This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
r > T) = J-TZl and the family of measures P*'1. This means that Et,x
/(x T ) T\
= E*'X
[/(^)
Xs
E's.X'
/ (*T)
T)t,X
a.s.
for all 0 < T < s < t < T and all bounded Borel functions / on E. It is easy to see that the backward Markov property for the time reversed
Transition Functions and Markov Processes
17
process Xs — XT-S follows from the Markov property for the process Xt. Therefore, if Xt is a Markov process with transition function P, then the process Xr-t is a backward Markov process with backward transition function P. In general, any property of Markov processes can be reformulated for backward Markov processes. We simply let time run backward from T to 0. For instance, if P is a backward transition function satisfying the subnormality condition, then we can use the construction in Section 1.3 to extend P to a backward transition probability function on the space EA.
1.5
Space-Time Processes
A transition function P is called time-homogeneous if P(T,X;t,
A) = P((T + h)AT,x;
{t + h)AT, A)
(1.16)
for all 0 < T < t < T, h > 0, x e E, and A E £. The values of a time-homogeneous transition function depend only on the time span t — r between the initial and final moments. We will write P(t—r, x, A) instead of P(T, X; t, A) in the case of a time-homogeneous transition function P. The minimum (T + h) A T appears in formula (1.16) because the parameters T and t vary in a bounded interval. Definition 1.5 Let (CtjJ7) be a measurable space. A family of measurable mappings fls : Cl —> Cl, s > 0, such that i W . = 0t+., #o = I
(1-17)
for all t > 0 and s > 0, is called a family of time shift operators. The symbol / i n (1.17) stands for the identity mapping on CI. If a stochastic process Xt with state space (E, £) is given on a probability space (CI, P), then it is natural to expect the process Xt to be related to the time shift operators 6S as follows: Xt o # s = X(t+s)/\T
(1-18)
for all t £ [0,T] and s € [0, T]. Equality (1.18) is often used in the theory of time-homogeneous stochastic processes. For some sample spaces, it is clear how to define the family of time shift operators {"ds}. For instance, if Cl = E$ and the process Xt is given by Xt(uj) = w(t), then we can define the time shift operators by -ds(u>)(t) = u((t + s) A T).
18
Non-Autonomous
Kato Classes and Feynman-Kac
Propagators
Let P be a transition probability function, not necessarily timehomogeneous, and let (Xt,J-[,FT>x) be a Markov process on (Q,F) associated with P. Our next goal is to define the space-time process Xt = (t, Xt), t G [0, T]. It is clear that the state space of this process must be the space [0, T] x E. However, it is not clear what is an optimal choice of the sample space for the space-time process and what is the transition function of the process Xt. It is natural to expect that the sample space for the process Xt has to be the space [0, T] x CI. We will show below that this is the case under certain restrictions. Let B be a Borel set in [0, T] x E. For every s G [0,T], the symbol (B)s will stand for the slice of the set B at level s, defined by (B)s = {x G E : {s,x) G B}. Consider the following function P (t, (r, x), B) = P (r, x- (r +1) A T, (B)( T + t ) A T ) , where (T,X) G [0,T] x E, t G [0,T], and B G be rewritten in the following form: P (t, (r, x), dsdy) = P{T, X; S,
B[ 0 ,T]X.E-
(1.19)
Equality (1.19) can
dy)d5{T+t)AT(s),
where 6 is the Dirac measure. If the transition probability function P has a density p, then formula (1.19) becomes P (t, (r, x), B)= f
P{T,
X; (T
+ t) A T, y ) ^
-' ( B ) ( T + I ) A T
= 11
P(i~,x;s,y)dyd6(T+t)AT(s).
(1.20)
The time variable in the definition of P is £ and the space variables are {T,X) G [0,T] x £ and B G £[0,r]x,ETheorem 1.1 The function P given by (1.19) is a time-homogeneous transition probability function. Proof. It follows from (1.19) that the function P is normal. Next, we will show that the function P satisfies the Chapman-Kolmogorov equation, which in the case of time-homogeneous transition functions has the following form: P ( ( t i + t 2 ) A T , ( T , a ; ) , B ) = [[
P(t1,(T,x),dsdy)P(t2,{s,y),B)
Jj[0,T]xE
(1.21)
Transition Functions and Markov Processes
19
for all (T,X) € [0,T] x E, B E B[0>T]xE, 0 < h < T, and 0 < t2 < T. By using the Chapman-Kolmogorov equation for P and equality (1.19) twice, we see that P ((ti + t2) A T, (r, x),B) = P (r, i ; (r + h +12) A T, ( B ) ( T + t l + t a ) A T ) = / P(T,x;(r
+
ti)AT,dy)
JE
P =
((T
/ /
+ ti) A T, y; „.
(T
+ ti +1 2 ) A T, (B) ( _p(T>x''s'dy)p(s'y''(s+t2)hT,(B)is+t2)AT)ds{T+tl)AT(s)
[0,T]xE
J he[o,r)xB
P (ti, (r,x),dsdy)P(t2,
(s,y),B).
This establishes (1.21). Therefore, the function P defined by (1.19) is a transition probability function. • Note that even if the transition function P has a density p, the measure B — i > P(t,(T,x),B) on B[otx]xE is singular with respect to the measure dtdm. The function P will play the role of the transition function of the space-time process Xt. There are several possible choices of the sample space fl for the spacetime process Xt- For instance, one can choose the full path space, that is, the space n = ([0,T\xE)[o'Ti, to be the sample space of the space-time process Xt. space-time process is defined by
(1.22) In this case, the
Xt( (c + t) AT as the first components of the elements of fl* and the paths u> from SI* as their second components. Note that we can identify the space fi* with the space [0, T] x fi*, using the mapping j : £1* —> [0,T] x Q,* where j (y c ,w) = (C,LJ) with c G [0,T] and wGfi*. L e m m a 1.5
T/ie following equality holds for all (T,X) G [0,T] X £ : P^ TIX)
(n*) = i.
(i.26)
Transition
Functions and Markov
Processes
21
Proof. Given 0 0.
(1.35)
The operators -d3 are backward shifts by s with respect to the time variable T. The time shift operators ds are connected with the space-time process XT as follows: Xrotfs = X(T_s)v0
(1.36)
for all s > 0 and r G [0, T], It is also possible to define space-time processes on smaller sample spaces as it has already been done in the case of transition probability functions. For instance, let P be a backward transition probability function, and let (XT, F[, P*'x J be a backward Markov process defined on the sample space Q and with P as its transition function. Assume that there exists a family of time shift operators i?5 on the sample space Cl such that XT o d„ = X ( r _ a ) v 0 for all s > 0 and r G [0,T]. Then the space-time process
(1.37) (xT,TT,P{t'xA
can be defined on the sample space [0, T] x fl by Xr{c,w)=
((T-C)V0,XT(C))
where (c, w) G [0, T]xCl and r G [0, T]. The time shift operators ^ s on the sample space [0, T] x f2 of the space-time process are defined by ?a(c,w)= ( ( T - C ) V 0,0,(2)) where s > 0 and (c, w) G [0, T] x Cl. It is clear that condition (1.36) holds for XT and i? s . 1.6
Classes of Stochastic Processes
In this section we introduce and discuss various classes of stochastic processes. Let (Xt, FJ, PT,X) be a stochastic process on (fi, T) with state space (E, £). A sample path of the process (Xt, FJ, PT,X) corresponding t o w e f i is the function s\-^> Xs (LJ) defined on the interval [0, T]. For a given u G O, the sample path s >—> XS(UJ) is often called the realization of LJ, or a realization of the process Xt.
26
Non-Autonomous
Kato Classes and Feynman-Kac
Propagators
Definition 1.6 (1) A process (Xt,?J,FTtX) is called right-continuous if its sample paths are right-continuous functions on the interval [0,T). (2) A process (Xt, J-J, PT,x) is called left-continuous if its sample paths are left-continuous functions on the interval (0,T]. (3) It is said that a process (Xt, !F[,FT,x) is right-continuous and has left limits if its sample paths are right-continuous functions on the interval [0, T) and have left limits on the interval (0, T]. (4) It is said that a process (Xt,!Fl,'PTtX) is continuous if its sample paths are continuous functions on the interval [0,T]. If the process (Xt, FJ,fr,x) is right-continuous and has left limits, then the following notation will be used: \imX3(u>) = Xa-(u).
(1.38)
Since the process Xt is right-continuous, we also have \imXs(uj) = Xs(w). sit
(1.39)
Definition 1.7 A process (Xt, J^,FTiX) is called stochastically continuous if for all x G E, T G [0, T], and e > 0, , lim
FT,x(p(X!l,Xt)>e)=0.
Recall that the symbol p in Definition 1.7 stands for the distance on E x E. For e > 0 and y G E, put Gc(y) = {xeE:
p(x, y) > e} .
(1.40)
Then an equivalent condition for the stochastic continuity of the process Xt is as follows: for all x G E, r G [0, T], and e > 0, •
, Km
t — sl0;r<s(a). If there exist Ai and A 2 such as in (b), then we set A = A\ and B = A2\Ai. It is clear that A £ T, B £ J7, A\A C A2\Ai = B, and H(B) = 0. (b)=>(c). If there exist Ai and A2 such as in (b), then we put A' = A\ and B = A2\AX. It follows that A' £ J7, B £ J7, AAA' C B, and fi (B) = 0. (c)=*>(b). If there exist A' £ J7 and B £ J7 such as in (3), then we set Ax = A'\B and A2 = 4 ' U B. It follows that Ai e T and A 2 £ J". Moreover, A\ C A. Indeed, if a; £ A\, then a; £ A' and x £ B. Suppose that x £ A. Then x £ A'\A C AAA' c B, which is a contradiction.
30
Non-Autonomous
Kato Classes and Feynman-Kac
Propagators
Hence x £ A\, and the inclusion A\ C A holds. Next, we will prove that A C A2. Let x £ A and x £ A2. Then x £ A' and x £ B. Therefore, x £ A\A' C AAA' C B, which is a contradiction. It follows that A C A2. We also have A2\Ai C B, and hence y, {A2\A\) = 0 . • Remark 1.3 It is not difficult to prove that the family A consisting of all sets A satisfying any of the equivalent conditions in Lemma 1.7, is a cr-algebra. Using condition (a) in Lemma 1.7, we can show that A = T^. Indeed, since the inclusions T C A and H C A hold, we have T^ C A. Conversely, if A £ A, then condition (a) in Lemma 1.7 holds for the set A, and hence A £ F*1. Definition 1.13 Let (X, J7) be a measurable space, and let V be a family of measures on T. The completion Tv of the cr-algebra T with respect to the family V is defined as follows: nev A cr-algebra T satisfying T = J-v is called V-complete. Let (X, T, fi) be a measure space. Then the measure /x can be extended to a measure [IQ on T^ by setting fio(A) = fi (Ai), where A £ J711 and A\ is a set such as in part (b) of Lemma 1.7. It is not difficult to see that the number Ho(A) does not depend on the choice of the set A\ in (b). We will often use the same symbol \i for the extension \i§ of \i. The measure space (XJJ^J/J.) is called the completion of the measure space {X,!F,^) with respect to the measure /u. If V is a family of measures on T, then every measure fj. £ V can be extended to the cr-algebra !FV as above, and we will use the same symbol V for the family {/io : H £ V} consisting of the extensions of measures ^L £ V. It is not hard to see that the cr-algebra Tv is ^-complete. Next, we will discuss completions of nitrations generated by Markov processes. Let P be a transition probability function, and let Xt be a Markov process on (fi, J7) associated with P. The process Xt generates the following families of cr-algebras: J=rt =a(Xs:T<sx U A'lx
and
F.,x (A's,x) = Fs<x «
x
) = 0.
(b) A set A belongs to the a-algebra J-^+ if and only if for all (s, x) G [0, r] x E there exist sets Aa%x G F[+, A!ax G [T^f"-X, and A!'sx G [JFf]P",x such that AUA'S<X = AStX U A's\x
and
P s , x ( i ^ x ) = Fs<x (
Xt(u>). Denote by ME the space of classes of equivalence of Tj£-measurable functions from the space Q into the space E, equipped with the metric d(f, g) = inf (e + P [w : p(f(u), g{u>)) > e]). Then the convergence in the metric topology of the space ME is equivalent to the convergence in probability. Moreover, if / € ME and / „ G ME are such that oo
£d(/,/„) f(u>) P-almost surely on Q. Any #[o,r] ® -F/f-measurable function / generates a function / : [0, T] —> . M E defined as follows: for t 6 [0, T], /(£) is the class of equivalence in ME containing the function U) H->
f(t,Lj).
A simple function is a function from the space [0, T] x Q into the space E assuming only finitely many values, each on a S[o,r] ® ^-measurable set. A simple product-space function is a simple function such that each value is assumed on a set that can be represented as a finite disjoint union of direct products of sets from S[O,T] a n d T. An elementary measurable process Yt is a stochastic process on the space (fi, J7, P) such that there exist a partition
Transition Functions and Markov Processes
35
{Afc : k > 1} of the interval [0, T] into Borel measurable sets and a sequence {fk • k > 1} of T/E -measurable mappings of the space ft into the space E such that Yt = fk for all t e AkDenote by DE the class consisting of all $[O,T] ® ^"/5-measurable functions / , for which the function J: [0,T] -> ME is B[0,T]/BJ^ -measurable and has a separable range. Here Bj^ denotes the Borel cr-algebra of the space ME- An equivalent definition of the class DE is as follows. A S[O,T] ® .F/f-measurable function / belongs to the class DE if the function / can be approximated by a sequence of elementary measurable processes in the sense of pointwise convergence on ft uniformly in t G [0,T]. Lemma 1.10 The class DE coincides with the class of all B[O,T] ®T IEmeasurable functions. Proof. Let us first prove the lemma in the case where E = R. It is not difficult to see that the class £>R is closed under pointwise convergence of sequences of functions and contains all simple product-space functions. By the monotone class theorem for functions, Lemma 1.10 holds for E = R. Our next goal is to prove Lemma 1.10 for any finite subset of R. Let Ro = {ci,c 2 , • • • ,Cn} be a finite subset of R equipped with the metric inherited from the space R. Next, using Lemma 1.10 for E = M, we see that if / : [0,T] x ft — i > Ro is a JB[O,T] ® F/Buo-measurable function and i s N , then the range < f(t) : 0 < t < T > of the function / can be covered by a countable disjoint family A\, k G N, of Borel subsets of the space MR so that f~l (A\) G B[otT], and the diameter of any set A\ is less than \. Therefore, there exists a sequence / , of elementary measurable processes such that fi{t) G MR0 for all t G [0,T], and moreover sup d(f(t,-)-fi(t,-)) EQ be a B[O,T] J 7 //BB 0 -measurable function. Then for every 1 < j < n, we have g(t, w) = Xj on a set Bj e S[O,T] ® J7- The sets -Bj may be empty. It is also true that the nonempty sets Bj are disjoint and cover [0,T] x ft. Let us consider a #[O,T] ®T/B^o-measurable function defined by f(t, u>) = Cj on the set Bj with 1 < j < n. By the previous part
36
Non-Autonomous Kato Classes and Feynman-Kac Propagators
of the proof, there exists a sequence fa of elementary measurable processes such that fi(t) £ MRQ and inequality (1.51) holds. Replacing Cj by Xj in the function /,, we get a function g^. Taking into account (1.51) and the fact that p(xm,Xfc) < c cm — Ck\ for 1 < m < k < n, where c > 0 is a finite constant, we see that lim sup d(ff(i,-)>ffi(t>0) = 0. °te[o,T]
,_>0
(1.52)
Since gi is an elementary process and (1.52) holds, we have g £ DE0- It follows that any simple function s : [0, T] x Q —-> E belongs to the class DENow let / be a S[O,T] ® T/E -measurable function. Then the function / can be approximated pointwise by simple functions, and since the class DE is closed under pointwise convergence, we have / £ DEThis completes the proof of Lemma 1.10. • Let us continue the proof of Theorem 1.4. By Lemma 1.10, the class DE coincides with the class of all B\O,T\ ® .F/f-measurable functions. Approximating 6[o,T]/Sjq-measurable functions with separable range by simple functions, we see that for any measurable process Xt, there exists a sequence Y" of elementary measurable processes such that
1} of the interval [0, T] into Borel measurable sets and a sequence {/£ : k > 1} of T/E-measurable mappings of the space Q, into the space E such that Y™ = f% for all t £ A%. Fix n > 1 and 5 £ (0, T). Our next goal is to modify the process Y™ as follows. Put si = inf {t : t £ A%}. If sn £ An, then the new process Z? is denned for t £ Al by Ztn = X s n. If s£ £ A£, then we fix f£ e A% such that *£ - sfe < *> a n d P u t %t = -XtJJ f ° r a u * G -4fe- It i s c l e a r that the new processes J?" are elementary measurable processes. Since Xs is an adapted process, it is easy to see that for every t £ [0, T — 5], the restriction of the function (s, w) i-» Z?(w) to the space [0, t] x £1 is B[o,t] ®Tt+s/£-measurable. Moreover, inequality (1.53) implies d{XuZ?) 1. By (1.50), Zt"(w) -> X;(w) as n -> oo for all £ G [0, T] almost surely on (1 Fix XQ £ E, and put I t (a;) = lim Z"(w)
Transition Functions and Markov Processes
37
if the limit exists, and Yt(u)) = xo otherwise. Then the process Yt is a modification of the process Xt. We will next show that the process Yt is progressively measurable. It is clear from the definition of the process Yt that it suffices to prove that every process Z " is progressively measurable. Since the process Xt is adapted, we see that for every u £ [0,T — S], the restriction of the function (s,u>) t-> Z™(u>) to the space [0,u] x 0, is #[o,u] Fu+s/S-measurable. Now let t € [0,T]. Then using the previous assertion with um = t — ^ , m > mo, we see that the restriction of the function (s, w) — i > Z™{w) to the space [0,t) x Cl is S[o,t) Tt/^-measurable. Since the process Xt is adapted, we see that the process Z " is progressively measurable for all n > 1. This completes the proof of Theorem 1.4. • Every sample path of a measurable process is a Bp^}/^ measurable function. The next assertion provides examples of progressively measurable stochastic processes. Theorem 1.5 measurable.
Every left- or right-continuous process is progressively
Proof. Let X be a right-continuous process. Fix r and t with 0 < T < t 1, define a simple process Xk on [r,t] as follows: X% = X oo, if s € [SJ ', Sj + '), and X$ = Xt. It is clear that the function (s, a;) — i > X*(w) defined on [T, t] x Q, is B\Ttt] F[ /£-measurable. It follows from the rightcontinuity of the process Xt that lim
X*{LJ)=XS(W)
fc—+00
for all s € [r,t] and w e fi. Hence, the function (s,w) i—»-X"s(w) defined on [T, i] x fi is #[T,t] ® .F t T /£-measurable. This means that the process Xt is progressively measurable. The proof of Theorem 1.5 is thus completed for right-continuous processes. The proof for left-continuous processes is similar. • The next result concerns separable processes and stochastic equivalence. Theorem 1.6 Let P be a transition probability function satisfying condition (1.41)- Then there exists a separable process (Xt,^,PT,x) on (Q,J-)
38
Non-Autonomous Koto Classes and Feynman-Kac Propagators
with state space (E, £) such that the transition function of Xt with P. Proof.
coincides
The following lemma will be used in the proof of Theorem 1.6:
Lemma 1.11 Let P be a transition probability function, and let Xt be a Markov process associated with P. Then for every (r, x) G [0, T] x E there exists a separable P r , x -modification XSI r < s 0, then there exists tk+i £ [r, T] such that sup P T;X [XtjeB:l<j
^.
te[r,T)
z
Put Nk(B) - {Xtj £B:l<j 00. Since P r ? x [N(t, B)] < 7^ for any k > 1, we have P r , x [N(t,B)] = 0 for all t G [T,T\. This completes the proof of Lemma 1.12. •
Transition Functions and Markov Processes
39
Now let Bi be a sequence of Borel sets in E. It follows from Lemma 1.12 that for every i > 1 there exists a sequence t\, k > 1, such that PT,x[N(t,Bi)]=0
(1.55)
for all t G [r, T\. By enumerating the set {t\ : i > 1, k > l } , we see that the sequence tk can be chosen independently of i. It is not hard to prove that the sets N(t,Bi) constructed for this sequence are subsets of the similar sets in (1.55). This means that (1.55) holds for the new sequence. Next we will show that more is true. Lemma 1.13 Let r G [0,T], x G E, and let Bt be a sequence of Borel sets in E. Then there exists a sequence tk G [T,T], k > 1, and for every t G [r, T] there exists a set N(t) G Tj- such that Pr,x [N(t)] = 0,
(1.56)
N(t,B)cN(t)
(1.57)
and
for all t G [r, T] and for all sets B which can be represented as a countable intersection of elements of the family {Bi}. Proof.
Put N(t) = ( J N(t, Bt) (here we use the sequence tk constructed
after equality (1.55)). Let B = f|. Biy N(t, B)c\J
{Xtk zBfk>\,Xti
Then Btj}
j
c\J{Xtk 3
G B^, k>\,Xti
Bij}=\jN(t,Bij)
c N(t).
(1.58)
3
Now it is clear that (1.56) follows from (1.55), while (1.57) follows from (1.58). • Let us return to the proof of Lemma 1.11. Denote by Cj the family of all open balls of rational radii centered at the points of a countable dense subset of E, and put Bi = E\d. It is clear that the family of sets which are representable as countable intersections of the sets from the family {Bi} contains the family of all closed subsets of E. Applying Lemma 1.13, we see that there exists a sequence tk G [r, T], and for every t G [r, T] there exists a set N(t) such that Pr,x[7V(i)] = 0 and N(t, C) C N(t) for all closed subsets C of E and all t G [r, T\.
40
Non-Autonomous Kato Classes and Feynman-Kac Propagators
Let Yt be a separable process on ft, and define a new process Xt as follows: Xt(w) = Xt{w) if t G {tk} or u £ N(t), and Xt(u>) = Yt(w) otherwise. Our next goal is to show that
Xt=Xt
=1
(1-59)
for all t G [T, T] and also to prove that Xt is a separable process. Indeed, if t G {U}, then lxt = Xt\ = ft. If t € {**}, then lxt ^ Xt\ C JV(i). This gives (1.59). Now we are ready to prove the separability of the process Xt- If t = ti for some i > 1, then Xti = Xti, and hence for all UJ G ft, the corresponding sample path is minimally continuous with respect to {U}. lit $. {tk} and ui G -/V(i), then X t coincides with a {ij}-separable process. If t £ {t,} and w $. N(t), then Xt(u>) = X t (w), and we proceed as follows. Suppose that X t (w) cannot be approximated by a subsequence of the sequence Xti{nv). Then there exists a ball C centered at Xt{u) such that Xti(w) £ C for all i > 1. Moreover, for every i > 1, we have Xtt{u>) G B and Xt{uj) £ B where B = E\C. Hence, w G N(t,B) C iV(i), which is a contradiction. Therefore, Xt (u>) can be approximated by a subsequence of the sequence Xti(uj), and this implies the separability of the process Xt. This completes the proof of Lemma 1.11. • Note that we have not yet employed the stochastic continuity condition in the proof of Theorem 1.6. This condition is needed to guarantee that any countable dense subset of [r, T] can be used as a separability set. Lemma 1.14 Suppose that P is a transition probability function satisfying the stochastic continuity condition (1-7), and let Xt be a corresponding Markov process. Fix (T,X) G [0,T] X E. Then any countable dense subset of [T,T] can be used as a separability set in Lemma 1.11. Proof. By Lemma 1.11, there exists a separable process Xt on [T,T] associated with a separability set {tk}- Let {SJ} be any countable dense subset of [r,T}. Next, we will show that for PTiX-almost all u> G ft and all k > 1, the element Xtk{w) of E belongs to the set A{UJ) consisting of all limit points of the set < XSi (w) >. By Fatou's Lemma and the stochastic
Transition Functions and Markov Processes
41
continuity of the process Xt, we see that for every k > 1, liminfp(xtt,X,4) >0] < lim P x
\iminfp(xtk,XSi)
< lim liminfP TiX n—»-oo i—>oo
> >
= 0.
(1.60)
Condition (1.60) means that for P r x -almost every w € Q, and every k > 1, the element Xtk(uj) of £ belongs to the set A(LJ). Therefore, the process Xt is separable with respect to the set {s;}. This completes the proof of Lemma 1.14. D The next result (Lemma 1.15) will allow us to get rid of the dependence of the process Xs in Lemma 1.11 on the variables r and x. The conditions in Lemma 1.15 are as follows. A family of stochastic processes X\T,X parameterized by (r, x) 6 [0, T] x E is given, and it is known that the sample paths of these processes possess a certain property. Our goal is to construct a single non-homogeneous process Xt from the processes X\T so that the sample paths of Xt possess the same property. Lemma 1.15 Let P be a transition probability function and suppose that for every pair (r, x) G [0, T]x E, a stochastic process is given on (fi,!F). Suppose also that X,
(r,x)
eB
P{j,x;t,B)
(1.61)
for all t with r < t < T and all B £ £. Let F be a class of E-valued functions defined on [0,T], and assume that the sample paths of all the processes X t ' T ' x) belong to F. Then there exists a Markov process Xt such that its sample paths belong to the class F and P is its transition function. Proof. Consider a new sample space (l = [ 0 , T ] x £ x ( l . This space will be equipped with the cr-algebra T=
{ACQ
: AT>X G T for all (T,X) G [0,T] X £?} .
Here AT^X = iu> : (r, x, ui) € A >. Define a stochastic process on il by Xt(r,x,u)
X\T
Ft = * (X.
T
{UJ) and consider the family of ^--algebras given by
1 - 2"
for all s G [0, T], T G [0, T], and x e E. Therefore,
nul'W.*-)^} j>ln>j
for all s G [0, T], the sequence X\n Without loss Let A be the set
^
'
r G [0, T], and x G E. It follows that for every s G [0, T], converges P r , x -a.s. to Xs for every r G [0, T] and x G E. of generality, we can assume that in = n for all n > 1. of all (s, w) G [r, T] x f2 such that linin-joo X™(ui) exists.
44
Non-Autonomous Koto Classes and Feynman-Kac Propagators
It is clear that the set A is (fi[o,rj ® ^-measurable. Moreover, for every se[0,T], PT,X{W:(S,W)G.4} = 1
(1.64)
for all (T, X) € [0, T) x £\ Indeed, if s G [0, T] and (r, x) G [0, T] x E, then the set {ui : (s, u>) G .4} contains an ^"-measurable set AST such that ¥r<x(As(T,x)) = 1. This follows from the fact that if s G [0,T], then X£ converges to Xs P T)X -a.s. for every (T, X) G [0, T] x E. Define a new process Xs on Q by X,(w) = lim X™(UJ), if (s, u>) G ^4, and
Xs(bj) = xo for all (S,UJ) $ A, where xo is a fixed point in E. Since for all (r, x) G [0, T] x E, the process X* is an P^-modification of the process Xt, it is clear that PT)X Xt e B = P (T, X; t, B) for all 0 < T < t < T and all B G £. Moreover, since the processes Xn are .^-progressively measurable and (1-64) holds, the process Xt is ^-progressively measurable. This completes the proof of Theorem 1.7. • 1.9
P a t h Properties of Stochastic Processes: Continuity and Continuity
One-Sided
In this section we continue our exploration of the properties of paths of Markov processes. Specifically, we will study the processes with continuous sample paths and the processes for which the sample paths have only jump discontinuities. Theorem 1.8 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that for all (T, X) G [0, T) x E the following condition holds: lim
PT,x-ess sup P (s, Xs; t, Ge (xs) ) = 0
t-si0)T<s 0, where Ge(y) is defined by (HO). Then there exists a Markov process (Xt,J-[,FT,x) on (Q.jJ7) such that Xt is right-continuous, has left limits, and the transition function of Xt coincides with P. Remark 1.5 It can be shown that condition (1.66) implies condition (1.41). Indeed, if a transition probability function P satisfies condition (1.66), then we have / P (s, y; t, Ge(y)) P(T, X; S, dy) < sup P (s, y; t, Ge(y)), JE
y&E
and hence condition (1.41) holds. Proof. The structure of the proof of Theorem 1.8 is similar to that of Theorem 1.6. We start with the following lemma. L e m m a 1.16 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that condition (1.65) holds for a fixed pair (T,X) 6 [0,T) x E. Then there exists a ¥T>Xmodification Xt of the process Xt, r < t < T, which is right-continuous and has left limits. The process Xt depends on T and x. Proof. Without loss of generality, we may assume that Xt is a separable process on (fi,.F) (see Lemma 1.11). Fix r and x. The following random variables will be used in the proof: tp(e,r,8,t) = P TlX [p(Xs,Xt)
> e | J7]
(1.67)
where r < r < s x -ess sup t, and p ( / (%_-,) , f (tkk)) > e for all k > 2. This completes the proof of Lemma 1.17. D Let us continue the proof of Lemma 1.16. For every e > 0, k > 1, and any Borel subset H of the interval [r, T], define the following events: Tfc(e, H) = {w : the function t •-» Xt(uj) has at least k e-oscillations on H} (1.73) and roo(e,H)=f)Tk(e,H).
(1.74)
fc>i
Our goal is to prove that Pr,x[roo(€,[r,T])] = 0
(1.75)
for every e > 0. Then • T,X
|Jr0O(e,[T,ri) =o,
and hence, by Lemma 1.17, FTiX {u> : the function t >—> Xt (w) has no discontinuities of the second kind on [T, 2*1} = 1.
(1.76)
Let I = {ti < ti < • • • < tn} be a finite subset of the interval [T, T], Put $fc(e,/)=PT,x[rfc(e,J)|J71]
(1.77)
(3k{e,I) = ess sup $fc(e, J)(w).
(1.78)
and
L e m m a 1.18
For every k > 1, the following estimate holds: /3fc(e,/) e for some j and m with i < j < m < n. In addition, let us denote by Blk_l(e,I) the event consisting of all u G rfe_i(e,/) such that w has at least k — 1 e-oscillations on the set ( -0 = r fe-i (e> -0> a n d moreover,
i=l n
rfe(e,/)cU[£i-iM)n4M)]. i=l
For every 1 < i < n, put /* = {tm : i <m pf (u, x; s, B) and m. It follows from the reasoning above that Pf(u,x;s,B)=
pf(u, x; s, z)dz
(1.133)
JB
for all x G A0 and B G £. In (1.133), pf(u,x;s,z)=
/ Pf(u,x;t,dy)q{u,x;s,z;t,y). JE We will next prove that the measure D >-> P? (u, z; t, D) is absolutely continuous with respect to the measure fit for /i u -almost all z G E. Indeed, we have Ht{D) = f dfj,u(x)Pf (u, x; t, D)
(1.134)
for all D G £. Suppose that C G £ is such that /ut(C) = 0. Then it follows from (1.134) with D = C that Pf (u,x;t,C)=0
for /x„-almost all x £ E.
(1.135)
Using the Markov property, we see that for every D £ £, Pf (u,x;t,D)=
Pf (u,x;s,dy)Pf
(s,y;t,D)
(1.136)
/uu-almost everywhere on E. Let us denote by A\ the set of those x £ E for which (1.136) holds for all D £ £. It follows from the separability of £
72
Non-Autonomous
Kato Classes and Feynman-Kac
Propagators
that /u„ (E\Ai) = 0. Fix xi £ A0r\Ai such that (1.135) holds for xi. Note that xi depends on the set C. Then (1.136) and (1.135) with x = x\ show that Pf(s,y;t,C)
=0
(1.137)
for Pf (u,xi; s, -)-almost all y G E. Since xx £ AQ, equality (1.137) holds for m-almost all y G E. Now we see that for all XQ G AQ n A\, equality (1.137) holds for P ' (u, x0; s, -)-almost all y & E. Then (1.136) with D = C gives Pf(u, xo; t, C) = 0. We have thus completed the proof of the fact that the measure D H-» P ^ (U, Z; £, Z?) is absolutely continuous with respect to the measure fit for /i u -almost all z G E. Indeed, the equality Pf(u, xo; t, C) = 0 holds for all C with Ht(C) — 0 for all XQ belonging to the set AQC\A\. This set is independent of C and such that fiu (E\ {AQ n A{)) = 0. It follows that there exists a function P such that Pf («, z; t,D)=
[ Pf (u, z; t, y) d(it(y)
(1.138)
JD
for all £> G £ and for /i u -almost all z £ E. Now it is not hard to show that for every u G [r, £) there exists a function (z,y) >—> P* (u,z;t,y) such that it is £ (g> immeasurable and coincides with the function (z, y) i-> P (u, z; i, y) /xu x jut-almost everywhere. Indeed, using (1.134) and (1.138), we get / dnu(z) JE
/ P (u,z;t,y)dnt(y)=
Ht(E) = l,
JE
and hence, there exists a probability measure v on £ ® £ such that iv(Ax.D)= / d/xu(z) / P"'
(u,z;t,y)dfj.t(y).
JD
JA
This measure is defined as follows. For any set G G £ £, v{G)=
P~f
/ d// u (z) / JE
(u,z;t,y)dfit(y)
JGZ
where Gz = {y G E : (z,y) G G}. It is easy to see that the measure v is absolutely continuous with respect to the measure fiu x fit. Let us denote An
~
the Radon-Nikodym derivative —. r by PA Then for all u with d (/xu x lit) T < u < t, the function (z, y) H-> P-f (U, Z; i, y) is £ ^-measurable, and moreover, / dnu{z) \ P JA
JD
(u,z;t,y)dni (y)
Transition Functions and Markov
73
Processes
= f dfiu(z) f Pf(u,z;t,y)dtH(y) JA
(1.139)
JD
for all Borel sets A and D in E. It follows from (1.139) and from the separability of the cr-algebra £ that for every u with r t>y)dvt(v)El>.tUx.,y)[F] Hr,t),n [? \t(y) J {z-n-i) = '—~ fEP(Sn-l,
Zn-1] t,
y)dvt(y)
s
JEP {sn-i, Zn-i; t, y) dvt{y)Q ( s n - i , Zn-ii n, An; t,y) dz, IEP(
;t,y)dvt{y)
(1.143)
It is not hard to see that (1.129) and (1.143) imply • (r,t),A»
•X-Sr, G Ann \ J•> s„ a.
fEp(sn-i,XSn_1;
t, y) dvt{y)Q (sn-i,XSn_1; fEp(sn-i,
sn, An; t, y) dzn
X8n_1; t, y)dut{y)
IEP{s^-^XSn_1]t,y)dvt{yyS'(sn_1,t),(xSri_1,y)
[XSn <S An] .
(1.144)
JE P(«n-i, ^ „ _ ! ; t, y)dut(y) Therefore, (1.142) follows from (1.144) and from the Markov property of the process Xs. A formula, similar to formula (1.142), holds in the case of a general probability measure \i on £ x £. Lemma 1.23 Suppose that the conditions in Theorem 1.14 hold. Then the following are true: (1) For all T < r < s < t and all J-* -measurable random variables F,
%,*),, [F I JT] /
JEXE
P{T,X;r,Xr)p(s,Xa;t,y)E{s,tUxs,y)
[
JEXE
[F]
p(r,x;r,Xr)P(s,Xs;t,y)^^
d x
^ 'f>
P\J-,x\t,y)
P{r,x;t, y)
(1.145)
Transition Functions and Markov
Processes
(2) For all T < r < s x° defined by (1.163) and (1.164) and with the final condition 5Vo. If p is a transition density, then we denote by Po,x0 the measure on Q. associated with the density p and the initial distribution SXo. If p is also a backward transition probability density, then we denote by P r ' y o the measure on Q associated with the backward transition density p and the final distribution 5yo. By Theorem 1.15, the measures P(o,T),(xo,2/o)> ^o,'xa' anc ^ ^o,'xo° c ° i n c i d e on the (7-algebra Tj.. It is not hard to prove that if p is a transition probability density, then the measure P 0 'x° is absolutely continuous with respect to the measure Po,x0 o n every cr-algebra 3-^ with 0 < r < T. Similarly, if, in addition, p is a backward transition probability density, then the measure PQ'"" is
82
Non-Autonomous
Koto Classes and Feynman-Kac
Propagators
absolutely continuous with respect to the measure FT'yo on every c-algebra T^ with 0 < r < T. Suppose that under certain restrictions on the transition density p the following conditions hold: (a) There exists a continuous P(O,T),(X0,I/O)" modification of the forward Kolmogorov representation of the process Xt on the half-open interval [0,T); (b) l i m t j r X ( = Vo ]P(o,T),(:ro,!/o)~amiost ev~ erywhere. Then the same restrictions guarantee the existence of a continuous modification of the reciprocal process Xt on the closed interval [0, T]. Similarly, suppose that under certain restrictions on p, the following conditions hold: (a) There exists a P( 0) T),(xo,!/o)" mo ^ mcat i on °f t n e forward Kolmogorov representation of the process Xt, which is right-continuous and has left limits on the half-open interval [0, T); (b) l i m t j r X t = j/o F>(o,T),(x0,2/o)"ahTlost everywhere. Then the same restrictions imply the existence of a P(o,T),(x 0 ,!/o)" mo< ^ mcat i on °f the reciprocal process Xs that is right-continuous and has left limits on the closed interval [0, T]. We will use these ideas in the proof of the following assertion. T h e o r e m 1.16 Let p(r,x;t,y) be a strictly positive function that is simultaneously a forward and a backward transition probability density, and let XQ € E and yo £ E be given. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrddinger representation of the process X t (w) = w(i), u> £ fi, 0 < t < T, on the space £1 — J5[°>T] with respect to entrance-exit law (1.160) and the reciprocal transition density q. Suppose that for all e > 0, lim / *i°
lim
p(0,x0;t,y)dy
= 0,
(1.165)
JGt(x0)
sup /
p(r,x;t,z)dz
— 0,
t-TlO;0(t), u> G Q., 0 2/) P (r,t),(x,y)-
Remark 1.11 If the density JD is normal, then in addition to (1.181) the following formula holds:
/ 4 * (XT eAo,xaieAi,...,xSn G A„,xt e > W i ) s = XA0 0*0x,*„+1 (y)Er,x b ( n, -X"Sn;«, y), XSl e Ax,...,
x3n e A n ] . (1.182)
The pinned measure ^yx is defined on the measurable space ( E [ ° ' T 1 , !FJ). However, if Xs is a Markov process on a smaller path space, e.g., on the space of continuous paths or on the space of left- or rightcontinuous paths, then certain difficulties may arise at the endpoints r and t. To avoid these difficulties, the measure fi^x is usually restricted to the CT-algebra F[_ = a (Xs : T < s < t) in the case of right-continuous processes Xs having left limits, while in the case of left-continuous processes having right limits, the measure /4'x is restricted to the cr-algebra F{+ =a{Xs:r
<s Q(r,t) is differentiable from the right on (0, t). T h e o r e m 2.3 Let Q{r,t), 0 < T < t < T, be a strongly continuous backward propagator on B, and fix t with 0 < t < T. Then for every x e D+(t), the function U(T) = Q (T, t) x is a solution to the following final value problem on (0, t): ( d+U
-(r) = dr limzi(r) = x.
-A+(T)U{T),
(2.16)
T\t
Proof. Let x € D+ (t). Then, using the strong continuity of the backward propagator Q, the Banach-Steinhaus theorem, and the definition of the set D+(t), we obtain d+u ,. _,. —— =hmQ(T,T OT
MO
,^d+u + h)—— OT
d+u Q(r + h,t)x-Q(T,t)x dr h Q(T + h,t)x - Q{r,t)x + limQ(T,T + /i) Mo h Q(T + h,t)xQ(r,t)x — lim Q(T,T + h) Mo h Q(T, t)x - Q(T, T + h)Q(r, t)x lim Mo h Q(r,T + h)-I = —lim Q{r,t)x. Mo h
= lim Q(T,T + h) Mo
(2.17)
Propagators: General Theory
109
It follows from (2.15) and (2.17) that Q(T, t)x G D (A+(T)) and the equation in (2.16) is satisfied. In addition, the equality limu(r) = x follows from the T"T*
strong continuity of Q. This completes the proof of Theorem 2.3.
•
Now we turn our attention to propagators on B. The generators in this case are defined exactly as in the case of backward propagators. Suppose that W is a propagator on a Banach space B. For every t with 0 < t < T, consider a linear operator on the space B given by 7 /^
i.
W(t
+ h,t)x-x
.„.,„,
A+(t)x = lim — i -^ . (2.18) v Mo h ' The domain D (A+ (t) J of this operator is the set of points x £ B for which the limit in (2.18) exists. The operators A+(t), 0 < t < T, are called the right generators of the propagator W. For every T e [0,T), denote by D+(T) the set of all x £ B such that the function t H-> W(t,r)x is differentiable from the right on (r, T), and by F(T) the set of all x € B for which limt^(T + /i,T)a; = ar. (2.19) Mo Theorem 2.4 Let W be a propagator on B, and fix T with 0 < r < T. Then for every x £ D+(T) n F(T), the function u{i) = W(t,r)x is a solution to the following initial value problem on (r, T):
^
w
= i + ( « 0 there exists B £ £ such that H(E\B) < e and \gn{x)\ < e, x£B,
n e N.
(2.38)
Moreover, since every measure vn is absolutely continuous with respect to the measure fi, and lim vn(B) = v(B) for all B € £, the Vitali-Hahn-Saks n—>oo
theorem (see Section 5.5) implies that lim sxxpvJB) = 0 .
(2.39)
M(B)iO n
Now it is not hard to see that equality (2.34) follows from (2.35), (2.37), and (2.39). Next, using (2.34) and the properties of backward propagators, we see that
[p*,JE
dT
lim /r(x, T + M r ( T + " - ' ' / - n r ' t ) / ^ h
n->°°JE
= lim
n
/ Y(T,t)f-Y(T,T
+
hn)Y(T,t)f
dv
n->ooJE
~ ~."St/, YiT-rlK)-'Y(r,
W,
(2.40)
It follows from (2.40) that Y(r,t)f € Dw (A*£(r)) and that the equation in (2.32) holds in the space (Lf, a (Lf, M)). In addition, the equality lim / u{r)du = I fdv follows from the continuity of Y on the space T T* JE JE
{Lf, £ Ct. By Q, will be denoted the space of all E-valued functions defined on the interval [OjT], which are right-continuous and have left limits in E. Then we have Q C ft. Put Xt{w) = u(t), w £ ft, t £ [0,T], and let J J , 0 < T < t < T, be the er-algebra generated by Xs with T < s < t. For every r € [0, T) and x £ E, denote by P T|X the probability measure on TT determined by FT,X [Xtl £Bu...,Xtn£Bn)=
PT,X [ x t l 6 B i
Xtn £ £ „ ] .
(2.60)
Here r < h < • • • < tn < T and Bj £ £, 1 < j < n. Let us denote by !F^+, t £ [T,T), the cr-algebra defined by Tl+=
Q
T[.
(2.61)
t<sQ.
(2.69)
Y (r, u) f(z)du = 0. Then the equality
Y (r, u) f(z)du =
Jr
du Jr
f(y)P (r, z; u, dy), JE
and the strict positivity of the function / imply that for almost all u € [r, T] with respect to the Lebesgue measure, we have P (r, z; u, E) = 0. This contradicts the definition of transition probability functions. Therefore, inequality (2.69) holds. Next, fix a sequence sn € [T,T] such that lim sn = s where r < s < T. We will show below that for a sequence zn G E, the following two conditions are equivalent: (1) There exists a compact subset C C E such that zn £ C for all n > 1. (2) The inequality inf /
Y(sn, u)f(zn)du>0
(2.70)
holds. Indeed, if condition (1) does not hold, then there exists a subsequence znk of zn such that lim g(znk) = 0 for all g G Co- It follows from the strong fe—>oo
continuity of Y on Co and from the condition lim sn = s that f
Y(s,u)f(znk)du-
Js
f
Y(snk,u)f(znk)du^0
(2.71)
JSnk
as k —» oo. Next, the strong continuity of Y on Co gives lim f
Y(s,u)f(znk)du
= 0.
(2.72)
By (2.71) and (2.72), we see that inequality (2.70) does not hold. Hence, the implication (2) =*• (1) is valid. On the other hand, if condition (1) above holds, and condition (2) does not hold, then there exists a sequence nk such that lim znk = z where k—*oo
130
Non-Autonomous
Koto Classes and Feynman-Kac
Propagators
z € C; moreover, lim /
Y(snk,u)f(znk)du
= 0.
(2.73)
k
^°°Jsnk
Since Y is a strongly continuous backward propagator on Co, and (2.69) holds, we have lim /
Y(snk,u)f(znk)du=
Y (s,u) f{z)du > 0.
k—>oo /„
/.
This contradicts (2.73). Therefore, the implication (1) =>• (2) holds. The proof of Lemma 2.5 is thus completed.
•
It follows from Lemma 2.3 with e = 0, Lemma 2.4, and Lemma 2.5 that for all r G [0, T) and all cc G £ , the equality PT,X [TT] = 0 holds. Therefore, PT,X fi\rr = 1. Moreover, for all w G f2\r T and r G Q ("1 [r,T), there exists a compact set C C E such that Xj(w) G C for all s G Q n [T, r]. The set C depends on r and u>. Given r G [0,T), put
S r = n \ (AT u r T ), where A r is the complement of the event consisting of all u> G tl for which the limits in (2.65) exist for all f € Co and all t with T
