((zl ..( .).q'.,q(.!:(j.',j.';.',l;jy(rjyj,gyygjy,.,,.j.,,j.. j((.y...,(.(....,.. The next step is shpw th,t the product .gjj jgy jss,jysjyysjssj y....... yjy zr# is a stationary hot that (4.6)iiriplies .. ..jyy.yyjyjjjy.y jj... ; !.j) COOr independently m ove so .. .qj.y!rj-(.k.;j.jyjy;.yyj.yrjj k)(j..g.jy. dinates y yj y .
(T:)1' 11. 11.111:E( 11::E 11. 111 (::11. z
.
:i y:
-
,
-''
is to note that if
:
....y..,y.,.,y.y.y..;yjjjj.jygjg.;(yyyy.j.y.y,,
.
to qhow that if
#A-+f'+3T(z1,
:
g q r j; E .... j.g.j.y...j.jjyj;jjjjjjjjjjj.j.jj j.jy.....j
-
.
-
. .'..i..i.,.!..rq.:(j....q,,.jyr..rj,q.j(.y..yy. . -
.
.
.;i!ii!ii;7t''''''.'
Our second reduction
y .
:
..
.
. .
'''''''''''''.
: .j
-
..
. .... with stationary
-
(4.6)it suces '''
:
.
'.
..
. .
j;.j.,).y,.(.j .
q(q;.yy;;.y,yjqr(j;;y,;.Ejy(,kyjyyyj
E(
'.j.jyj.j..jjig.i-Egjgg.yy.. E. !qj . . g...E(jqi.y.jjqjjyj.y.g(.-jj.j.yy.y.j.y.j (......,..;...g.(yy(yj;.jjqj)..(j.;.!.y,r.j . (' ..'E';yj.g.y.EyE. yygjjq j.y,,.yy.. .. . ;... j.j. .. yg.y. jy
'''':'''''''''i'k''.
'-'
.
.
.
i!E'.r.'.).;jjj(iyEyE..j(.qjyij.4;.yyj,.
...
mp.
14::::::,11::::::),-
.'''''.'''',.'i!,--.
:
.g j jy;,,, .j ..Ey,'.j'.y.j;y;,j,gjiiy,,r,j,jy,,,y,,)y,;(,jyj,yyyyyqyjy(;yy;,.yyyyy
.
.
,. apd
(
.
.
E: . . g.y . .
. Remark. Here 1111 denotes the total variation distance between the measures. (4.4) guarantees th.t a.e. z satis:es thehyythesis, while (4.5) (4.7), d (4.8) imply g' is absolutely continuous with respect to P
=
.
'iiiiy,lijqllryq.jjlqqiliyjjrriqi,jl
-
.
-
.
noW ready to pove .
(1995).We are
. 11. 11. (::11. 11. 11::E( (::),' 111 . ''''''''''''''
:
:
.
411@:k-
:'''':'.'
-
-
(y . 'E;y.'qj',r'.'E,g:,.'jiI',ijE!.i, j. .g . j.gj j yy r..y 'jt, .
.
. :
.1::::::,..::::::),.::::::::::::::: '
,;i!r::;;''
-
'
,EE'(@i)(qE!Eiy!@,i(;l'1''i),(El.(E(E,@y,.iyq(r'g
''''
Let Xn be an aperiodic recurrent Harris chain r:lliir'' Elll:::li,. !;p,'';I!;. (:111,.. (111:':11.. ,.-;i!!:E:::'I'';::.!:,,.
.--.
.
y
.
.--1:::::::::q
.::;,-117-..
.1:::::),1
:
:g r.. yg. .y.. . .. E. E !. - . .. ;q. yq . y : : q (' y. y .-' y -' i. - . - - ! . i. -(qqyyjjjjyjtjjjj ! ...j.y(,;,; .. E- yyyg y, . .
q.;:ij.k .. ....,.,..E.y,y;E..E(.irj.,.!jy(.E4j..E,r. y
Chapter 5 in Durrett
.,.;i!!::iii!!:.
.111:;-
. -
.
-
that #*'ta,
so
. -
. . . .
.y.
-
=
.
-
-
-
Section 7.4 Harris Chains 267 lg.liIj j jjj .' . ;''.lyyi'kyjyj:jjj1yiy..y. yj...y...y j... ' . ')ik )(.y.lj((. &)) and We claim that the product chain is a Harris chain with .j....:)..) . .yjyyjyj.. ((a, j...y:jy.jy..,.j.s;jssssj. y.gy h .. .t yjj = check B x B. Clearly (ii)is satisqed. To . );.y,jt.j( (i) we need to show that for all (z1., z2) there is an N so that N((z1, za), (a,&)) > 0. To prove this lt ff and (j . j jk!j.jjjj(j(yj jj ( g .. L be such that #X(zl, &) > 0 and #f'(z2, (z) > 0, let M k ma, the constant in ti ,.,r..,.,...,:,,,,.i.,.,,,,,.,.,,k;.,,. E (4.9), and take N = ff + + M. From the deflnitions it follows that ..
.
11(::::).1
.
y
(. . ir...r(j. yg. E.
.
.
-
-
-
'
:
-
.
:
-
.
If I is a set of positive integers, we let g.c.d.(.0 be the greatest common divisor of the elements of f We say that a recurrent Harris chain Xn is aperlodlc if g.c.d.((n k 1 : p (a,a) > 0)) 1. This occurs, for example, if we can = of take B in the deqnition for then pta, a) > 0. A well known consequence :5ii.. (iii.. (::!i,?'''' ;1!!5111-- '1Ij2;.4
i. EEE.
.
.
. --
'11It;:.
'll.r'':l.:.
-
i
:
.: .. . -
-
(lllizErll... .11::::),.-
Onverg .111::::::::::.
c.
W
E: .E!' E: . .
' kk;y! !y . y .'..-.. l.)y)j.
j;jq'.;y.rj.j j.y.y'.).yy;iy ..j..y( EE(ij';jj@ . . .@(yij(!;jj;yy.jjgjy .!. jrE.j ..jEj.yjyjjjy. ;)((rj ;!(il;qy j .yj.j . .kjj..g.j..y.(..
-
,
.
-.
-
-
-
- .
E y
-
E
-
.
.
-
-, y
y
.
i i -. E -E i
.
. .tj)),y,,,) ,
.
-
. y jy y -g y Ei - . . y . y E EE- y y . -. --ty gy y- . . -. . . - -. .y y. - ; - yy- -j..y. --j . . - - y-. -j-. E . E.. E , E. q . - . ( yj'ij g: ; g. -- -. - -i -. g. . -- .
-
!. - . E - y .
..
-
y
.
-.
.
.-
. -
'
.
-
-
.
y .
.
-.
.
y- -.. . y;. y .E
jy
. . .
. .
-
-y
(
-
-
-
-
.
-
-
.
'
.
-
-
.
.
.
c) s
c,
s
n)
.
Capter
268
7 Disions
as Marov
ConvergepceTheorms
Section 7.5
Pzocesses Letting n foltows.
-+
x and using the monotone
269
theorem the deskqd result
convergence
In '
Taking utzl
&S
n
'-#
Though
Theorenas
In this section Fe Fill apply the theory of Harris chains dkeloped i'l th previsection to difpsion processes. be able to use thattlyoiy we wilt szppse 'tb
us throughout tlii
sec iton
'
t af
q
MP(, c) i? (5.1) Assumption. (HC) and (UE) locally.
norma
=
Proof
:
.
.
By
.
l . .
.
... '
(3.8)there
is
gives the following
.k'. .
i
..
.
F= 6
'
.
. .
.
.
function pftz, p)
>
0
: .'
. do
. . .
i '. .
th>t
=
for
-e
ff c
z 6
=
.
(z : IzI>
r)
sharp. -c/z
=
=
=
=
-1
=
-
Erxv
=
Rearranging, then letti g t gepce theorems, we have
s.(qz,)
-
and using
(:x7
-+
(z, s) aj
.Q'r(1,)
pftz, y) dy
.)
e
'
(z) K
.
.
Exnmple 5.1. Consider # 1. Suppose c(z) 1 and (z) for lzl 1 where c k 0 and set ff 1). 1/2 :7 = 2c- 1. To with > then If holds c (-1, (5.4) compute ENTK, suppose c # 1/2, let : = 2c 1, >nd 1et utzl = z2/:. Ln in (1, ) so u(a'V)+ t is a local martingal util tife (t,l p inft : X3 / (1, )). Using the optional stopping theorerp at time q1,,) A t w have
-.,
rilmsie on Itarris chain.
2z
to prove! it is remarkably
emsy
21
' .
lower semicprtpnuop:
.F'.(a'
(5.4)was
2
.
From this we see thqt .P.(S continuity implies #:(z, p) k #(a, a) > 0.
a and b satisfy
coecients
(z : lzIK and p is Lebeskl pewsure ther Xn is p.n aperiodic
=
a
th
well ppsed ard
r)
TheoreG. If X B lized to be a probability .
lzl2/E
(5.4) Corollary. Suppose tr c(z) + then ErTK ; IzI2/f.
X.
7.5. Convergence
(5.2)
=
.
2: =
-
2
A t)
the monotone /.x2
-
bounded
and
conver-
r(,,s)
X
1) > 0 for each z so > 0 wtzen z, p E d so
(i) holds.
(it)holds
Lower
yemi-
and we have (:I
theorem (4.10),we To flnish checking the hypotleses of the convergence distribution zr exists. Our approach will be to have to show that a stationary ielii ili (4.5)ptduces a statihary show that ER < x, so the constructio sligllt check (x7 generalization ER < with fnite total mass. To we will use a of Lemma 5.1' of Khasminskii (196)k '
E
for all Suppose k 0 is C and hws Ln i (5.3)Theorem. = inf : Xi A-). Then > Tx compact set. 0 Let a k is (t e -1
'u
'tz(z)
z
To evaluate the right-hand side, we need the natural convenient to let start froz 1 rather than 0:
/
ff
where
/(z)
z
=
(3.2)in
=
ff
EmTK.
z
2
-
dy
=
CJ(p2c+1 -
1)
(
=
1 it follow ttiat
j2c+1
z2c+1
-
-
t:
zcdz Z
1 y2C
this it is
1
Chapter 6 with c
>Qq1,,)
-
1
=C Using
p
exp
and doing
scale
ac-j.z
-
1
z2c+1
1 '
-
-
52
1
-
-
'
2.+1
r
-
1
:
The second term on the right always converges to ms b x. The third term on the. right converges to 0 when c > 1/2 and to cxl wlien c < 1/2 (recall that e < 0 in this case). -1/6
-+
supermartingale
Proof It's formttla implies that 14 u(.;V)+ t is a local Tx). Letting Ts 1 Tx be a sequence of times that reduce 14, we have (0, =
'u(z) k Ezun
h Tn) + Ecn h Tr,)
k Ern
h Ts)
on
Exercise
5.1.
Use
(4.8)in
Chapter 6 to show that
ErTK
=
cxl when c
% 1/2.
Chapter 7 Dillsions
270
Pzocesses
Marov
t'zs
1 and (z) K -Vz: for z k 1 Exerclse 5.2. Consider d = 1. Suppose c(z) where : > 0 and 0 < J < 1. Let T1 inft : Xt IJ and show that for z k 1, 1)/6. Ezl' K (z1+J =
=
=
-
5.2. When d > 1 and c(z)
Exnmple
I the condition in
=
(z) % IXt 1,and use
To see this is hms
1et K
sharp,
=
+
.
8 SVeakConvergence
6)/2
-(d
z
(5.4)becomes
lt's
fz)
with
formula
Izlwhich
=
,
Dif
zf/lzl
=
Diil
z?z/lzl3
1/IzI
=
-
to get Xa
'.%
y
5$ =
, We have
+
c
.
1.x,I
(xa)ds +
t
be a one dimensinal Brownian and write -4d
.(z)
the
=
condition . .. ..
-
z
.
.
.
motion B
1
(z)
(3.6)implies
0
wherep71(z, p) is the transition probability for the process killed when it leaves the ball of radius r+ 1. From this it follows that PM > m) < (1- 0l=. Since (:I Um F'm % 1 we have EUM < (1+ Czll&z and the proof is complete. -
J
=
E
(1.1) TheoreG. (i)
r6W
M, we observe that if rn is the exit time from H then Pa,trx
=
-.->
=
-
=
=:$
< x.
Proof Let Uz 0, and for r?z k 1 1et V'm = inftt > fm-l : Xt E ff ), inftm k 1 : Um E Z). Since Um inftt > Fm : lml / H or t e ZJ, and M Xumj E .A, R S Uu. To estimate EJR, we note that .X'(Jm-1) e .A so E
We begin with trextment 9 we>k pzyergene on a general space S with a wittk and i.e., me tri p a tunction (i) plz, z) 0, (ii)plz, y) py, r)z),witlk (iii) z). yj-py, z) ptz-, ptz, yj : balls < Open deflned by (g r > 0 are ptz, k and the Bcgelr sts S ar the g-fleld genr>ted by the ppep balls. Throughopt this sectton F Fill supppse S ij a mtric spae arld S, is the cpllectiop of Brel sets, even though several results are true in much greater generaltty. For tater if there i a countable dense set, and use, recall that S is said to be separbl complete if every Cauchy squence converges. A sequence of probabilit# msures #n on (S, Sj is said to converge each cpntinupuf if fpr bounded ftpction JJ(z)pn(dz) r..-, J(z)>(dz). weakly situattons it will be coveniept I n this cmse We write pn to deal p. I many di reC tl y with randont variablrs X a r>ther thn with $he Msociated distribuE .A). We say that Xu onverges weakly to X and tions JIn(.A) Px ale if EfX). X X for any bounded continuous function f, .E'J(.Xk) write called the Portpanteau result, sometirrws gives Our flrst Theorem, fve ' t ( equivalentdeqnitions of wak' onvergene. qa
1
-
Spaces
n j
.
Let ff then ER
#s
Ix a j
.$
. .
=
(x)
1
-
intiim,
The lmst detail is to make the transition from Erlk
(5.5) Theorem.
()
(itis a local martipgale
(.Xz) +
.
reduces tltis6)/2
1+ c)/2
j
d
t x dptz I.X,I ()
=
. Xt 1.Xh l
dyk =
t
1
+
,
'
l
z
()
tjj,s
.
1/7 here for a d dimensional Brwnian
written
let
If
Xs 1-X-, I
EJCXA)
The fllowing staterents EJX)
-+
sets G, ltminfa-x
(iii) oi
all
(iv) For
all sets .A with P(X E t'9.)
pn '
(v) Let D)
PX
e
D))
= =
;
'
=
Pxtt e t?) Px .
.
' '
'
'
.
'
-+
E1X4.
''
.
0, limsups-xPtAk
the set of distontinuities of J. For
0, E1Xu)
functibn
E ff ) K PX
Px
,
.
quivalet.
are
for any bounded continuous
(ii) For a1l clqsed sets ff limstlps-.x
'
a11
J.
E ff ).
e G). 6 W) = PX
.).
e
bounded functions with
Section 8.1 In Metric Spaces
Con vergence
Chapter 8 Wa
2T2
Remark. To help remember (ii)and (iii)think about what can happen when Pxn = za) = 1, zs z, aud z lies on the boundary of the set. lf ff is closed fpr we can have za $ ff for al1 n but z e ff lf G is open we can have zs E G all n but z # G.
The deflnition of the cu implies
Proof We will go azound the loop in roughly the order indicated. The last step, (v)implies (i),is trivial so we have four things to show.
and this inequality holds with X in place of Xn Combining haye limsup I.E'J(.Xk) .E'J(X)l K
273
-+
0 s J''laf/l-'%k e =1
Efxn,
.)
.
-
sr our conclusions
.
we
'26
-
(1) lmplles (i1) Let ptz, ff ) = inftptz, y4 : y e ff ), z-b= maxtz, 0) is the positive part, and let h (z) = continuous,and k lx so
(1 Jr)+
'thrj
=
-
where
neX
ff )). h is boundedj
'thpz,
but '''Ii'' ..'iII1''.
n'ex
Now
h t lx
as
lim Eh (Ak)
e ff ) K
limsup Pxn
j 1 x, so letting j
ELxj
=
R**X
(ii).
cxl gives
-+
'only
.
interior of
tiiilnd
that OA = ff
nte
.,
.
.
.
..
.
...
.
.
E
..
.
:
.
of the . .
(1.1 is e.
''v
.
.
'
.
'
.
.
(1.j)cotlnuous mapptng theorem. juppose pn on (, ) opvrrge Fpily gbp) 0. to /I, au d let v : S1 S' have a discontinuity set Dp wit
Jlp
P sl
O
':. .
.
:E'
.
ren
=
-
# oW tF:A
'
.?...
.
..
Prpof t t follows from
J
i
:
(v)in
imyly
so our assttmptions
G,
::
=
=
-
.: .
E1
-+
(ii) ls equivalent to (iii) This follows immediately from (a)#(.4)+#(u4c) z if ylc is closed. 1 and (b) is open if ad f Aj & Ae' the the closure txibly(iv) tet zf (11)and =
so the proof is complete.
An important consequence
'
' .. .
.
is arbitrary
6
..
.
.
.
c.y R be bounded and coptinuous,
sice
(1.1)tltat
.
.
DJOV?
c
Dp, it
.
Jpa(dz)T(w(z)) Jp(dz)T(w()) -'+
E ff
PX Using
)
(iii)nov with the
(ii)and
Pxn limsup N*eX
E
e W) = PX
PX
=
last
liminf
6 G)
PCX,Z
PeX
n-
v
ff ) E G)
%PX
y
=
=
.P(
6 .A)
PX
6 .A)
At this point w have shown .
..
'
'
nrex
n
.)
e .A) k Px
which with the triviality liminf
.' .
E
yg
.. ' .
.
'
.. '
.
o
s-ltgajytzj
sincethis holls
for any bounded
k
e
. . .
i
.
lipqup Pun amx
%limsup gives the desired
.
'
:
6 .A)q
In checking useful.
convergence
result.
' '..
(t.2) convergingtogether
i probability
then 'F,z
z:z).
.k
s
.y
o p
y
jjjgjyj
. . . .' .
continuous
.
. ..
lim )n f Px
.
Changing variables gives
e ff ) K PX
.'
liminf #(Ak E X)
(
we have
equality
Pu %limsup N=+x
a4)
G4
6
J, the desiretl
result follows.
in distribution the following lemma iq sometimes
lemma.
Suppose Xn
:::).
X and p(.Ya,t)
: pty, ff Proof Let ff be a closed set and ff: (g .JfJ ) K ), where jtp, ff ) lf and only z) if there ls a y > : inftptg, z E ff ). It is emsy to see that z / ffJ closed. and = y) With this little is so that Bz, n ff 0, so (ff,)cis open of detail out the way the rest is emsy.
f
aipuh =1
E y1f)
aipx
-+
i= 1
0
-+
X. =
=
;
l
.4)
6
g
.
.
#tys 6 K ) S Pxn
y. .
E ffd) + #(p(Xn
. .. .. . 's) ,
.
>
)
274 Letting
Capter n
-+
aud using
8
Weak Convergence
Section 8.1
the second term (ii)in (1.1)we have cr, noticing
limsup .P('Fs n
oo
-+
e ff ) K limsup n
the right goes to 0 by assumption,
PCX. E ff4)
e f)
.
E ff,)
K PX
,
.
t PX
ta'
j
'i'a
Let Fa(z)
rmsy.
Fs-1(g)
=
inftz
:
=
Fs(z)
.
.
z)
yzatsx,
k
'
...
.. . ..
.
'
'
(fJ)
x
t
So
(1.1)implies
that Xk
:zty
K p(.f1/k)
.
ff it follows that
txkE ff ) %ptff )
limsup
y and it follows that X has distribution y.
D
Proof of (1.4) Construct a decomposition Xk of the space S ms in the preceding proof but this time require that each Ak hws Ilnrluhk, ) 0. For ttnluik ?J. each n construct decompositions (I'II of g0,11 so that tf2z ) arrange the intervats in an appropriate way we introduce an ordering in wich tlj if b, < (c, % c and demand that IL' < 4 if and only if fr. < ITj. Let , )E Aktj and Xknwj deflne limk-.x Xkn k,j for td e Iknj As bfore, Xn zk t/ and hms distribution pn. exlsts EJ pn(.AJ) IlnAkj 1 1 0, we have Since
yj
ro
=
, tJ
-
'*'
''
'
N
.*
)J
' .
@
*
'-''
'4
'-''
'#'
.
.12
)
l'
We will warm up for the real proof by treating the special case pn
.
.
. .
H
. .
.
.
'*.-1
'v
=
-
I(.J:l)
-
i
(.2g)l
'
=
-
lpw(.k,J)
=
p,z(.Ak,./)I
=
i
=
l
,=
.
.
p.
/4
'
,
'
'*
=
=
=
*'
..
,
Proof
J'-! IlAk,jj
=
JE.S(k,.&')
nd noting ffl/k
-4
'
'i'x
Gls
(-n(=) e .,,J)
=
be the distribution
-+
=
Letting
'k'
td
=
E ff ) K (.X@) E Lhsk,xjAk,jj
izskbhej
e'
for tl e (0,1) be a random variable that is uniform on (0,1), almost jurely. (For a proof, see (2.1)of F,71(J). Then Fa W invite the reader to cohtemplate the question: Chapter 2 of Durrett (1995).) there an emsy way to do this when S R.2 (oreven S (0,1)2):?'' while we givethe klneialprof. The details are somewhat messy and hot important fr the developments that follow, so if the reader fnds herself msking ttWho cares?'' she can skip to the beginning of the next section.
let IJ() and let 'Fs
e
s J'-! = J''!
'
..->
1
EEL'7
(1.1).
oGplete septheorem. Suppose l (1.4) Skorokhod ' s representatlon ::::> random variables JIs metric Fa on px then we can deine space: If arablr with the Borel se (0,1) (equipped ts and Lebesgue zeure ) with distributi surely. # Fx almt pn so that
R tis is
,
jzskbKj
'
Remark. When S funC tion let
.
proof of (1.3).
) as 8 t 0 so we have shown
E ff
The next result, due to Skbrokhod, is useful for proving results about weakly, because it converts weak convergence into that converges qequence '''' a ' . . . . . . tinst ure conveigce.
=
.
=
#(Frz e ff ) K PX e ff ) limsup n-G) The desired conclusion now follows from
275
Let zk be some point in Ak and 1et Xktjwj = xk,j for (J E Ik,t Since Xk A%+1. is contained in some element of Ak which by dehnition hs diameter K 1/, it is a Cauchy sequence. So X() = limk A%@) exists and satisfles p(Xk, .X') K 1/. To show that X has distribution y we let ff be a closed set, 1et Sk, ff ) = fj : Aktj n ff # 0J, and 1et ff: (z : ptz, ff ) % JJ ms in the ,
OQ
-+
Memsure theory tells us that PX
on
Tzlhketric Spaces
2 7(#x(W,J)
pn(W,J))+
-
J
E
'
where #+ = maxtp, 0) is the positive part of y. Since the px oAktjj c 0, (1.1) implies pa (.4k,/) pnAk,j) and the dominated convergence thepre!n implies -.+
(1.5)Theorem. Eor each defned on (0,1) with .P(F
memsure p there pro babtlity = p(.4). E
ts
.)
variable y
radom
J7 Itfr
(E 1 6) .
of S iht k Proof For each k construct a decomposition Ak (.Ak,1, i.e., eac disjoint sets of diameter less than 1/ and so that Zk reflnes set .Ak-1,f is a union of sets in Ak For each k construct a crresponding = itAkkjj and arrane decomposition Ik of the unit interval so that a4k-1, and only if Ik-,i D Ik,j D Ak,j if things so that .)
=
.k,2,
.
.4k-1.
,
.
tfk,j) .
,2f
)
(Q )1
0
-.+
-
,;f
as n
(k7
-+
.
t4'
Fix k and lt aa be the left endpoint of IL% and let Sjzj :r:z j : 11 4 < Iljo which by our construction does not depend on n. (1.s)implies that .,
'*' t4
ax
=
JE Sioj
lfkx 24 l
=
-
u
,
1im
n-cxl
'
lzn? I 1,
JES(Jn)
=
v
lim cen
n--.cK7
'''r
''
ct that is closed under intersection it is esy to see /z(tJ,=1.f Given an open set G and an e: > 0, choose sets that patu,xzAl ). al.k t'. Using the convergence just proved it so that /Itus=zAl > p(G) -1 .4
-+
,
.
.
-
.
followsthat
2/
;&G4
set of points that is x we see that if (J is not one of the countabl wich I::1 of Xx , proves (1.k). some IT,.i7we have Xn the end point
Letting k
Prokhorov's Theoreins
-
e < ptuf-l.f)
lim /,stukzzfl
s
s liminf
/z,,(c)
-+
-+
Sine
is
6
arbitrary,
(iii)in (1.1)follows and
the proof ist/omplete.
C1
( .
roof for Rd This is a fairly straightforward generalization of tlte argument k of d (See 1. e.g., (2.5)in Chapter 2 Durrett (1995).) for In jjt ls cae we can rely on the distribution functions F(z) yfy < z) where for two vectors y K IR.J wit ratinal coordinates. z means yi K zi foi each i. Let Qd be the points in By enumerating the points in Qd >nd yhen using a diagonal prgument, we an flnd a subsequence Jk. so that Fak () converges for eac h q e Qd Calt the limit G and deqne the pzkoposed limiting distribution functiop by .
8.2. Prokhorov's
Theorems
.. .
.
.
.
::.::
Let 11be a family of probabilit# measures on a metric space (S, p). We call 11 relatlvely compact if every sequence prt of elqments of 11 contains a subs!r that converges weakly to a limit p, whic may be / I1. W call 11 quence pnk 6 for all each for tight if e: > 0 there is a compact set. ff so tht Jltff ) k 1 proofs of section devoted is the This to y E I1. -
If 11is tight then it is relatively
(2.1) Thorem.
(2.2) rheorem. Suppope S is pact, it is tight.
coinplete
compact.
and separable.
lf 11is relatively com'
but th secohd okl is The frst result is the one we waut for appliations, comforti: since it says that, in nice spaces, the two notion are equivalent. .
the result successivelyfor Rd rtx a couwtable finally, a general S. This approych is somewhat union of compact sets, lengthy but to inspire the reader for the flbrt, we note that the llyst two special cmsesare important examples, and then the last two steps and the converse are rem arkably emsy. Before we begin we will prove a lemma that will be useful for the first two exqmples.
(2.1) weshallandprove prool'ol-
)
1
.
!E!)
.
F(z)
=
inftGtl
:
qG
d
Q
and q > zl
where > z means qi > zf fpr each f. To heck that F function we note that it is an imrediate a distribtion cpnsequence of the deqnitiop that (i) if z K y then F(z) % F(p)
ts
(ii)iipuw#(p),
=
F(z)
.
where y t z mns yi t zi for each i. The third function is emsier to prove than it is to state:
conditim
d) (iii) let = (c1, 1) x x (cd, be a rectangle d)., >nd sgntr) 1et (-1)n(c,v)where . atx (tld, .
t
.
distribution
a
with vertices I/' = (a1, x = ntc, is the number of a's appear ir b. The inclusion-exclpsion formula implies that the piobability .
.
.
j
for being
of 2. is
Evuvsgn(1?)F(r)
,l)
.
k 0.
.
.
'p)
'
'. .
.. . ..
.
y
. .
The inclusion-exclusion formula says m
P(UP1,-1.i)
.P(.f)
=
f=1
#(.
-
i<j
n+)
+
.
.
.
+ (-1)m+1.P(nt,-l.f)
. . .
j ' .. .
y
The fact that (iii)holds for the Fn implie.s that it holds for G and by taking limits we see that it holds for F. imply that F is the distribution function of a measure Conditions (i)-(iii) on Rd, which in our case will always have total mmss K 1. Define the ith where the ltmit is m arginal distribution function of F by Fi@) =.: lim Fyj = coordinates takn through vectors y with yi z and the other yj x. F is a one dimensional distribution function and hence its discontinuities X are a cotmtable set. Call z a good point if i / D for each i. We claim that F is continuous at good points. To see this, let w < z < y, 1et zi be the vector which uses the flrst 's
-+
Proof
. :
y
.)
-
.
=
g
g
F(p)
Fwj
J7F(z)
F(z-1)
-
S
f=1
J-lF(w)
ptz,p)
i= 1
and
of F now follows from the monotonicity
The continuity the Fi
F('tz4)
-
continuity
the
9
Our next
p,
9d with
r 6
,
p < q
F(z) Sihce . .
ted where- Fatff ) is short for the probability of ff under the mewsure witi ire., one with the distribution function Fn. Now pick a good rectangle the ibabilitk of ai bi $ D for all i, so that ff (:: A. The formula in (iii)foi about pointj good at imply convergenc .A and the previou, claim
J-lleld
,
gnerated
by the
'
(z :
P#of To argue that d D 1t* observe that if Gi are open then Gi for 1 K K J is open. For the other inclusion note that
zi
e
-.+
n
l!/
K 51
p(z,p)
:
=
nrzzz y
.,
,
F(d)
ruli
Finally. we
t
wltich
=
Eav to
of interet
lim Fas(.A) -yx
prove
k
thatJks
Fntff timinf '
n''ex
. : .
.. .
)k i
-
'
F. For thi w)
=>
in its oWn right.
6
.
.'
' '
' .
E
:
te
.
.
. .. .
=
EI
-
Rd be the projection zutzl Let g'd : R* (zl, zd). Our first claim is that if 11 is a tight family on (R= then fy o g-i t ;t e ll) is a tk ht family on tRd, 7zd). This follows from the followhlg general result. -+
=
.
.
.z
'
-2*)
,
a -1
tight family :
y
e
l1) is
Proof Given E, choose in S a compact ' c'= ? If ff ) then ff is compact and all p E Il.
tff
theE
:::.
/ n' 1/n) e 'J=.
: pz,y) n') us==z.(p
p(z, y)
.4.
.4
Using (2.6),the result for Rd, and a diagonal argument we can, for a given converges to a sequence ps pick a subsequence Jlnj so that for a11 d, Jlst o limit vd. Since the memsures va obvlously satisfy the conslstency conditions of Kolmogorov's existence theorem, there is a probability measuve v on (R* j T*) g'- 1 vd. so that v o We now want to check that ynk v. To do this we will apply (2.3) lz(:./1.) with the finite dimensional sets W with 0. Convergence of fnite ?z(.) and dimensional distributions for al1 .A e #. (1.1)imply that ynk (.) 'md
,
=
:z:)*
.4
=
=
-+
280
Captez
8
Weak Convergence
Sectfon 8.2 Pzoorov's
The intersection of two fnite dimensional sets is a fnite dimensional set and t'9(.An Bj c: (').Atl OB so is closed under intersection. To prove that any open set G is a countable union of sets in we observe ptz, yj : > < there 8 that By, if G is 0 (: E a so that p ) G. Pick J ) (z CG 2J) large that Gc R= By, 6 can 0. so we suppose G' # 0.) If n # so pick N so that 2-N < J/2 then it follows from the deinition of the metric we that for r < /2 the flnite dimensional set =
.4
.(p,r,#)
(p : Iz
=
!&1< r for 1
-
.N'Jc
S
By, J)
c
G
.
.'
.
.
.
.(g,
.
.
.
yi,
=
=
-y)
-
,
.
=
completesthe proof
of pug
z::)-
(7I
v.
Before passing to the next case we would like to observe that shown
have
we
-+
=
,
,
-+
The boundaries of these sets are disjoir!t for diferent values of r so we can pit r, N) hms v measure 0. r > J/4 so that the boundary of of th be an enumeyatipn Let y pa, As noted above R= is separ>ble. dense set that 1ie in G and let Jl members of the countable rf, #) G suppose be the sets chosen in the last paragraph. To prove that Ufyk and pick G, that B > G 0 let C a point yi sp yhat so y tlf-f z g (z, Bz, yi) < yl9. W claim that z E Wf To see this,note that te trilpgl 719, which inequalityimplies Byi, 8v/9) c: G, so & k 4v/9 and r > J/4 ,
.
tzw)
; i
tzn)
=
,
.4,
281
lf the points zn p(z, qi) and hence z in S then limptzn qi) (z). Suppose z # z/ and let 6 = ptz, z/). If ptz, qi) < V2, which must be true for some i, then p(z?, qi) > E/2 or the triangle inequality would lead to the contradiction :7 < p(z, z9. This shows that # is 1-1. Our fnal task is to show that /z-1 is continuous. lf za does not converge :7 > 0. lf p(z, qi4 < 6/2 Fhich must be twe for to z then limsup ptza z) f) ptza again then limsup the triangl ineqality would jiv spme qi k V2,oz contradicition, and hence does not converge to (z). This shows that a. (zn) /z-1 if continuous. then is zu z so C1 (z) -->
.4
=
Teorems
-+
lt follows from (2.6)that if 11 is tight on S then (p o : p E 11) is a tight family of measures on R& Using the result for R= we see that if yn is a sequence of memsures in 11 and we 1et vrt pn o ?-1 then there is a convergent subsequence vug Applying the continuous mapping theorerp, (1.2),now tt? the = function weakly. it follows that puk o vng converges v -1
.
=
.
=
The g eneral plii) k 1
Whatever S is if 11 is tight and we 1et Ki be so that case 1/ for a1l y C 11 then a1l the measures are supported on th Uff and this caze reduces to the previous one. set Sz (a ,
'
-
J-compact
Proof
=
of
(2.2)
We begin by introducing the intermediate statement:
(H) For each t', J > 0 there is a flnite collection W1, so that ptUksa.Afl k 1 :7 for al1 y 6 H.
.
.
.
.,z
,
of
balls
of radius
J
-
(2.7) Theorem. ln R* weak convergence is equivalent dimensional distributions.
to
Proof for the c-compact We will prove the case reducing it to the previous one. We start by observing
result
,
flnite
of
convergence
in this
cmse
by
We will show that (i)if (H) holds then 11is tight and (ii)if (H) fails then 11is not relativley compact. Combining (ii)and (i) gives (2.2). Proof of (i) Fix e and choose for each k flnitely many balls d 10 da k of radius 1/ so that p(Usapd,) closure all 1 for l1. Let K be the k ;t e V2 of n2'.1Uuns .Av Clearly ptff ) 1 1 &. ff is totally bounded since if Bb is a J ball with the same center as A) and radius 2lk then G?,1 K j %nk covers ff Being a closed and totally bounded subset of a complete space, ff is compat and th proof is complete. I::I .
lf S is c-compact
.
,
-
.
(2.8) Lemma.
.
-
then S is separable.
.
.
.
'
. '. .
.
Proof Since a countable union of countab le sets is countable, it stzces tp of radius t/n, let znZQ, pr ove this if S is compact. To do this cover S by balls and check subcover, of tha t the m of balls fnite the the centers be a m K ms E1 countable set. dense are a .
i
Proof of (ii) Suppose (H) fails for some t: @nd J. Enumerate the memers of the countable dense set :1, q2, let A Ulkjdf Bqi, J), and Gn For each there is a measure yu E 11 so that Jza(Gs) < 1 6. We claim that yu hs no convergent subsequence. To prove this, suppose pnu :::t. v. (iii)of (1.1) implies that for each m .
.
.,
=
=
'rt
-
(2.9) Lemma. If S is a separable metric space then it can be omorphically into R*
embedded
homc
.
Proof Let 1, 2, from S into R= by .
.
.
be a sequence of points dense in S and deqne
(z) (p(z, =
z), p(z' :a),
.)
-
.
a mapping
'
zzttpml %limsup pnk (Gm) -.cxl
s
limsup paptG,,sl k-+ oo
s
1
-
e
.
Capter
282
Weak Convergence
8
However, Gm 1 S as
8.3. Th
so we have vsj
1x
m
Section 8.3
l
J-feld
,
dimensional
K )
the
,
1 /* 0
Theo4p. Let yn 1 K n k be probbility memsures on C. lf the flnite r:xn distribut (ons o y pu converge (o oy yy aud jj tjw yg as (jgu then y'k z::> #x.
(
.
for 1 %
K 1 and A
tk
> c)
M)
yn is tight if and only if for each e > 0 there
< .: for
all n
k
Proof
From
no
'
'
. .
(3.5) The Arzeia-Ascoll
and
only if supwux
(see .
.
.
:
.E
page (1988), :
'
.E
..
.
and lim,-.c supwix
:
...
AL ubset .A of
'heo/em.
< x 1kv,(0)1
Royden
.g.j
.
p)41 +
169) .
p,
a.s
.ts(u)
c
'
'
.
:
.
Piqk N so thay Jf2-N/(1 tt foltows that >(,t 7k e)
8.4k
'
.
'
.
.
To prove the necessity of (i) and (ii),we note that if p i tijht and E > 0 we can choose a compact set ff so that pntff ) k 1 6 for al1 n. By (3.5), ff C (X(0) %M) for large M apd if e > 0 thp ff c: (ug:%E) for smarll J. of (i)arnd (ii),chopse M sp that pn (IX(0)I To prove the suciency lt M) S e7/2t1 for all n, It we let f / 2 for all n and choose %.so that ynlnk > 1/:) % ff be the closure of (X(0) % M, K l/- for al1 1-) then (3.5)implies ff is all n. CMITIPaC t- an d yn 1 EI foy e k (ff) -
-
.l
-
E '
.
K 2-N/(1
-
l.Xk(U Xn (r)lS
.
..
tz
.
.N
.
L
..
Vkorokhd's
*.
q
E
!
-
.
E
'
: is triv ja j w jjeyj For example zt Condition (i) is usually emsy to check. result, .Xk(0) cp za and 4a The next useful z as n -..A x. (3.6),will b J'andpm condition result of formulate in checking Here, in ih we terms (ii). variables Xn taking values in C, that is, to say random processes (Xa (f),0 K t K 1). However, one can ealy rewrite the result in terms of thei: distributions ps(.) Px 6 .A). Here, d E C. -+
=
(3.6) Theorem.
enough so
rl-t
for
2-)
s
ksc.
:
Exlstence
p) > 0
.
then
s
2-(1-n)N
Theorea
2-*/) then with
-
t, r c Qa f3 (0,1) with
-
285
that
3 2(1-n)-/(1
=
we have
-
(1+
-
./1.
2-)
2-(1-9)X
so jhat A%
%6
and
o
for SDE
In this section, we will describe Skorokhod's approach to constructing solutions of stochmstic dilerential equations. We will consider the special case (*) dXt = 'uk
)dft
.
'tzl4k
Existence Theorem for SDE
py)
-
Chapter 1 it follows that if
probabilityk 1
'
.
(1.7)in
's
and pick p > 0 small -
'
.
Skorokhod
= (1
=
We begin by recalling
p) S ( for n k nn. Proof
Let y
&/#,
E
.
e for all n
s
Section 8.4
where o' is bounded and continuous, since we can introduce the term (.X) dt by change of measure. The new feature here is that c' is only assumed to be continqous, not Lippchitz pr even Hlder continuous. Examples at the end of Section 5.3 show that we carmot hope to have uniqueness in this generality. Skorokhod's idea for solving stochastic diferential equatipns wms to discretize time to get an equation that is trivial to solve, and then pass to the limit and extract subseqential limits to solve the original equation. For each n, defne .Xk(t) by setting Ak(0) = z and, for mz-n < t K + 1)2-n,
tm
AQ(f) Xa(m2-n) + c(Xn(m2-n))(#t =
-
#(m2-''))
.
E
Suppose for some a,
#> 0
Since Xn is a stochmstic integr>l Fith respect to Bzownian motion: the formula for covariance ot stottkmstic intgrals implies (.A-n
1
xijt rt
0
k
then (ii)in (3.4)holds. Remark. The condition should remind the reader of Kolmogorov's ontinuity criterion, (1.6)in Chapter 1. The key to our proof i the observation that the proof of (1.6)gives quantitative estimates on the moduls of iontinuity. For a much diferent approach to a slightly more general result, see Theorem 12.3 in Billingsley (1968).
t =
tc.rc.jkltxatgzrzsj/zallgs
t
= wherems usual' c = O'O'V. If followsthat if s < then
cfj(A%.((2nsJ/2n))ds
0
that cjtz)
we suppose
< M for
,
I(A'n' -Yjlt ,
-
(.Xk' Xni ,
lal ;
Mtf
-
s)
all
,
j
and
z, it
Chapter 8
286 so
(5.1)in E sup uE(:,)
Taking p
=
Weak Con vezgence
Chapter 3 implies
Section 8.5
Skorkhod's representation theorem, (1.4),implies ! that we can onstruct processes 'F with the same distxibutions s the Xgk) on some probability space in such a way that with probability 1 as k converges to F(t) x, Fk (@) uniformly on (0,T) for any T < n. lf s < >nd g : C R is a bounded continuous function that is msurabl with rspect t ,F,, then
E
llk (u) xni(s)IF s cpl'l(xj ) -
-
'
(x,$),1F/2 s
cp(.&J'(t
-
s))#/2
.-+
-+
4, we see that E sup lAk (u) Xni(s)l4K CM, 2(t j1 ' uE(.s qf: =
'.'
-
s)2
E :(F).
E '
T(K) T(K) -
J'
-
@
,
Uing (3.6)to check (ii)in (3.4)and then noting that .Xktl so (i)iik (3.4) is trivial, it follows that the sequence Xrt i tight. Ivkink Pfohorov's Theorem (2.1)now, we can conclude that there is a subsequence Xnk) that converges weakly to a limit X. We claim that X sqtis:es MP(0,c). To pypve this, we will show (4.1) Lemma.
If
16
fAt
C 2 and
IAj
-
k,
J(z) cp
J
j
-
Since this hols
theoremshows
is
a
local martingale
'
;
izj. '.
J(z)
lxnj
=
J''! i 1
lL'ilutd-Mj t
--
+ i' )! -
ij
So it follows from the definition of Xn
.
,
xplr
.
'
l.'
.
0
,
-
1
xjlr
that
-
.
.
s
sm/.zriifif t linear
=
t
a.f(Ak(E2nz(2-'z))du
Thorem.
rovptanmption:
As
n
m/n
=
e (m/n,(m+ Bn
x,
-r,
singt
' .
'
..
.
'lz
1
n ta
-
n
'l.l
,
.
*
*
.
n
B Fhere B is a ptapdard
:::>
,
'
'tis
(32) .
Convergenc f flnlte dlmnslonal vatiables, sum of (nt)independent randrr theorem that if 0 < 01 < K 1 then .
1)/p1
y
,
We w ill p rove thi iesult .
rj r12
.
.1'
B
Prof
))s it ytwa ; z-sztr) )7cf3(Ak(E2 it Lnfr) dr is a local martingale. J(.Xk(s)) =
.(.
0
=
.
indexed s s
( hn
so if we let
then J(Ak(t))
=
/,'
.''z
Let &,&, be i.i.d. with E = 0 and 1. Let Sn = (1 + + a be the partial nth sum. The most n>tural way to turn Sm 0 S m % n into a process = sLntjly-.n where (-z) by 0 t 1 let is the largest integer z. continuoud To have a trajectory we will instead 1et
(5.1) Dopsker's
r
(Ak
.
te
.
.
ihigs
are t*o
dlstributlons. lt follows frin
.n
n
-
m-z
,.a
)
Z:'
(B tl
3
B t2
-
Xt
1. ,
*
*
-
B tm
To extend the lmst conclusion from hn to Bn, we begin by rn nt (nt)then Bn hnt + ra knjj-vl-n =
to do:
Since Syj is the the centrl limit
'
-
B tm-z
.a
-
lmss '
Theorem
(r)
,
f th motpne
z-ytyrlkrj z'al
-
c
,
'
Dijlxrjjdqv'l
7,(y,)
-
.
-
=
'z?
=
Proof lt suces to show that if f, Dif, and Dijf are bounded, then the martingale. above It's formula implies that is a process .f(.Xk(t))
zsytyr) dv
la
8.5. Donsker's follws by applying the result to
/'
-
'
(4.1).
which proves fz.f(A%)ds
gnk)
-
g, an appliction
foi any cotinuous
.s
t
3
.
(J()
then a Efyait o fy g
#r))
T(Fr)
) y(x)
,(y
= lims
J.
=
conclusion
Once this is done the deslred z and
.
Donsker's Theorem
-
=
pbserying
) that if
!jl'..
t
! Capter
288
8
t!
Weak Convergence
Section 8.5
Since 0 % rs < 1, we have rnyq-j-l/v-'n 0 in probability and the converging together lemma, (1.3),implies that the individual random variables B7 u:). Bt To treat the vector we observe that
whereC
-+
.
CB
f''
B
-
i
in probability,
''.f=
z
) (t''. =
1'.f-) (ft''.
-
B: 1
-
,
Btnm
.8y.
-
,
.
.
.
,
tn.
-
)
,')=
CB
-
.,'',-
-
z
z
)
0
-+
..-
z
) c:),
B3
-
,
, ,
.
.
.
,
Bt
'
Pwof
Bt
-
.
m-
(iii)needs
-+
. '
.'
..
'.
.
'
'.
..
.
'
.
.
.
'
.
,
.
..
. .
.
.
.
.
.
.
.
,
'.
'
Tightness. The L2 rpaximal inequality for mprtingales Chapter 4 of Durrett (10j5))implies 2
E Taking '
P or
writing
oslsz
.. .
.E
:
'
xpax oKlKa/m
(seee.g., (4.3)in
things in terms of P
,
.4-
(i) P
# i.
(ii) 1, (iii) lf M k -
,
'
f)
From this,
(4ponly
(ii),apd
.
.
= .
.
Iltn
B,n I >
be i.i-d. with
s M-2.E'(lfI2;
K ftlflj
M
-#'?)
N
'. .
-
This gives tlze term Cz'2. To estimate '
1y E
...
it follows that wssumptions
our
(
E
-
c1
3- pu t
putp
-
Ef- J
4/m:2
-
:
pwzl
integer
y
max >XtX>+l/m
(.2) Lemma. Let l (2 S T Fl: ' let a(.T) .t. #. Ten w have
t
an even
E
Since we need m intervals of length l/m to cover (0,1) the lat stimte is not good enough to prove tightness. To improve this, we will truncat d cozrtpute . fourth moments. To isolate these details we will formulate a general result. .
lf p is
('
15'I> EWn
E
.' .....
.
-
S 4E%2
-k
z
(V
j1)
289
jlwzj.
+
=1 4 z = XEfi- gwz)+ 2
'
.
z
=
.
and using Chebyshev ) s inequality it follows tltat
n/m
=
max Sj
.
.
' ...
to be proved.
.&t-jz tpvl4
za) Using the tct that (zl,z2, + zm) is a zl + (z1,zl + za, continpop? i.papping and invoking (1.2)it follows that the finite dipensionql t dlstristln . . of Nn converge to those of B. .
' .
. . .
-
.
6% = 24
.
Only
)
,
and
. ' '' ' .
independent Bt z Btu
(#4ay
i7zT)
+
Remark. We give the explicit values of t hey only depend on pu and pv.
'
(1.3)that
so it follows from
(fn l BjnQ
=
,
'
.
32
(.
.
-
,
=
Donsker's Theorem
=
0(./)
0 0 part
(ii)implies
Capter
290
8
Weak Convergence
Section 8.5
x. Using the L4 maximal inequality for mattingales as n (againsee e.:., (4.3) in Chapter 4 of Durrett (1995))and part (iii),it follows that if ' = n/m and n is larte then (hereand in what follows C will change from line to line) -+
E sup
1,9k wzI4 < C'flst -
oKkKf
'#wzl4
-
sup
0KK;
lxi
l k EWn/9
kpu
-
B = Jsj/v-n and Ik,m probabilitiesin (5.5),it follows that Let
limsuoP -x
a
....
max
0< < m
I.n max E Iy '
,
.
@
-
m
To turn this into an estimate
J
H*
-
r
+
1 =
rqo
-
e/3
limsup n-X
> e)
f
of
Let
Donsker's theorem.
maxttt)
=
max osmsn
k
c)
: -+
0 K t % 1J. It is easy to see that R. is continuous, and (5.8)inplies
SmlWn M =>
J5(Tc
=
that it is continuous E
-
15(M1
-
e7
R has the property
-.+
S
max Bt cstsl
Y ,
To complete the pictuye, we observe that by of the right-hand side is 1)
(3.8)in Chapter 2.P0(11
=
k
1 the distribution
c)
txnmple 5.2. Let #(tJ) = suptt < 1 : (f) = 0. This time # is not continuous,fpr if oz has tJE(0) = 0, E(1/3) = 1, :(2/3) = :, w(1) = 2, and linear on each interval (,/,i + 1)/3) then #@c) = 2/3 but #@,) = 0 for E > 0. lt is ewy to see that if #(w) < 1 and (f) hms positiv and negative values in each mterval (#@) &,#@)) then # is continuous at tJ. By arguments in Example 3.1 of Chapter 1, the lwst set hms memsure 1. (If the zero at #(*) isolated left, would
J(,)I
-
.&
K e7 contluity
-+
the triangle irtequality implies .p(o,/m(n)
'*oj
we note that
g : it follows that
To convert this into an estimate the modulus of by observing that (5.3)ard (5.4)imply P
5.1.
-
-
-
If we pick m
> e)
the proot
1'/(*) #()I E 11 #11so 4 : c(0,1)
i (k+ 1)/m K t i (k+ 2)/m
I/(i) /(s)1K I.f(t) lk + 1)/m)l + IT(( + 1)/m) T(/m)I + IJ(/m)
than the maximum
so
ws-v,inhn)
s
completes
(5.8) Theorem. lf # : C(0, 1) Ptra.s. then lBnj @ 4(.8). Exnmple
1
-
i
291
,
'tt'1/m(J)K 3 ctrjaxwtyyj,rrj: a.x..IJ(s) J(/m)I since for example if klm
(ii)in (3.4)and
smaller
Theorem
The mairt motivation foi proving Donsker's theorem is that it gives as corollarie? > number of interesting facts about randpm walks. The key to the vault is te continuous mapping theorem, (1.2),which with (5.1)implies:
1)/rn1. Adding up rrl of te
el9
'
of the modulus
:2
C%-*.
hunt-I >
-
uqnnj
j is
E
l/rlz,(k +
=
(5.7) This verifes
Usiug Chebyshev's inequality now it follows that P
,
.Ptu)j/amtfnl limsup n-O
n2
-
.
of Bn over
oscillationof hn over Ilnsl/n,(gnf)+ 1)/n) and it follows that
2 1 s c(zM2 + z2)< c mo+
(5.5)
Now the maxhnum oscillation
Donker's
sc
wms of
hn
we begin
on
sptm
the
it
not be isolated on
%n Sm-l Sm K Qjlt o L :
.
The distribution of L, given in 0
(4.2)in
=
the right.) Using
suptt
; 1 : Bt
=
=
c)
I
0J
Chapter 1, is an arcsine law.
Bxnmple 5.3. Let /@) = j(t C (0,1) : tJ(f) > tz)l. The point that # is not continuous but it is emsy to see that # is continuous with l( e g0, f) : () = clj = 0. Fubini's theorem implies that
.EI( e (0,1J: Bt
(5.8)now
1
=
jo .!aa(.st
=
c)
dt
=
()
H a shows at paths tu
292
Capter
8
# is continuous
so
Weak Con vergence Pn-a.s. With a little work
llmf Before doing that that I(t 6 (0,1) : Bt
n
work
Proof
:
Sm > c/fl/n
(5.8)gives
m
ca/fi)l
I(fe (0,1) : Bt
=>.
l(mK n
Chapter 4 shows
:
Sm
>
i'orsxedu5.1, Example
0
>
E
EW)
K nptllml
>
Kn-
Evh'j
>
'
wtmaxmss Ixmls (1.1)it follows that
we have apd
The right-hand
(Bllv-ntk
' .
:-2.E'(xm2 ;
Ixml> :W)
.
.
and on
convergence,
.
IAk l K EVRJ, w (maxmsa
.. .
.
.
.
5 ' .' . .'
hv
.
l .
'.
.
.
.5 .
max
() Kj< 1
dt
'
.
in probability
(5.s)so
using
M
n
and it follows from the converging
n
-.+
0
together lemma
n
j'''lsk m
-l-(/2)
m
m=
=
0 by
m=1
.
c.zr;)1 I(t e (0,1) : B;%> (c+ e)4:J)l / ll(m s ?z : sm > n 1) : B1 > (c e)./fi)l K I'(t6 E0,
l# l
c).l
cW)Iwe
-
1
s
by dominated
/-
,,-1-(,/a)
dt (stnl/.z,;lk
that for any c,
a result about
Ptm!.x IAkl
.
1
c)I
(9.2)in
we would like to observe that 0Jlhas an arcsine 1aw under
I( e E0,1) : B.1 >
(5 9)
I( e
we have
(0,1) : Bt > .&.
>
Application of
To convert this into then
zc.)-
(5.8)implies
Secton 8.6
1
1 0
(1.3)that
skt dt
It is remarkable that the lmst'result holds under the assumption and EX,? 1, i.e., we do not ned to assume that .E'l.X'/l< x.
that EXi
=
0
=
Combing
this with the flrst conclusion of the proof and using the fact that a with probability one, we arrive -+ l( e (0,1) : Bt > JI is contluous at conclusion. emsily at the desired =
Exnmple cotinuous
5.4. Let #@) = J(a,ljtJttld so applying (5.8)gives
where
rl
1
j (f1'/W)
dt
=>
j
k
>
0 is
an
integer.
# is
Bt dt
To convert this into a result about the original sequence, we begin by observing that if z < # With lz pl K 6 and Izl IpIK M then -
,
1z From this it follows that c,,(M)
=
-
p
:.#.r+1 p lzl+l dz S k+ 1 k+ 1
ls
'
'
8.6. The Space D In the previoussection, forming the piecewist linear approzmation' was n >L noylng bookk-eeping detail. In the next sectlon vWhen w consider proces that jump at random times, making them piecewise linear will be a genuine nuisance. To deal with that problem and to educate the reader, we will introduce the space D((0, 1), Rd) of functions from (0,1) into Rd that are right continuous and have left limits. Since we only need two simple results, (6.4)and (6.5)bc'Chapter loW, d e an flnd this material in Chapter 3 of Billingsley (1968), and of Chapter Jacod Shiryaev 3 of Ethier and Kurtz (1986), VI or (1j87),we will content ourselves to simply state the results. W begin by deflning the Skorokhod topologyon D. To motivate this consider u
Kxnmple 6-1k For 1 %n K x let
on
max lxml M-k+ztv-n, mKn
s
max mKn
< MWn I,$'m
A(f)
=
t
0
t 1 t
e (0,(n+
s
1)/2n) 1)/2n, lj + g(n
,
294
Capter
8
Wak Convergence
where (n+1)/2n= 1/2 for n 1 for all n.
=
Section 8.6 The Space D
We certainly want
x.
f,t
Jx
-+
llA-/k 11
but
y. .
.
g
Let A be the class of strictly incremsing continuous mappings of (0,1) onto itself. Such functions necessarily hate (0) = 0 and (1) = 1. For J, g E D defne df, #) to be the infmum of those positive : for which there is a E A
so that
sup
l()
-
l< e
and
fu and
=
= ,
tm
d(Jn,Jm)
r?z+ 1
-
=
2m
2n
+ 1)/2m so
;
-
1 2n
fm in Exampl '
'
Exnmple
6.2. For 1
Kn gutj
ln
order
to have
and ((n+ . .'
.
=
,
:
1)/2n)
,'
< cx) 1et
=
=
:
gml
dgn, T e po in twise limit of gn is g. to show
H
=
(6.1)Tliikl. sie tfliky 'E i ' . .. .
(1/2,(n+ 1)/2n) ((n+ 1)/2n, 1)
1
s
llogtn/mll) under
the metric #n.
' ..
we must
:
' .
:
.
=
:
'
1
'
.
..
.
'
(
..
dfj are
equivalent,
d we rtoti
i.e., they give
(.
.'
:
r
(0,1). For
.'
jgj
gjjg
.
:cx
0
x (t) o
0.
.
(q
.
To fix the problem with completeness be close to the we require that identity in a more stringent sense: the slopes of a11 of its chords aTe loqe to 1. lf C A 1et (@) (s) Illl sup log f #t -
=
-
.9
are
tc
=
of
au
tl
get the main result
we ned.
'
in
(ii). If we remove
'
the we
Caper
296
(6.4) Theorem.
If for each
6
> 0
there
> M) K : for all n k (i)pr,(I*(0)I E) (ii)pst'trj > < .7 for all n k no
then pg is tight and
are nc
M,
,
=
where Bz, c) due to Stroock
:
.
'
.
.
'
.
' . .
..
.
E
.continuouq
we
time time
=
:-;r1
I
(g : lp
=
yi
Ip-rIS1 :f 1
(p
f).
clfftz,
-
dp)
zflzfstz,
-
dy)
:)c)
.s(z,
zl
radhan, about usion procede. ere i ur t': mez ttie will * and of discuss Varadhan but Stroock Chapter that 11 follows in (1979) both discret and continuops time in parallel. Our duplicity lepgthepp thv proof but is pzeferable we betieve to te usual yomewhat kangvous)copsomewhat, jt e other case. i) with changes argument. minor hanlles t the out: same of transition probabilities ln discrete time, the bmsi data is a sequen jz values in Ss (: Rd j. chain takug 0, dy4 M>rkov r?z for a 11?j(z, m?j e 'A.d z) Tl,(z,.A) for z ss,A .P('F(m+,), .
ct(z)
to DiFuslons
we will prove a result, convergence of Markov chains to
' . .
=
1) (z) = Ij,
section
297
and deflne
(1968). (3.3).
of yn has a further subsequence
(6.5) Theorem. lf each subsequence then p r, cz;> y. convergej to y
In this
(z,dy)
1.
(6.1)-(6.4) are Theorems 14.2, 14.1, 15.2, and 15.5 in Billingsley We will also need the following, which is a consequence of the proof of
8.7. Convergence
l1?&(z,dyjlh
'E
Jf limit p hs yC)
to DiFusih
in the limit then we hav weak convergence. To make a precise statement need some notation. To incorporate both cases we introduce a kernel
8 so that
and
nc
subsequential
every
Con vergence
Section 8.7
Wak Con vergence
8
=
Xf
:
t
f
bixjj
-
ds
Xtivji
and
=
aij
-
(X,) ds
((
,
,
.
.
,
.,
.lym,
e
=
local martingales.
are
,:
=
,
e
=
g'
(7.1) Theprem. Suppose in that (A) hol tls an d for each
(i)linutp
In this cse we deflne Xgh = Flpyy i.e., we make Xth constant on intervals (''n+ 1)). E'zz., dp) In continuous time, the bmsicdata is a sequence of transition rates Q?&(z, Rd, values Xth i.e., in Sh C for a Markov chain k 0, taking ,
' . .
W
e
s Al-'L
=
z)
=
Qsx,
.)
for
z
e &,
.,1.
e
7
d
w kth z
,
y
.x
(iii) limta If Xt with
i
E
.
.
(B) For any compact set
z'f supwox Qs(z,Rd) ,
< x.
. . ..
''
,
< x,
IctJtzl cj(z)l
=
-
and
in
either
case
0
.
A)
suplxjss z
-.+
j R
then
we
lzll
=
(z)
=
have X)
=
0
.
0 =>
X3 the solution of the martingale
problem
z.
::.q .'
is excluded from the interpretation, we will allow Note that z e hich ccur at a ar invilble) case jumps from z to z (which Q(z,(zJ)> 0 n w chain ttpse will well Markov behaved we positive rate. To have a although
zts
= a'
..
.
.,1.
.' .
,
(ii) limhtn suplzlss I)(z)
,
d PCX,?,
supjxlsa
. . . ..
time that (B) holds, and 6 > 0
continuous
..
.
.
tlere
denotes convergence in D((0, 1)tRd) but the result can be triviallygnetalized to give convergencr D((0, T), R d) for all T < x. In (i),(ii),and (iii)the sup is taken only over points z e Sh. Condition (iii) annot hold unless the points in Ss are getting closer together. Howevei, there is no implicit mssumption that Ss is becoming dense in al1 of Rd. Renaz-sk
=>
'
Proof The rest of the section is devoted to the proof of this result. We will and 6 > 0 frst prove the result under the stronger mssumptions thas for all we have .i
,
Chapter 8 Weak Convergence
298
sup. la,ttzl (ii') lizl?jl.tlsupwlltz)
c2J(z)I 0 (iv)sups.. 1at(z)I< cxl f(z)l 0 (v)sup.. I)()I k & zs',(z) 0 (vi)sups,xl) (z) < x
(i') linutc
and in
continuous
=
=
Q(z, Rd) k
time that (F) supr
'
.
x.
Lhlzj
E
s vJ(z) +
a. Tlghtness If
J is bounded
and memsurable,
Lhfz) As the notation
may suggest, we expect this to converge
ZT(z)
=
1 x...-x aiiq:blhjfq 7. i,i
+
;;
(7.2b).1n continuous
Llxb fxh kh ) E-l j=() Fz
fxj ?1)
f
-
JcL
:
7.a(z)DT(z)
r .
f is bondd
(
and
?t
T(Xa
) is
ls
) is a
martingale.
'*
what
The first step in the tightness proof is to see
Lh to
A,y(z) k',lz =
-
g) where
#(z)
=
A(z)
(1 -
()
=
z2)(1
g(lzl2/e2),
-
z)3
0 z
hppens
E
wh
dyj
i I'?zl 11 =
1
j
yi
uini i
js 1.u11.p1
llruj'11gives xijyj
-
zjlfhj-flzlj
-
=
inftn
k0
=
mintra
=
maxtlftt)
=
lg zI2Ik (z,dy) -
the oscillations a N
(7.4)
uimijut
'
j
ad
K ; 1 k1
ztlihjfzr,yllLz,
-
it follows that
e
y
zijyj
jj
sup
(7.5)
,
(7.2a)follows emsily from the deqnition of Lh and inductipn. Since we ?. E1 (B have ), (7.2b)follows from (2.1)and (1.6)in Chaptr 7. Proof
zfktz,dy)
Noticing that the Cauchy-schwarz inequality implies
(7.6)
martindale.
z)
i
.
'
and the deinition of
-
time
=
x-x
that if
J'ltp j -
Ip-=lKl
Ilmlll (( '
we note
ly-rIS1
-
over the set
=
to
'
To explain the reason for this definitin measurable then (7.2a) In discrete t ime
Let Ae = supz
J(z))
-
y
.
(z,dyj
299
B, 1)c) + 2llJIlxff?&(z, where lVJ(z)l, and Bz supa llDuT(z)Il
1et
ffstz, dyjfy)
=
ffN
-
-
=
-
supa
wherezr,v z + crbvy z). Integrating with respect to lp zl K 1 and using the triangle inequality we have
=
-
(iii')limstc
Section 8.7 Convergence to Disions
a -
-
:
t : Ju
ra-l =
la' > IJ -
:
X.h
1; n
x(t-)1
jk
R-1
c/4)
K Nj :
0
6/4 occur at a rate smaller than supxllhtz, B, E/4)c) and k 1 z (7.9b)
by (iii/).The first step in estimyting do this we begin by observing that
or rearranging
in
One of thq prob>bilities
tlme.
discrete
(:I
-
-
< 1)
26/4
'ra.f.l
-
Tlghtness proof, trivial to estimate
PyN
(7.12)
l.f(t) J(s)1 f lT(t) J(a)l 2.
%6.
'
< raml -
cwsB
' '
this we note that if o. > & and 0
Iterating and using the strong Markov property and
we will now prove
.
(7.8) Lemma. If
Convergenceto; Dihsiotjs
Section 8.7
Weak Con s'ergence
'
Since we have assumed that the martingale problem hms a umque solution, it suces by (6.5)to prove that the limit of any convergent subsequence solves
the frst
martigle
ptoblem
is to
show
(7.13) Lemma.
If
step
fe
to cnclude
Ci then
that the whole
IIEA.J Lf 11x 0. -
-+
seqence
converges.
The
Prof 1: -
zl
Using
Section 8.7
Weak Convergence
Chaptek 8
302
(7.4)then
to ffhtz, dy4
integrating with respect
the set
over
r(e)
(z)ofy(z) zytz) J'lt', i
+
Ip-rl
(' '.
..
(J(p)
>1
T(z))Ikx,
-
.. .
.
zt4
-
-
+ . . .. .
.-.
zityt )7(!n j
Ip-rI S 1
.
.
o.J Jz.
,)zf,(z,dp) ,
. .. .
. .
j .
E
. (
r
biztihlzt
-
I)(z)
K sup
i'.
. ..
.
J
r
;i
(7.15) + (7.5)the
gr
z
.
uniformly in z. Now the diference between the second term in smaller than is
By
.f(z)I I2J(z)l
-
aijztlhil) . .
Ip-rl %1
.. . .
E
yi
'
fJ zf)(pJ
-
' .
j
'
(
-+
0
r
'
(714) .
nd
jbitfz)
E ij
lcf.f(z) sup J
d#)
c,blzll
hnL gj
n
-
y
Ixjlj,
To pmss to the lizit, property implies
# (xn)
=
0, 1, 2,
.
.
.
is a martingale
property by introduciug a we rewrite the martingale R which is memqurable with respet g : D = + 1 and n = ; Lilhnj + 1 the rparting>l Lslhnj
J(.X'ha zaa
-+
y
:
) Jxkhn ) -
na
ln -
-
1
)-')Fzaznytxk j
)
a
j=ks
=
()
theorem (1.4)implies that we can Let x 0. Skorokhod's reprqsentation P with cxi the same distribution as the Xh= 1 construct processes %n % 'O almost surely. Combining this with (7.13)and using the that Fn so bounded convergence theorem it follows that if f E Cx2 then =
'
,
ij
If%J(z)I
:(10)
J('0)
.
(p
. -
s r(e')
zityi
.
. -
ztt
Ip -
-r1
1:
X
E
.
tfhJT(zr,p) lhilx. -
zlzfflzlz,
dp)
-
,1
.f(A-,0) -
Llxr'j
'';z:
dr
0
4
Since this holds fpr all continuous followsthat if f E Cx2 then
fft,
'
g
:
C
-+
R meuurable
respect to Fs it
with
t
fxjnj
LfX,0) ds
-
0
(7.16)
k
J=0
bopnded continuous function to F qnd obkerve that if kn
E
which goes to 0 uniformly in z by (iv).To stimate the secod teiz iil (7.1) zl unifprmly we note that for auy e the integral over (p : e < 1g- K 1) goes to 0 in z by (vi)and the fact that the integrand is uniformly bounddk Usin: (7.6), the last piece
i.i
.
conklfgs
.
th>n
smaller
-
Ip-rlse
0 so that Xlt=
-%
s
-+
Iirst term is
.
' .
j
k-1
s ) Ttxkla
E
-
ij
.X'O.
,;
,
flhilzz,vt fhJT(z)l ffhtzl
zJ)
-
-+
f discrete time. Let (7.2a)implies
PrOO
..
ct(z)DJT(z)
-
-
.
C Onvergence weakly to a limit
dp) .
llfh.fJ(z.,,) DJJ(z)lI
..
Recalling the dflnition of A1y ahd using (vi)it follows that the third terrft goes to 0 uniformly in z. To treat the flrst term we note that (7.5)apd (v)imply )(z)D/'(z)
sup Ip-rlKe
=
Now zz,v is on the line segment connecting z apd y, Dijf is continuous and hms 0 as T 0. Using (7.7)and (iv)now the quantity compact support, so r(:) in (7.16) is complete. (:l converges to 0 and the proof of (7.13)
=
(7.14)
303
where
1 we have
S
Convergence to Disions
is a martingale
'' .
.
dpl
Applying the lmst result to smoothly truncated versions of fzj z and J(z) .X0 is a solution of the martingale problem. Since the follows that it zizj martingaleproblem has a unique solution, the desirqd conclusion now follows =
from (6.5).
=
1Zl
Capter
304
.XO.
to a limit
verges
Jtxa )
* Le t tlme.
continuous
proof,
Convergence weakly
Secton #.8
Weak Convergence
8
(7.2b)implies
Lh%% Jtlaa
-
0
) ds t k
0
.
is
.
.
that Xhn
so
con-
a martingale
yetj j s a so
uuou o
(e
marC1
c. Locnlization To extend the result to the original msjumptions, we 1et pk be a C* function that i 1 o (, ) and 0 ort X( j k + 1) and dejj ne l
g
,
.#' *' %'
dy)
Ik,k,
yutzlffptz,
=
dy4 +
(1 -
pklj&mdy)
,
& it
a
fiht N
''''
irs
'
.'
It is
at z. 'N
asy
'flrst
#'
vth
='wk(z)c(z)
ctz)
=
We have supposed that the martingale problem for c and has a unique to show that nonexplosive solution A'q In view of (3.3),to prove (6.1)it suces X#'a 10 To subsequence W that is there 0 iten any sequence a n a so each sequence Xhn is tight, argument diagoyal since that this note a we pove 10,#., ?n so that for each k, Xa' shows tllat we can select a subsequehce tf.J 1). : set Gk is 0 open Gk Let so if an ), K t K (f) e 0, ' a' implies th>t then d X X an If c C s open (2.1) .
:::).
-+
.
'k
::4.
.8(0,
=
:::ty
'n'
liminf PX n''-x
e Gk n #) k
PX
0jk
e Gk n
ff)
( .
=
PX
0
e
.8(0,
.
'
.8(0,
a
aVe
Ptxi nf lim n'etr i
: #)
sincethere is no explosion audthe proof is complete.
k
liminf #(.X$ n'ex
the right-hand
: Gk side
z + n
.k ya
When e' > n -1/2 Ll% r an d (ii)We note that =
r
=
n S) k .P70
incremses to
PCXD
G Gk
n Hj
e #)
as
t- 1 x
with th
Efollowing
/n c1
() z
-n
zg 2n z
-
111/a (z
z
,
=
+ zW :n
check
conditions
n
n-f/2)
-
n
1
-
n
.
n
z
-
Wn -
(iii)holds.
0 for all z so
=
1 =
'fkl-n
'
(z)
,
l/n(z)
)
. '*
Wn
u-n
uu e' .
N
L
+
1
n
.
z
-
Wn
1
'''
2p.
n
=
-
2n
.
u
y
.
'-
.
2n -
To
Wn
ug V
-z
E
-
=
27:
n
=
(i)
1
The limiting coecients 1 are Lipschitz iptiuous, jo (z) yln the martingale problem is well posed and (A) holds. Letting Xjt/a sjj/s and applying (7.1)now it follows tat =
and
-z
a(z)
::.;q
=
1/n (8.1) Theorem. As n x, Xg converges process Xt i.e., the solution of .->
weakly
to an Ornstein-uhlenbeck
,
Gk rl 1I)
1-), Xo' is a solution of the martingale problem since up until the exit from and and hence its distribution agrees with that of 10 Now Jlp to the for b a k a, of exit k-) X ? hms the same distribution so we from flrst wsthat X
'
'he
/k(z)(z)
tz)
(7.1)beginning
of
applications
Exnmple 8.1. Ehrenfest Chain. Here, the phycal system is > box flled with air and divided in half by a plane with a smll hole in it. W iitodet this mathematically by two urns that contain a total of 27: ball, which we think of as the air molecules. At each time we pick a ball from the 2n in the two urns at random and move it to the other urn, which we think of ws a polecule going t h roug h the hole. We expect the number of balls in the left urn to be about n + Cv-n, so we let Z be the number of balls at time r?z in the left urn minus 'm/a i stite t space S1/s = < k < n) n, and let 1/a i Zlv-n. while the transition probability is lIl/n(z,
the to see tht if Ii stisfles (i)-(iii) tw oarts' yf the' prof it (i/l-ttii/) aud (ivl-tvi). Frorft the IL 1, sati:e limit solves follows that for each k, Xh,k is tight, and any subsequential ' martingale problem with coecients F hre
305
8.8. Examples In this section we will give very simple situation.
.
the discrete time proof shows tht and the desired result follows.
and repeating tingale problm
0
-->
n
Examples
dXt
-Xj dt + dB3
=
Next we state a lemma that will help us check the hypothses two examples. The point here is to reflace the trucated moments moments that are emsier to compute.
ju(z)
=
yi
lz)
=
yi
yvh (z)
=
-
-
lp -
zijyj
-
dp)
zilliz, zlrff
zllffhlz,
?,(z,
dy4
d
in thr next y
nary E
.
Clpter
306
(8.2) Lemma.
8
Weak Convergence
If p
k 2 and for
(a)limptnsuplalss ItJtzl (b)limhlo suplwjss lJ)tzl (c) limhto suplmjss .
'
'
.
5
.
.
..
ot)
'
Section 8.8
a11 R
0, Ek>,s1-2,2 = for larg it
0
=
then (i),(ii),and (iii)of (7.1)hold. Proof
Taking the conditions
lJ(z)
-
bt
order, we note A)
in rverse
(z)I
1 -
lp-rl
zl
a
=
fflz,
yk
-
zJllffplz,
d
S
-tb
zk)2
K Ig -
z12
1/n 74 (z)
dy)
=
n
st?z-4tzy
.E'((Z2
-
nztl
+
and
pnlnjj4kzl
=
nz)
,
date back to '
of br>nching Consider Processesk sequence which probability children of p2 hms mean the k processes(g;)jm k 0) in 1 + pnln) and variance %2 Suppose that
Exnmple
To bound Efl
Branchlng
8.2.
Efl
-
(A1)
pu
(A2)
c.n
-+
-.+
p
:
r :
(-x,x), (0,x),
(A3) for any J > 0, Following the motivation Our goal is to show
(1+
/k/n))4
j
=
nJ
22 P(IC
-
2p2
--$
lH
3252,25 n 74 (z) S
0
in Example 1.6 of Chpter
5, we 1et Xtgjn
=
y; Zwjln.
(8.3) Theorem. If (A1), (A2), and (A3) hold then X3l/a converges wea kjy to of Feller's branching difusion Xt i.e., the solutio ,
=
pXt dt +
o.
A't dBt
nzEL.l
=
,
..j.
=
nz)
j
(1+ /k/n))4
yyjzy
note that if 1 + nl
-
ajyy
-
z)
we see that if + 960252n-1 n
dz
y yjyjjjy
.jj
Xjn S
4z3#(l(J1 (1+
(1+ #s/n)l
Combining the last three estimates
Ekx,a
dXt
S
2(n)2
#a/n))4, we
-
,
.
(1+
#a/n)J4Ig;
nz(1+
-
that
we note
The desired conclusion follow. easily from the last three observations. Our flrst two examples are the lmst two in Section 5.1, of the subject, the begiihgs ee Feller (1951)t
nzl
=
s 16n-3(z#r,)4 + 16n-3F((Z2
To bound the second term
(z)
rlxjz;
-
nl
yk/nlj
> z)
dz
%252%2n2
IzlS R then + j6(z/k/n)4
Since 6 > 0 is arbitrary we have established (c) and the desired conclusion and followsfrom (8.2) (7.1). To replace (A7/) by (A3) now, we observe that the convergence for the special cmse and use of the continuous mapping theorem as in Example 5.4 i ITIPy l ' .
n
..2
n-1
m=0
1
znm
.x
=)>
,
0
d&
Chapter 8
308
Secbion8.8 Examples
Wak Con vergence
(A3) implies that the probability of a family of size larger thau nl is o(nr?). Thus if we truncatze the original sequence by not allowing more than 6rtn children (A3J) will hold and the probability where % 0 slowly, then (A1), (A2) and will (:1 converge to 0. of a diference between the two systems
(8.5) Lemma.
all
R
1-t) property gives
=
1 + @1.T)
=
o
o
. Ineans :iz
:
i
y'
:
)
and
j
1(a>1+:). Using
thp.t 'F
S il.f)
'z1
o
h
=
s
pl
(z,y)
gives
Jlsi
2.7. Since
1
t)
-
y4
=
p,(0,
pt(0, p)Pp(Tn > 1
-
=
at time n gives
=
Pen (T
>
tr,Bv ) dr
Fs
let 'F@)
- >
=
hs +
ul
'u,
dn
and note
#ts +
os
=
jo*clfr)
,u,
#a+u) du
c
,
implies La
Br) dr F,
tr,
=
EB. ()
dr C Ss,
we
hs +
'a,
#u) du
have
s,) /'c(s-)(s) (
e'xptc-ls-tztstlexp -
J
4
lj k--.Ci
gives
) dy
'o
0n jFa)
,
't
./a
...
j-,
=
=
dr
ctgul d'uj and To evaluate the second term we 1et y@) J(#-,) expt/ .. = c(#,+u) fB3) dul F the prpperty implies note o % so Markpv rxptl -J
Ez =
o
s.(z(s)exp(c,)Is-)
n.' By mssmptiim it holds for 2.5. We will prov th result by induction 1(v'>1,.,Ex). 'F@) Applying the it is fo Suppse 1. n. now tet true n o
z
'
1(zs, (Tp >
and: zecalling
.15tf-S )
.E'.(y
so the Markov property
pllz, :48.35(Tc> f) dy
=
and npte
Jantf,
y)
=
we
.
(T > t)
PB
tr,#r)
.
E=
> 1+
Taking expected value now
With
'
1 so
-
now and recalling .J7.(f1
J%(A
Markov property
)v
/a
.'5
->
,
> .17.4.2
Taking expected
R
=
the second term
evaluate
a
are indep endent norfflals
1(g'o>f)'and nte
To
BiB: s
=
(
.sf
.,
have
under
sice
with i # J' 2.1 becomes
izj
=
the desnition of
and the desired result follpFs.
+ E ir
.E'.(.I.'&)
2.1 becomes
,
with mean zi. When ..
since un er .P
e Xa,we
Jvh-aj
=
.
d.
dv
f
0
=
side is smaller than #(.a+1)
'
2.1. Txke F(td)
),
.s
313
.E%I'Fo pr&;.r,) k a.P(.rz)
the Borel-cantelli lemma implies
so
= (T > n, Bn E ff recalling oyer and uslng our mssumption show
c2-n
.
..-
The right-hand
is summable
.P
Cheby-
and
The last result
1KmK2R
The right-hand
for Chapter 1
Ansuws
1, B3 G ff )
J(#f) expl .
c(#r)
tk)
Fa
=
.Eb(,)
JBt-aj
c(=y) du4
expt g
:
2.8. (2.8),(2.9),and the Markov property impl# that with probability e there are times sl > tl > s2 > t2 B (c) that converge to c so that S(sr,) s,zl there is a local maximum. and S(s) > #(c), In each interval (s,z+1 .
.
=
.
,
2.9. Z H limsup/n #(t)/J(t) 6 Ft so each c, which implle: that Z is constant
(2.7)implies surely.
almt
Pzz
> c) E
(0,1)
for
314
Solutions to Exercises
Azlswrersfor Chapter 1 for some n k CVLLrelation (1.3)imply
2.10. Let C < x, ta t 0, Ax (#(ta) k X = xAx. A trivial inequality and the scaling =
#n(.N)
k
.Fb(#(fN)
CW7)
k
.Pn(/(1)
=
#)
3.3. Since constat time. are syopping times the last three statements from the frst three. (5' A T K f) (S % t) tl (T K f) 6 Ft T (,$'v S tJ (S S tJ n (T S J q Ft fS+ T < t) = u,rEq:x-rxctlxs> < :) n (T < r) h
and
=
k C4 > 0
e
3.4. Defne
=
, so
osasl
it follows from symmetry J
,1
< e/2
ISa l < e/j,
1/,1
) e h by (i),so (5'< T) e Fs. (5' k T)n(T < ) U T) irz Fs n Sr. Takihg complements and intersections we get (S k T (S K TJ, and (S = TJ ate iu Fs F =
.
'
'
' .. . Uctlffn where ffn are ctsed. 3.1. . . Suppose Let .Sk . U'R cll'6. ffn 't rn In view closed is inf (t : Bt E Hgj is a stopping tlme. so (3.4)implles Ts of (3,2)We cap complete the proof now by showing that Ta Tx ms n 1 t Clearly, Tx K lirztF'n for all n. To see that < cannot hold let t be a time at which Bt e Then Bt 6 1% for some n and hence limo ; Taking the inf over @with Bt e we have limTk %Tx :
=
.
.
:
.
.
.
=
=
.
..
=
n
,
.
.
./1.
.
3.2. If m2-n
..
On the other hand if .A n (S < t) e Fg and the qltration is right 1/n)) then .An LS K ) (S < + t (. n e nn Ft-vlu = &.
4 > 0
=
.
.
sup
'.'
.jk
SitKe
tw
limsup Ta a
the bottom one for z ; 0 shows that if
Exercise 2.5 gives that if .
Pc
fl Ts t
''''''
''''
.
.
.
.4 n (5'< t)
'
.
.
0
'''
Using the top result for a)
.
''''
.
.
0 S J KJ
=
,
3.5. First if W E Fs then
nsasd
= P0
=
'
2& > 0
H
that
sup
T1 Rn Sn-1 V Tk Repeated use of Exercise 3.3 & by that 1?,ais a stopping time. As n 1 (x7 Rn 1 sups Tn so th frst conclusion follows from (3.3).Define Sa b# S = T1, Sn = Sn- A Ta Repeated use of Exercise 3.3 shows ttwt Su is a stopping time. As rt 1 x, Sn infn Tn so the t second conclusion flloWs from (3.2).The last two coclusions follow frm the .&1
' shows N
2.11. Since cooydinates are independpt it suffice. to prove the result in ne dimension. Continuity implies that if 8 i small
1
.
:p
-+
sup
follow
.
(x) and notitk Letting N Ag t .A we have .J%(/) k #p(.81 k C) > 0. Since S1 it follows from (2.7)that /:(.A) 1, that is, limslp-.c f(t)/W k C . e probability with the prof is cmplet. one. Since C is arbitrry
.Pp
315
'
ds dy
1
p1(0, g)
2
()
s)
=
15(Tn
s) dy ds
=
()
()
by Fzbini's theorem, so the integrand gives the density .J%4.2 Pv (% = t) = .J%(Tp f), (4.1)gives
y
6 Sq and the fact that Ta i,s >
.Pp(T0 g
1 + t). Since
=
'
j. .
(j
=
E
xu...x,u
wu
m(0,yj X
= ;, uo
t mI.o y
Letting
.
x
< = u B and hence a and result hint. 1 as n follows from the the is discontinuous in (0,n)) cxl -+
Taking expected values now gives the desired result.
+
=
-+
-->
-+
Solutions to Exercises
318
Answers for Chapter 2
inftt > 1 : Sa2 > .G). Since Let M1 = maxcsasl Bs2 and T rfl the strong Markov properiy and BI (4.3)imply that with probability #g2 at is discontinupt!s M. This shpws Ca yo s one # 1 s % (x) ahd the result follows from the hint. is discontinuous in (0,n)) 1 as n 4.6.
=
nJ. By (2.3)this that Xn S n. Jensen's inequality for conditionl implies in chapter4 of Durrett (1995)) Eexj
ar fltration
w.r.t.
.A)
=
. .
.. .
'
..
.
.
'
but this is 0 for r < s and true for r Fs then Fsaa it follows that if .
.4
n IS
,))
A
S'rl
k
j
s. If
=
-
'
is a martingale .
with respect . .
l.Tk+a)
2.5. Let Ta be a sequence that reduces X. The martingale property shows that if .A e J;tl then E t.Xagk 1(za>a); X) E (A 1(gk>c); .A) '
'
=
C
Fr
.E'txtamtlarrk 1(za>a)l.Tk+,) =
Letting n orem gives -+
XaS
-
= 1(n>a)f lX(amtlan 1(n>a) Yshsn1(sn>c) = .Xtam,lan
a
.
>
,
.0),
=
..
to .
x and using Fatou's lemma and the dominated convergence ECXL
; d)
%limf n-
Ezhn
1(gk>()) ; .)
=
.E'(.X;.)
f pe see that EXt K EXz < x. Since this holds for al1 Taking ECXL 1.T) To replace 0 by s apply the lat conclusion to K t t which by Exercise 2.4 is a local martingale.
E Fz we
.4
=
other hand
ECXtSCAn ('S S
.x.
.:1)
=
f(A1(s>o);.
n (S S
s))
=
ExssA
n (S.K sl)
.
$
=
X,yi
,
14 is increasing, we only have to rule out jump discontinuities. 3.1. Since Suppose there is a t and an 6 > 0 so that s < t < u then F(u)-- 7(s) > 3: for -+
Adding the lmst two equations gives the desired conclusion.
the-
.)
lim f(A1(,zk>n); S n-G)
on the
Note
(1.1.b)
.fk+,
ftasa 1(s,x>n)lG)
. .
XL. .g.,
.&)+.
.A)
.
(d n IS
1
wtxTal
=
G is that
.E'IJ',S;
will reduce expectation (see
sequence
gk)I.faa,) k e s (xnt js nx' )) =
.
.
-+
If s < t 2.4. Let Ta % n be a sequence which reduces X, and Sn (Ta the (Ta > > R Ta involved S on (& 0J then definitions (note SR (:: and imply fact stopping the 1(gk>.a) theorem, optional e
.
,
(3.5)in
l
x
dy + (log.R).P.II#f l > R4 k (2'mt)-d/2 :f l 1og 1p1
Ecx,
and
Chapter 1 implis that Fshs C Fs so if X( is Irtinale with reject t Ss it is a frtartihgle With respect t: Fshz To agr the tlii .. directin 1et E F. Weclaim that n (S > s) E Fsa,. To prk thi w hk#e t check that fr alt r ./1.
j jog jpjjpdy
0 so that if Irvl < &then lX Xr l < E. Assuming sn and un have been deflned, pick a partition of lsn un) not containing t with mesh < 5 and variation > 2E. Let sr&+1be the largest point in the partitiop < t and shosvs that the be the smallest point in the partition > t. Our onstruction ua) always of @a+1) variatih X 6, > is over the r,+1, variation ttl so (sn, 'tIp) contradiction which is in:nite, implies 14 is contluous. t a over (sn,
Let Zt
3.2. (aY+ 'Flz
3.8. By stopping at Ts inft : IXtl > n) and using txercise3.5 it suflices to prove the result when X is a bounded martingale. Using Exercis 3.7, we can t suppose without loss of generality that S 0. Using the L max imal inequality optional stopping and throrem martingale the Xj61n, on the martingal on the (X)r it follows that X;
-
'upl
-
' ..
=
-
3.3. .-XCZL iq a.,local martingale, so lf 5$ desiredyeslt follows from Exercise 3.2.
martingale.
3.4.
ah
)(X)
abx,
F)
c tXt$
=
(X, F)t) is a local
-
((Ar)s+t
. .
. .
.
we get .
.
. .. . .
.y
:
. .
.
.
.
-
-
-
=
='i
-
stopping theorem and Ekercise
whr thelast equality follows from thq optional 3.6. This shows that Zt is a martmgale, so (F)t
/ '
'
d
'
ihe
(vY)s+f =
=c
(X)s
.
=
.
s
-
. .
=
-Xo then (F, Zj3
=
.E'(-Ys2+, A'ja.a
-
-
.-2
(a'LZLt (F, Z) is a local
-
. .
=
=
'arzml
-
-
(I)s)I.'&+a)+ l.Xma 4 1,)2 A%)2 -Y.l+, + (X)s + (.X+, f(-Ys2+t (-Y)s+zlFs+a) = -Ya)2 Zs -(X)s+, + (-Y)s + (Xs+a
f(ZtI.T+a)
,
-.+
-
'
. .
martigale.
it i'otlows from (3.t) and 3.5. If T is such that X v' is a bounded rartigale the dennition of (A-) that (-YT) = (XIT. For a geperal stopping time .-?Tlet Ta = inf (t : 1.XLl > n), note that the last result implies (.X'TATa) = and then let n x. The result for the covariance process follows immediately from the definition and the result for the variance procqss.
,
-
E
sup
t Xn
XLv
%4ELX;hnj
ELLX,ltarr 0 =
=
(
'
(XITAT,Z
.-,
3.6. Since X1
-
(.X) is a martingale E (.X)t
so letting t dominated by
(x)
-.+
3.7. The
an
optional
=
5X/
we have .E'(.X)x
integrable stopping
random
l.Xtl K M for
and
EXol
-
; M2
variable
.
all
Letting n
-+
3.9. This is
(x)
now gives the desired cpnclusion.
an
immediat
4.1. It is simply a matter of patiently 1. s < t K c. '.K= Y E Sft so
K M1
This shows that XI (X)t is and hence is uniformly integrable.
(3.8).
of
consequence
a11 the cmses.
checking ,
-
2.sa)) s E.M+ z7(A4x
E
.
nd it
1(n>c)
'
Clearly, .4S1 .X') (5.4)implies
323
E
IlznllS Ilztl+ Ilz,z IIso
Ta
uzpalV X)1
S 4E
'
-
llzllk lirnsup Ilzall
Letting n
jo
Sf2 d(.X)t
x now we can conclude
-+
4.5. Let Hn e 11l with Ilffn Jfllx 0. Since Ilt#n .X'):=zCH X)II2 0, using Exrcise 4.4, the isometry property for l1l, and Exercise 4.4 again -+
-
.
:
Ilff Xlla -
-
.
.
lim ll#n X112= limll#n
=
.
-
lIx Ilffllx
With this in hand
=
we can start
5.1. lf L, f/ E M2 have the desired properties then taking N L f/ we have = (f,, L f/) Uj H L Using 0. Exercise 3.8 so (f, now it follows (f/, that L U is constant and hence must be Y 0. =
-
EH
.&)
-
-
-
.
with
.X')z2
=
a
E
'
-
FT
ltaszl then X = S k X and 5.2. If we 1et ff, H 1 and ffa ff F. (It is 'for this remson we have assumed v'L = Fp 0.) Using (5.4)now it follows that =
=
j
=
't
1(,sc-) #(.X',F),
=
(X,
F)J
5.3. If we let S'a = 1(,sz) and Jf, 1(,>rr) then XV .I X and 'F 'FT K F. Using (5.4)now it follows that (vYT, F H 0 'TIf so the deired rsult follows from (3.11). =
=
.
.
x
-+
to get
and ff ,
=
ff al +
.rf,2
where
d(a'):
result.
-
6.3. By stopping we can reduce to the cmse in which X is a bounded continuous martipgale alzd (.X)f K N for a1l t, which implies # 6 112(X). If we replace S and 1aby Sn and Tn which stop at the next dyadic rational and 1et Il'k c C for Sn < s K Ta then H.: e lI1 and it follows easily froz the de:nition of the integral in Step 2 in Section 2.4 that
=
j
-
6.1. Write .Jf, = ffal + #,2
thedesiied
.;:2
.
=
(Ah FT)
and let n
j
Ta
.,.7
dxa
=
cxn
-
xs-) .
the proof now we observe that if 1C@)1
'ro complete
ff then
:
H
;
=
ff 41
=
Jfaltssaso
=
Jfa ltssasz)
IlSk
-
Jfllx
S ff2(.E'((X)n
-
(-Y)g') +
.
.
ttxlsn(X)s)J -+
-
0
Solutions to Exercises
324
Answers for Chapter 3
theorem. Using Exercise 4.4 and (4.3,b) x by the bounded convergence as n it follows that 11(.Sk vY) CH vY)IIa 0 and hence H.n dxa zzadxa. Clearly, CLXLU Xsj Cxk A) and th desired iesult follows. -.+
.
-
J
-+
.
-->
-
=
XI XJ
6*4.
=
aY2(jP y) '+
=
-
.f'
-.+
.X'2(jP) t
of the Riemann approximating sums for the x by the convergence as n of integral. Convergence the means and variances for a sequence of normal random variables is sucent in distribution (consider the for them to cnverge characteristic functions) and the desired result follows. -+
t
.
7.1. supf l((ff ''
=
(-Y'(t7+z)
-
)l2+
-Y(P,
i
2Ar(t7)(A-(t7+T)
-
A-(tP,
)J
i
(3.8)implies that the flrst term
l'cAti
=
converges in pobability to (X). The prkious t shows second exercise the term converges in probability to 2X, dXa
6.5. The dtfl-erence betwen evaluating J7'(-Y(f7.it)
2i
at
the right
X(t7))?
-
l(t2-nt) and Dk = Bk ck n Bk + 1/2)2-nt) a 6.6. lzet1/2)i-af). Bk + In view of (6.t)it sumces to shw tllat ms n =
-
Su
J7Ck,nck,u
=
+s
k,a
k
.
)
.-+
+ 1)2-nt) x
As 8 0, GaJ Exercise 7.1. -+
' .. . .
('
-
t/2)2
(
.
V Ck2 )
2-T:4/2)2
:
-+
implies (6k7) (:x$
we have
fty)
-
-+
0
i
in probability. variance
The left-hand
i
side
of
-+
(0,) with
t -
tP) t
j:
' .
.
.
uniformly
(j
-.+
0
=+
132ds a
the desired
so
0
mean 0 and
conclusion
from
'
.
E
E
and
+.A( is pntinuouj
.t
vl
0.
aPrB1,
=
c)
=
.
theorem implies that '
m/z
=
=
(
=
z)/('-
-
'
(1 PrBr
u) +
c))
=
-
c).
1.2. From (1.6)it follow that for any s and n Ptsupk, Bt fz) 1. This implie Ptsupfk, Bt x) z 1 for all s and hente limsup 4-. tkj B # cxn a.s. =
'
=
'
. . .
.
EE
..'
.
'
'
.
5'r(Y)
< x
with
'
.
'
.''
:
'
E
'
: .
positike probabilit#
t
.
'
.
.
j
en
.
..
.
#t(o,)()
-+
3.2. Let J(z, )
::.z
z6
-
=
E
01z2
a2
+ bz2t2
a4t
Dtf
=
=
-(Iz
(1/2)DraJ
W .
+c -
+ 52
n
*
=
.
+c
ct3. Diserentiating
4 + 2z 2 t = 3ct a 15z4 6cz2j + = -
j2
c
gives
0
=
3.1. Using the optional stopping theorem at time T A f we have EftB7)2 sy .E'c(T A ). Letting t convergece x and using the bounded and monoton theorems with (1,4)we have E 0T
with
(0,t)
on
dA'
.
1 @(td)which is 1.3k If a'ndwe have a contradiction.
peh
joC
=
'
.
s.,d.%
hms a normal distribution
(tP :,2 't ;% z+ 1
.'
1*
.
that if )' is a sequence of partitions ht:Bt:w,
.
Solving gives PrBv
.
If we let
(' . .
z
,
2ncj2-2n + Ck2a .D2 k,a S
-
1a
T(.o)
-
.
l
,
'and then use Chebyshev's inequality. 6.7: ms n
.
.
.
2-nj/2 + Ck nDkvg
-
T:
?. f (Xa)
-+
1.1. The optional stopping
E
k
V(C2 = E
.
.
E
=
I.AIK M.
when
suces
i
. ..
Chap er 3
2
E'n
0
-+
local mrtinxale 8.1. Clearly Mt + MIis > cotinuous nd hs dp + adapted, locally of boundd variation
-
in probability. EC2 ,n = 2-n,/2 and kck sfjk a = 0 so ESg = t/2. To complete the proof now we note that the terms in the sum are independent so ,
l
ff,
-
t
tjg
...
,
Ifn
supa
to prove the result then as in (7.4)we get
it
Jk+,))
T(A)
2(A-)t
-+
.
.
the left end points is
and
ff ) .A)fl % M
-
stopping
7.2. By
Gj
,
Ja
325
=
Solutions to Exercises
326
Answers for Chapter 4
Setting c = 15, 25 = 6c, and = 3c, that is c 15, we have 15, b = 45, c (1/2)D=.f martingale. 0, fBt is local t) Using the optional so DtJ + a stopping theorem at time a A t we have =
=
=
EnfBnht
4 lsfv.oazt'&
-
-+
x
the first term on the right
Letting t cxn and using the boun with the results in (3.3)we have
15./r3 fl
=
c6
2
)+
A
.-.+
E
ed and monotone c9(1
=
.G('D(.Wa); r >
.
!
A i)
h i) ) = 15c(a
15c4.&a + 45c25r2 J
-
,
2
45#aa(a j
,
3
and it follows that for large n,
theorrs
convergnce
15 + 75)
=
#(z)
gyt
so
61/15.
=
j
=
t'
Ltting lmt
n
(x)
-+
kulely
and poting that on d hnce Zt tks a..,
IT-U
-
x)
have Ztlt = csf/ the desired ilt fllows. =
y
+p
we
-.+
p
y
-+
=
*
J
=
=
.
.
,
=q
.
-
.
.
.
,
=
.
.
.
0z
.
y4 tfp
-
.EE
:
=
.
.
JJ
=
,
=
=
7.1.
(a)lgnoring
Cd and dilerentiating
Dz.k:
=
d
-y
gives
Dxa.:
(1z
-
y (Iz :1z + y a)(a+a)/a
-d
=
0i4y #1a + p a)(a+a)/2
2(zf .
.
Dyho
=
(lz
1 #lz + yzjalz ..
-
#)2p #(#+ 2)(z 2p)(d+4)/a #j2 + (jz -
+
-
,
'tz)
follows.
-
2
# (Iz #I2+ !P)td+2)/2 d(d+ 2)p/ # + 2yi + (d+a)/a z pI2+ y ) (jz #ja + ,a)(a+4)/2 o
,
-
,-
d
.
-
-.
.
-
-.
.
-d
=
.
(1z
4
-
4.1. Let r inf (t > 0 : Bt $ Gj 1et p E 0G, and 1et 4 inftt $ 0 :'# ciii p). (2.9) in Chapter 1 implies #p(Tp 0) 1. Since r % Tp it follows that y is regular. =
=
,
=
Adding up we see that
=
d-1
X-! f=1
Dxrp
4.2. .Fjt'r = 0) < 1 and Blumenshal's 0-1 law imply .J$tr = 0) 0. Since we yj 0 nd hence 6 1 EyfBpj > are in d k 2, it follows that PvB. Let trn = inf (t > 0 : Bt / Dy, 1/n)). a'n 0 ms n 1 (x) so Pycu < 1. t =
=
e
=
EvfBp)
=
.G(J(#.r);r % tzrjl
+
-
.
+
r > .G('??(#c.);
cu)
1) -lJ
-
N I r;z u pj2
(-d) wa(d+a)/2
.y..
=
'
d(d+2)p(Iz -
--+
-
((-d)(d
=
l
:+
'r)
1
+ Dvuhe
The fact that
u(z)
=
,
-
'
Dyyhe
Chapter
..
=
,
,
#r2dr: By mpdifying Fzaafter with (.X), = time t we cn suppose withopt loss of grnrality tht (A-)s ::: c then Xvtulls a BrWnln motibn. now we see tlkat if Sipce (A'), >nd hnce th time change n'(u)are deterministic, the dsired 4q!1t ,/,4u)
p))
-
(, 4.4. Let r = inft > 0 : Bt e ?(p, 1), a)J. By translatzon and rotatzon we can ::':: suppose y = 0, r (1,td0, 0), and the #- 1 dimnsipnal hyperplane is zd 0. Let Tp inf ( > 0 : B. 0). (2.9)in Chaptr 1 implis that .J%(Tc= 0) 1. If c > 0 thenfor som'e k < (x) the hyperplane can be covered by k rotations of J-/ so f%('r = 0) k 1/i and it follows from the 0-1 law that .J%('r 0) 1. F.:.Fz
-
Xa is a local martingale
Vp(p + p(z
-
.
4:2.
.
'vj
-
,
a
-
.
Jf
-
-
E E
Rd be piecewise constant and have 4.1. I#,I 1 for a1l s. Let = 'i E i 0 dXia The formula for the covariance of stochastic itegrals shows '.Kis a local martingale with (F)z so (4.1)implies that M is a Brownian motion. Letting 0 tc < t1' < < t,z and taking 0s for s E (tl-, t./) for 1 %j S n. shows that a'Vz X3o .X'z. Xgn- are independent multivariate n-llf matrices (tl tilf j normals with zean 0 nd covarince and (tr, it follows that Xt is a d-dimensional Browniap motion. Let 0 (0,x) :
K1
-
.
l-eartl
1=
r(z)
E
-
74g, Tgy), 1). Calculus gives us
=
1
1
-+
Continuity of Vg implies that if Iz p1 < r and z e U then Tgy + p(z y) k 0 so #(z) k 0. This shows that for small y)) 0z r the truncated cone U n Dy, r) C G so the regularity of y follows frprp (4.5).
3.3. Using 'the optional stopping
theorem at time T-c A n we have j Eo expt-tz#/J
cu)
infxEdotp,l/s)
4.3. Let y E OG and consider U -
Ezvt
0, so
-+
,
.
6
As n
327
0 follows from
( jz
(1.7)Aas
-'.
J
.
31# E
pI2+ y2)
p jz + y z)(,+.)/z
in the proof of
=
()
(7.1).
:
y
Anslvers for Chapter 5
Solutions to Exercises
328
(b) Clearly J do ptz, p) is independent of z. To show that it is independent of p, let z = 0 and change variables 0i = ypi for 1 S K d 1 to get -
jdossz,v/
(gpjwjca wdyyzl,,,
(c) Changing variables 0i
zi + rf y and using
=
D(.,4e
p)
ptz,
d0
1)
-
-
1
-,
Dz,slylr
1)
rto,
dv
-
...:11#11x,,
-
=
-
-
since for To extend the last result to Lipschitz continuity on sexvingthat if z e D1, y e 1)2 and is the joint of bD betweenz and y then
0
-.+
=
-
.
dominated convergence
=
: z E D lIIIx,f sup(I1-(z)I 1,and
I#(z) (p)I: IlJIIx,2l#() .(p)I+ I/(z) J(p)lIl.lIx,a s 1 czlz 7/1+ 1 !/1 C1z pl + l#(0)l/A)lz .%(26% z e D1, I(z)l K I(0)l + CR.
jdvsro,
.yd-dv
-
Combining the lmst two results, introducing using the triangle inequality:
,
(d) Since 9.1. r(z)
c(z) =
< x)
.&('r
-#
0 we flnd
pu
-
2p. Then we take B
=
,1Vwq
the lie
o
(z)l+ l(z) (g)l al + c2lz pI (72Iz pl Iz p1, This shows that
-
Bb2
=
coshtz)
PB
-
2
3.2. (a)Diferentiatint 0
=
-
=
h/(z)
=
is Lipschitz taking
-
argument
-czlogz
-llog
x&(z)
27) to satisfy the boundary condition.
1/ coshtc
=
have
we
x(z)
by obr segnient
-
=
-
-
egin
.
since C % C2 and lz zl + lz pl continuous with constant (72 on IzIK 2S. Repeating the last lzl % 24 and y > 2./ completes the proof. -
g
329
z
0 if
0
if 0 0
z >
< z < e-1
26:-2 And so x is strictly increasing and cpncave on (0,e-2). Since x(e-2) /P(e-2) = C x is stric'tly increastn: and concave o cr). When :7 K e-2 (0, =
Chapt
r 5
1
is Lipschitz 3.1. We frst prove that ..a-1l(0)Ion oa (.R lzl K 2J0. Let #(Xz/Izl).lf z, y e D( then
with
conttuous
s
=
J(z)
constant
2R
=
-
IzI)/S
Ca and
=
:(z)
1 IT(z) T(p)I IIpIJ, lzll S qilz
y
-
Since is Lipsch itz continuous of z onto D we have
op D3
(lzlK J0,
l#(z) .(y)I I(./lzI) -
=
c,
pl
-
=
j
R
-
Apj
and
Az/lzl is the projection
-
,
(r
=
y
5.1. Under
-1
-c-l
Cz logz dz
expl-l/tr)
.
g
t
(j)
Q the
=
with
p t j)-j.y
zje
0
=
x
Llyljp
1/@p ) = pg()(log(1/g(@)))
E
.
coordinate
maps
=
vL(u?)satisfy MP(# t
/, -'ez /' txa)
u't, x, g1
loglog
=
p > 0 then
expt-
=
(lp/lpI)I
Izl 1p1s cllz -
0
=
(b)lf #()
=
-
f
2C1 +
#(x,)
-
-
0
+
ds
(x,) ds
+
,
J).
Let
y
Solulons to Exercises
330
Since w
intrested
are
Anslsrezs for Chapter 6
in adding drift
let
we
-
c
c-l
=
ms
before, but let
so w(z)k z, w(x) = x and (ii) If (z)/c(z)k e ylzen
j
J
=
c(#,)
-
n
.
p
t
c(Xa)
-$
+
=
a-l(x,)
+
eXP
(a'L)ds
.
0
-yk
=
dX,
so
ds.
0
w(x)
-
-a#
#
.
:44)
y)
-
ap-l
y.x w#
ew
-+
=
=
T./aAT2.
l-2 ds
0
(Ta/a A T2z) o 0n
does not depend on z. Letting Sa Markov property it follows that the distribution S=
1-2 ds
=
(x)
-+
in
(4.7)of
j*
-
using the strong
and
(5.3)to
of
Lu
under P
siuce
R*.
does not depend on z. Pick J > 0 so that Plr k J) > 0. The fa-n are inde1 pendent so the second Borel-cantelli lemma implies that Pltfa-a k 6 i.o.) with < x) > 0. but this is inconsistent =
(z) i
Chapter
=
p
-x
>
0.
Chapter 6 we have
.FQqljxj.= 2(' (z) w(1))
5.2. We apply =
-+
7
m(z)
-6/z#,
=
v(z)
=
gjkpz)
1/fpJ(z)c(z)
=
z1+J
(1 + 8j8 2
dz +
w(1))mtzl
-
l
=
The result now followsfrom m(z)
under P,,
J
exe
-
5.1. Letting b
-1.
1. To do (c) lf # > 1 then .S > H so it suflices to prove the result when J at 1, starting motion = is Brownian z-Bnr, that if Bz this, we note a z then of distributiop so the
I
*
C
=
t (#)J(#)
0. From this we see that I < cx) if p < 1/2. When # 0, M(0) = Cz2# as z M(0) x. When p > 0, Mtzl 0 so J < (x7 if Comparing the poibilities for I and J gives the desired result.
sy so .J
Chapter since 6 < 1 implies 1
-
1
-1
5.3. (a)ffj K Ta < x. (b) (2.9)in Chapter 3 implies
333
Hexe and in what follows C is a constant which will change from line to line. If (p40) and f # k 1/2 then w(0) = Cz1-2# x. If p < 1/2 then w(z) ahd =
in Example 5.3 show that (a) I < t:xn when 7 < 1 and 5.2. The computations and hence J = x. Consulting the table (b) for 7 < we have AT(0) = of deinitions we see that 0 i an absorbing boundary for y
0 and a subsequence lrn': rl > t for all k. However it is impossible for a subsequence have a contradiction which proves rn r.
we
ltave the indicated
-+
Referencs M. Aizenmau and B. Simon (1982) Browniaa motion and a Ilyrnack inqpality for Schrdinger operators. Comm. p'tlrc All. Xf. 35j 209-273 P. Billingsley (1j68)Convevgence ofpvobability New York
Measures. John Wiley
Mariov Pvocesqs R.M. Blmepthal and R.K. detoor(1968) Tlteqvy. Academic Press, New York
anb
Sons,
ePofenicl
and
D. Buzkholder (1j73)Distribution function inequaltties for marhgatest Probab. 1, 19-42 D Burkholder, R. Gundy, >pd M. Silverstein (1971) A m>xtrat HP. class acterizatiop of the Trcn.s A.M.S. 157, 137-153 '' .. . '
.
'
K.L. Chung and Intvoduction .J. Wiltiams (1990) edition. Birkhauser, Boston second
K L Chung and Z. Zhao (1995) From Brownian Molion ion. Springer, New York '
g
. '
.. '
function char'
ochqstc
t/ tl
Integmtion.
Schrsdingev's fpzc-
'
.
Wnn.
.
.
First pmssage times and sojlly Z. Ciesielski and S.J. Taylor (1962) tipes for Brownian motion in space and exact HausdorF memsure. Tmns. .u s .
103
1
t34-450
!!
-
.
.
'
Semimartingates and E.. Cinlar, J. Jacod, P. Protter, and M. Sharpe (1980) Markv proceses. Z. fuv. Wahr. 54, 161-220
(1983)On Brownian slow point. Z. fnr Wahr. 64, 359-367 and P.A. Meyer (1g78) pvobabilities c?,d potentials ? xqrts oellackerie uol-
B. Davis
c
*
land, Amsterdam
.
C. Dellacherie and P.A. Meyer (1982) Probabilities Mavtingales. North Holland Amsterdam .
and
Poteniials B. T/lct?o of
1
C. Dol>ns-bae (1970)Quelquesapplications de la formule de changement variables pour 1es semimartingales. Z. fnr Wor. 16, 181-194 > ro a iility:Theovy and Ezamples. Second R. Durrett (1995) Press Belmont, CA )
edition.
de
Duxbury
336
Referencs
References
M. Kac (1951) On some connections between probability theory and diserential and integral equations. Pages 189-215 in Proc. 2nd Berkeley Symposium, of California Press U.
Some problems on random walk in space. A. Dvoretsky and P. Erds (1951) Pages 353-367 in Proc. 2nd Berkeley Symp., U. of California Press
(1965)Marko'u Pvocesses. Springer, New York A criterion for invariant measures of Markov processes. P. Echeverria (1982)
E.B. Dynkin
1. Karatzms and S. Shreve (1991)Brownian Motion Second edition. Springer, New York
Z.
kkr Wahr. 61, 1-16
Markon Processes: Ccrccterzcon S. Ethier and T. Kurtz (1986) and Sons, New York John Wiley wergence.
S. Karlin nd H. Taylor (1981) W Second Conrse in ' demic Press, New York
Con-
and
T. Kato (1973) Schrdinger 135-148 13,
W. Feller (1951) biflksionprocesses in genettcs. Pages 227-2246in Proc. 2nd Bevkeley ,S'pflzp.,U. of Califorhia Ptss G.B. Folland (1976) Xn Inkvoduction Princetn NJ U Press t *
)
E
d D 9gsion. Holden-bay,
an
Pariial Dilkrcnicl A Friedman (1964) Hall, Eglewood Clifs, NJ
Equakions
'
.
'
E.
.
' .
:
F
..
.
'
. '.
of
.
.
.
D. Gilbarg and N.S. (1977)Eiliic Scond Ovdr. Springer, New York
(1969)Reql and bs .
rccf Analysis. Springer, New
(1981)Stociastic
Drereztticl North Holland, Amstrdam Lprocesses.
Equations .
(1987)fimif Theorems
J. Jacod apd A.N. Shiryaev Spr in/er, New York P. Jagers
(1975)Bvanclting Pvocesses lnit
j
..
.
and
.D;#k-
.
for Soc/zcdc
rocesses.
Biological Applicaiions. Joim Wiley
and Sons, New York
Pavtial Dkrrenql F. John (1982) Ne:? York '
Equalions. Fourth edition. Springer-verlag,
s'occsic Pvoceses.
Aca-
Isvael J. J/.
and
Dlrkdon. American Math.
(1969)Siochastic Integrals. Academic Press, New York (1976)Un ours sur 1es intgrals stochmstiques. Pages 246-400
P.A.. Meyer Lcture Notes
bessel
kqnaiionsof
Calculns.
H.P. Mclfean
ln Vath. 511, Springer, New York
'
in
N. Meyers and J. Serrin (1960) The exterior Dirichlet problem for second order elliptic partial diserential equations. & fc. Meclt. 9, 513-538 B. Oksendal (1992) Stocltastic Dlrerencl New York
order eniptic
Partial Dlrrenicl
'ludinger
S. Watanabe
second
s'occc
with singular potentials.
F. Knight (1981) Essentials of Brownian Motion Society, Providence, RI
..
( . ' ' ' ' 1(' .. .
of Brownian motion and
o.
.
E
olume
diliccfons,
and
.
type.Prentice-
Paraolic
Gilbarg and ,. serriu(1s56)on isolated singutarities o diferetial equations. Ji d'Analyse n. 4, 309-340
and
.
' .
Excursions R.K. Getor and M. Sharpe (1979) 47, Wahri 83-106 Z. proeses. fnr.
E. Hewitt and K. Strom erg York
San Fran-
.
:
Stocltastic Dlrerentfcl Eqnations A. Friedman (1975) 1. Acadmic Press, Ne* York
N. Ikeda sion
'
,
Brownian Motion D. Freedrpan (1970) cicd CA
operators
and
R.Z. Khmsminskii (1960) Ergodic properties of recurrent diflksion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Tlteov. Prob. Appl. 5, 179-196
Partial Dllkrcnticl Eqnations. Prince-
o
337
Eqnations. Third
S. Port and C. Stone (1978) Bvownian Molion Academic Press, New York Stochastic Inkegration P: Protter (1990) New York
and
and
edition.
Springer,
Classical Jbenfcl
Tlteo'y.
Dlrrenicl
Equations. Springer,
D. Revuz and M. Yor (1991)Continnons Mcringcle.s and Brownian Motion. Springer, New York L.C.G. Rogers and D. Williams (1987) Dtkgsionst Markov Processes, and Maw tingales. Volnme 2: It Calculus. John Wiley and Sons, New York
(1988)Real Analysis. Third edition. MacMillan, New York B. Simon (1j82)Schrdinger semigroups. Bulletin A.M.S. T, 447-526 D.W. Stroock and S.R.S. Varadhan (1979)Mnltidimensional Dirkdo?z
H. Royden
cesses. Springer, New York
Pvo-
338
Refezences
additive functionals H. Tanaka (1963) Note on continuous Brownian motion. Z. kkr Gcr. 1, 251-257
A diflksion with a discontinuous loil J. Walsh (1978) 37=45
of on-dimensional
tiiri. Attviqne 52-53,
Index
of solutions On the uiqueness T. Yamada and S. Watanabe (1971) to stoihastic diserential equations. J. A. Klloto 11. 1, 155-167: lI, 553-563
adapted 36 arcsine law 173, 291 Arzela-Ascoli theorem 284 : 1aw 76 nxqociative averaging propertyl 148
D, the space 293 difeventiating under >n integral 129 difusion process 177 Dini function 239 Dirichlet problem 143 distributive laws 72 Dolans measure 56 Donsker's theorem 287 Dvoretsky-Erds theorez 99 q
E
basic predictable mocess52 Bessel process 179, 232 BichtelemDellacherie theoxem 71 Blumenthal's 0:1 law 14 boundary behavior of dilusions 271 Brownian motion continuityof paths 4 de:nition 1 Eldex continuity 6 Lvy's characterlzation of 111 6 nondilexentiilisy quadratic variation 7 write.s your name 15 ! Zeros f 23 Burkholder Davis Gundy theorem 116 '
,
;
esectiv dimension 239 exit distributions ball 164
Ein
.
.E
half space i66 exponentialBrownian motion 178 exponential martingale
108
Feller semigroup 246 Feller's branching diflkston 181, 231 Feller's test 214, 227 Feynman-lfac formpla 137 fltration 8 flnite dimensional ets 3, 10 Fubini's theorem 86 fundamental sollztion 251 .
C, the space 4, 282 Cauchy process 29 change of measure SDE 202 change of time in SDE 207 CiesielskivTaylor theorem 171 cone condition 147 contilmous papping theorem 273 contraction semigroup 245 converging together lemma 273 covaxiance mocess50
.
E
.
generator
of a semigroup
Girsanov's formula 91
246
Green's functions Brownian motion 104 one dimensional disusions 222 Gronwall's inequality 188
Tndex
346 Index harmonicfunction 95 averagingproperty 148 Dirichlet problem 143
zaxris chains 259 heat' equation 126
inhomogeneous 130 hitting times 19, 26
half space 31 optional process 36 in discrete time 34 optional stopping theorem 39 Ornstein-uhlenbeck proces 180
stopping time 18 strong Markov property for Browni>n motion 21 for dilusions 201 strong soltltion to SDE 183
pathwise uniqueness for SDt 184 theorem 11 'm Poksson's equation 151 potential kernel 102 predictable process 34 basic 52 in discret time 34 simp Ie 53 C ( predictable c'-feld 35' progressivley measurable process 36 Prokhorov's theorems 276 P unctured disc''146
tail c-fleld 17 tight sequence of memsures 276 transience, see recurrence transition density 257 transition kernels 250
-
5*
ininitesimal drift 178 infnitesimal generator 246 infnitesimal varianc 179 intjration by parts 76 isometry property 56 It' S fOrmll la 68 multivariate77 i !1 E
.
' E
'
:
;E
'
. .
.
' .
' . . . . . '.
.
.
'
.(
:
Kalman-Bucy fllti 180 Khmsminskii' lmma 141 Khsrnihskii's test 237) Kolmogorov's continuity theorem 5 Kunita-Watartabe inqtzality 59 '
''
.
E
.
law of the iterated lojarithr 115 Lebesgue's thorn 146 Lty' theorem 111 37; 113 local martigal local tiine 83 continuity of 88 locally 4uivlent memsures 90
qua dratic
variation
recurrence nd transience Brownian motion 17 98'. one dimensional difusions 220 Hafris chains 262 reflecting boundary 229 reflection principle 25 regular point 144 resolvent opertor 249 relatively compact 276 right continuous fltrti
E
.
.
E
E
'
E
'
'
1
'
.
Markov property 9 martingale problem 198 well psed 210 Meyer-Tanaka fothmla 84 283 modulus of cotinit# monptone class theorem 12
'
E
.
scaling relation 2 Schrdinger equati 156 semigroup pibperty 245 semimartingale 70 shift transformatios 9 simpte predictable process 53 skew Brownian motion 89 Skorokhod representation 274 Skorokhod topology on D 293 speed measure 224 E
.
natural scale 212 nonnegative definite matricts '
'E
occupatiori
.(
timsE
ball 167
i
198 . ..
Cr
E
'
weak convergence 271 Wrightisher diflksion 182, 234, 308 Yamada-Watanabe
50
E
i '
'
.
E
E
E'
'
'
.
.
variance process 42
'
'
.
.
uniqueness in distribution for SDE 197
.
E
'
i
('
'
q
theorem 193
341
