Loeb Measures in Practice: Recent Advances: EMS Lectures 1997

Preface This monograph, based on the author’s 1997 EMS Lectures given at the University of Helsinki in May/June 1997, ...

Author: Nigel J. Cutland

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Preface

This monograph, based on the author’s 1997 EMS Lectures given at the University of Helsinki in May/June 1997, outlines the Loeb measure construction (a way to construct rich measure spaces using Robinson’s nonstandard analysis) and discusses recent applications in stochastic fluid mechanics, stochastic calculus of variations (“Malliavin calculus” and related topics), and mathematical finance theory. The four lectures in Helsinki were designed for a general audience, as is the expanded version presented here. No previous knowledge of either nonstandard analysis or the fields of application is assumed, beyond the general knowledge of the working mathematician. The aim in Chapter 1 is to provide a brief but coherent account of the fundamentals of nonstandard analysis (NSA) and the Loeb construction that is sufficient to make sense of the applications of the later chapters. For each of these we have endeavoured to provide sufficient by way of introduction to the topics concerned to enable even the reader unfamiliar with them to appreciate the basic ideas of the field and then the particular contributions that can be made using NSA and Loeb measures. In fact, one of the major contributions that NSA has made to many fields of application is to aid in understanding of the basic ideas of that field1 – and it is hoped that among other things this will come over in this monograph. To cover both an introduction to NSA and Loeb measures together with applications to three advanced and diverse fields of current research in four lectures (and now in four corresponding chapters) is somewhat ambitious. Necessarily the treatment will omit many details. The present volume should be seen then as something of a trailer for in depth study of both Loeb measures and the way in which they can make useful contributions to mathematical research. The topics chosen for discussion in Chapters 2–4 are drawn mainly from work of the author in collaboration others, and bring together material that has mostly been published elsewhere but is scattered. In each 1

A classic example of this is Anderson’s construction of Brownian motion as an infinitesimal random walk, discussed in Chapter 3; at a more elementary level is Robinson’s original discovery of how NSA can be used to develop real analysis rigorously, using infinitesimals to make precise the informal ideas of differentiation and integration.

II

Preface

of the areas the applications include results that represent advances in the standard theory. The applications in Chapters 2–4 are in the three seemingly unrelated areas mentioned in the opening paragraph. The link between them, from the point of view of this monograph, is the common methodology of Loeb measure techniques. This stems of course from the fact that each field involves measures and integration – and in most cases there is the more specific common feature of stochastic analysis in a variety of guises. But there is also a less obvious unifying factor that is harder to pin down precisely. This involves the idea of passage to a limit in a very generalised way. NSA facilitates this because, having constructed or defined an object Xn , say, for each finite n in the standard world we have automatically in the nonstandard world an object XN for infinite natural number N (the meaning of this will be made precise in Chapter 1). Such N is called hyperfinite – that is, finite from the point of view of NSA but infinite in that N > n for all n ∈ N.2 The work is then to find some standard (real world) object associated with XN which will provide the solution to the problem in question. In the applications to fluid mechanics, for example (Chapter 2), the fundamental equations (PDEs and stochastic PDEs) are normally solved by solving finite dimensional approximations, which are ODEs and stochastic DEs, and then “passing to the limit”. In our approach, the passage to the limit is achieved by taking XN (in the terminology above) to be a solution for the (infinite but) hyperfinite dimensional approximation of dimension N . From the nonstandard solution XN a standard solution is obtained. The “Malliavin” calculus – treated in Chapter 3 – is at heart a kind of differential calculus for the Wiener space C0 [0, 1], thought of as a subspace of R[0,1] which is itself viewed as a product space generalising Rn , with its associated differential calculus. Conventional expositions do not make this so clear. In our approach we achieve the “passage to the limit” from Rn to C0 [0, 1] by considering ∗RN for infinite hyperfinite N . The Malliavin calculus can then be seen as a suitable projection of classical calculus in ∗RN onto C0 [0, 1]. Among other applications, this shows how Wiener √ measure “is” simply the uniform probability measure on the sphere S ∞ ( ∞), and allows a precise formulation of the experts’ intuition that the infinite √ dimensional Ornstein–Uhlenbeck process “is” Brownian motion on S ∞ ( ∞). The final applications, in Chapter 4, are in the field of modern mathematical finance theory, which, in common with the the previous topics, has stochastic analysis (particulary Brownian motion and Itˆ o integration) at its foundation. Here there is great interest in connecting the approach using financial models based on a discrete model of time, with the other main 2

As the reader may be aware, the starting point of NSA is to construct a field ∗ R ⊃ R that contains both infinitesimal and infinite elements – and this contains a corresponding extension ∗N of N. Then an infinite hyperfinite number N is an element of ∗N \ N.

Preface

III

approach, using continuous time. The latter is in some sense obtained by “passing to the limit” in discrete-time models – and again the NSA framework greatly facilitates this. In essence, if we have for each n ∈ N a discrete financial model Mn then we immediately have MN for infinite hyperfinite N . Then we can show that the continuous model is obtained as a suitable projection of MN , and as a result obtain some new powerful convergence results. Acknowledgments On a personal note, I should like to thank - the EMS for the kind invitation to give the 1997 EMS Lectures; - the Mathematics Department of the University of Helsinki, for their generous hospitality; - my collaborators Marek Capi´ nski, Jerry Keisler, Siu-Ah Ng, Ekkehard Kopp and Walter Willinger, who have worked jointly with him on much of the work that is described in these lectures. Many thanks too to my wife Mary, who has been so supportive, patient and encouraging over the many years during which the material surveyed here was developed. Hull, November 1999

Nigel Cutland

Contents

1

Loeb Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nonstandard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The nonstandard universe . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 ℵ1 -saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Nonstandard topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Construction of Loeb Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Example: Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Example: Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Example: Wiener measure . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Loeb measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Loeb Integration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Elementary Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Lebesgue integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Peano’s Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Itˆ o integration and stochastic differential equations . . .

1 1 2 2 7 10 11 13 16 17 17 19 20 23 23 24 27

2

Stochastic Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Functional formulation of the Navier–Stokes equations 2.1.3 Definition of solutions to the stochastic Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Nonstandard topology in Hilbert spaces . . . . . . . . . . . . . 2.2 Solution of the Deterministic Navier-Stokes Equations . . . . . . . 2.2.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Solution of the Stochastic Navier–Stokes Equations . . . . . . . . . 2.3.1 Stochastic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Nonhomogeneous stochastic Navier–Stokes equations . . 2.4 Stochastic Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The Foias equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Construction of statistical solutions using Loeb measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 31 32 33 34 36 37 39 40 40 42 42 43

VI

Contents

2.6

2.7 2.8

2.9

3

4

2.5.3 Measures by nonstandard densities . . . . . . . . . . . . . . . . . 2.5.4 Construction of statistical solutions using nonstandard densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Statistical solutions for stochastic Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attractors for Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . 2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Nonstandard attractors and standard attractors . . . . . . 2.6.3 Attractors for 3-dimensional Navier–Stokes equations . Measure Attractors for Stochastic Navier–Stokes Equations . . Stochastic Attractors for Navier–Stokes Equations . . . . . . . . . . 2.8.1 Stochastic attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Existence of a stochastic attractor for the Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . Attractors for 3-dimensional Stochastic Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45 46 46 46 48 49 50 52 52 53 55

Stochastic Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Flat Integral Representation of Wiener Measure . . . . . . . . . . . . 3.3 The Wiener Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 BM on the Wiener Sphere and the Infinite Dimensional O–U Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Malliavin Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The Wiener-Itô chaos decomposition . . . . . . . . . . . . . . . . 3.5.3 The derivation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 The Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 The Malliavin operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 64 66

Mathematical Finance Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Cox-Ross-Rubinstein Models . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Options and Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Pricing a claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Black-Scholes Model and Hyperfinite CRR Models . . . . . . 4.5.1 The Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 General claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Convergence of Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Discretisation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Poisson pricing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 88 90 92 94 95 95 96 98 99 99 99

69 72 73 75 77 79 83

Contents

VII

4.8.3 Incomplete markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.8.4 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . 100 4.8.5 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

1. Loeb Measures

1.1 Introduction Loeb measures, discovered by Peter Loeb in 1975 [71], are very rich standard measure spaces, constructed using nonstandard analysis (NSA). The range of fields in which they have found significant applications is vast, including measure and probability theory, stochastic analysis, differential equations (ordinary, partial and stochastic), functional analysis, control theory, mathematical physics, economics and mathematical finance theory. The richness of Loeb measures makes them good for - constructing measures with special properties (for example the rich probability spaces of Fajardo & Keisler [49, 50, 62]); - representing complex measures in ways that make them more manageable (for example Wiener measure) – see section 1.3.3 below; - modelling physical and other phenomena; - proving existence results in analysis – for example solving differential equations (DEs) of all kinds (including partial DEs, stochastic DEs and even stochastic partial DEs) and showing the existence of attractors. Later lectures will describe some recent uses of Loeb spaces that illustrate these themes – in fluid mechanics, in stochastic calculus of variations and related topics, and in mathematical finance theory. This lecture will outline the basic Loeb measure construction and give some simple applications, with a little of the theory of Loeb integration. From one point of view, Loeb measures are simply ultraproducts of previously given measure spaces, such as were considered in an early paper of DacunhaCastelle & Krivine [46]. The rˆ ole of NSA in their construction is to provide a systematic way to understand their properties, which opens the way for efficient and powerful applications; without this we would have a supply of very rich measure spaces but only ad hoc means to comprehend them. Necessarily these lectures will be somewhat informal and lacking in a great deal of detail. The aim is to convey some of the basic ideas and flavour of Loeb measures and how they work, as well as pointing to the literature where the topics can be pursued in depth. We must begin with a brief and informal look at NSA itself.

N.J. Cutland: LNM 1751, pp. 1–28, 2000. c Springer-Verlag Berlin Heidelberg 2000

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1 Loeb Measures

1.2 Nonstandard Analysis 1.2.1 The hyperreals Nonstandard analysis (discovered by Abraham Robinson in 1960 [83]) begins with the construction of a richer real line ∗R called the hyperreals or nonstandard reals. This is an ordered field that extends the (standard) reals R in two main ways: (1) ∗R contains non-zero infinitesimal numbers; and (2) ∗R contains positive and negative infinite numbers. This is made precise by the following definitions (where | · | is the extension1 of the modulus function to ∗R). Definition 1.1 Let x ∈ ∗R. We say that (i) x is infinitesimal if |x| < ε for all ε > 0, ε ∈ R; (ii) x is finite if |x| < r for some r ∈ R; (iii) x is infinite if |x| > r for all r ∈ R. (iv) We say that x and y are infinitely close, denoted by x ≈ y, if x − y is infinitesimal. So x ≈ 0 is another way to say that x is infinitesimal. (v) The monad of a real number r is the set monad(r) = {x : x ≈ r} of hyperreals that are infinitely close to r. Thus monad(0) is the set of infinitesimals, and monad(r) = r + monad(0). Of course, once a field has non-zero infinitesimals, then there must be infinite elements also – these are the reciprocals of infinitesimals. It follows also that R is enriched in having, for each r ∈ R, new elements x with x ≈ r (taking x = r + δ where δ is infinitesimal). One way to construct ∗R is as an ultrapower of the reals ∗

R = RN /U

where U is a nonprincipal ultrafilter2 (or maximal filter) on N. That is, ∗R consists of equivalence classes of sequences of reals under the equivalence relation ≡U , defined by (an ) ≡U (bn )

⇐⇒

{n : an = bn } ∈ U.

Sets in U should be thought of as big sets, or more strictly U-big, with those not in U designated U−small. The ultrafilter property means that every 1 2

This takes its values in ∗R, and is defined just as in R, so that |x| = x if x ≥ 0 and |x| = −x if x < 0. A nonprincipal ultrafilter U on N is a collection of subsets of N that is closed under intersections and supersets, contains no finite sets, and for every set A ⊆ N has either A ∈ U or N \ A ∈ U .

1.2 Nonstandard Analysis

3

set is either U-big (those in U) or U-small (those not in U). It is convenient to use the terminology U-almost all to mean “for a set A of natural numbers with A ∈ U”. Using this terminology we can say that the equivalence relation ≡U identifies sequences (an ) and (bn ) that agree on a U-big set of indices n, or that agree U-almost always. We denote the equivalence class of a sequence (an ) by (an )U (sometimes the notation [(an )] is used instead). The reals R are identified with the equivalence classes of constant sequences, so that ∗R is then an extension of R. The algebraic operations +, × and the order relation < are extended to ∗R pointwise (after checking that this is safe); strictly the extensions should be denoted ∗+, ∗×, ∗ 0. Definition 1.5 (Standard Part) If x is a finite hyperreal the unique real r ≈ x is called the standard part of x.

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1 Loeb Measures

For a finite hyperreal x ∈ ∗R there are two notations (both useful) for the standard part of x: ◦

x = st(x) = the standard part of x.

On occasions, when considering extended real valued functions (with values in R = R ∪ {−∞, ∞}), it is convenient to write ◦x = ±∞ if x is positive (resp. negative) infinite. Remark The Standard Part Theorem is equivalent to the completeness of R. The next two theorems illustrate the way in which real analysis develops using the additional structure of ∗R. For the sake of completeness we give brief proofs that provide a flavour of the nonstandard methodology, and especially the use of the Transfer Principle. For a full account of the development of real analysis using infinitesimals, see any of the references [30, 54, 47, 56, 58, 60, 69]. For the following, note that since a sequence s = (sn )n∈N of reals is just a function s : N → R, its nonstandard extension ∗s = (sn )n∈∗N is simply a function ∗s : ∗N → ∗R. Theorem 1.6 Let (sn ) be a sequence of real numbers and let l ∈ R. Then sn → l as n → ∞

⇐⇒

∗

sK ≈ l for all infinite K ∈ ∗N.

Proof Suppose first that sn → l, and fix infinite K ∈ ∗N. We have to show that |∗sK − l| < ε for all real ε > 0. For any such ε there is a number n0 ∈ N such that the following holds in R: ∀n ∈ N[n ≥ n0 → |sn − l| < ε] The Transfer Principle now tells us that ∀N ∈ ∗N[N ≥ n0 → |∗sN − l| < ε] is true in ∗R. In particular taking N = K we see that |∗sK − l| < ε as required. Conversely, suppose that ∗sK ≈ l for all infinite K ∈ ∗N. Then, for any given real ε > 0, we have ∃K ∈ ∗N ∀N ∈ ∗N[N ≥ K → |∗sN − l| < ε] The Transfer Principle applied to this statement shows that in R: ∃k ∈ N∀n ∈ N[n ≥ k → |sn − l| < ε] Taking n0 to be any such k proves that sn → l.

For the next result note that if f is a real function defined on the open interval ]a, b[ then ∗f is defined on the hyperreal interval ∗]a, b[= {x ∈ ∗R : a < x < b}, and takes values in ∗R.


7

Theorem 1.7 Let c ∈]a, b[ (where a, b, c ∈ R) and f :]a, b[→ R. Then f is continuous at c

⇐⇒

∗

f (z) ≈ f (c) whenever z ≈ c in ∗R.

Proof The proof is very similar to that of Theorem 1.6. Suppose first that f is continuous at c, and fix a hyperreal z ≈ c. We have to show that |∗f (z) − f (c)| < ε for all real ε > 0. For any such ε there is a number 0 < δ ∈ R such that the following holds in R: ∀x[|x − c| < δ → |f (x) − f (c)| < ε] The Transfer Principle now tells us that ∀X[|X − c| < δ → |∗f (X) − f (c)| < ε] is true in ∗R. In particular taking X = z we see that |∗f (z) − f (c)| < ε as required. Conversely, suppose that |∗f (z) − f (c)| ≈ 0 for all z ≈ c in ∗R. Let a real ε > 0 be given. Then taking Y to be any positive infinitesimal the following holds in ∗R: ∃Y ∀X[|X − c| < Y → |∗f (X) − f (c)| < ε] The Transfer Principle applied to this statement gives, in R: ∃y∀x[|x − c| < y → |f (x) − f (c)| < ε] Taking δ to be any such y shows that f is continuous at c as required. Before moving to the next section, it should be pointed out that there are several other ways to construct the hyperreals. Moreover, the conventional terminology is misleading in that different constructions do not necessarily give isomorphic structures. All versions of the hyperreals however obey the Transfer Principle, and this is all that is needed to do basic nonstandard real analysis. Indeed, one perfectly workable approach to the subject is an axiomatic one, which merely specifies that ∗R is an extension of R that obeys the Transfer Principle. (This approach would be parallel to a development of real analysis that proceeds without being concerned with any particular construction of R, using only the assumption that R is a complete ordered field.) 1.2.2 The nonstandard universe To use Robinson’s ideas beyond the realm of real analysis, it is necessary to repeat the construction of ∗R for any mathematical object M that might be needed, giving a nonstandard version ∗M of M that contains ideal elements (such as infinitesimals in the case of ∗R). M could be a group, ring, measure space, metric space or any mathematical object, however complicated.

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1 Loeb Measures

Rather than construct each nonstandard extension ∗M as required, it is more economical to construct at the outset a nonstandard version ∗V of a working portion of the mathematical universe V that contains each object M that might be needed. Then ∗V will contain ∗M for every M ∈ V. Such a construction has the additional advantage that the corresponding Transfer Principle preserves connections between structures as well as their intrinsic properties. Here, briefly, is the way it works. First, for most mathematical practice, an adequate portion of the mathematical universe is the superstructure over R, denoted by V = V(R), defined as follows. V0 (R) = R Vn+1 (R) = Vn (R) ∪ P(Vn (R)), and V = V (R) =

n∈N

Vn (R).

n∈N

(If V (R) is not big enough to contain all the objects4 required, simply replace the starting set R by a suitable larger set S, giving V = V (S).) The next step is to construct a mapping ∗ : V (R) → V (∗R) which associates to each object M ∈ V a nonstandard extension ∗M ∈ V (∗R). Roughly, we have M ⊂ ∗M with ∗M \ M consisting of “ideal” or “nonstandard” elements. For example ∗N \ N consists of infinite (hyper)natural numbers; if M is an infinite dimensional Hilbert space H together with its finite dimensional subspaces then ∗M will contain some infinite hyperfinite dimensional subspaces. The way to visualise the resulting nonstandard universe is as follows. External objects

U

? Standard objects

z

A

V = V (R)

∗ A

∗

V (R) (= internal objects)

R

∗

R

The Nonstandard Universe The nonstandard universe is in fact the collection 4

We are now taking the view that all mathematical objects are sets.

V (∗R)


9

∗

V = {x : x ∈ ∗M for some M ∈ V}

consisting of all new and old members of sets in V. Although ∗V ⊂ V (∗R), it is crucial to realise that ∗V is not the same as V (∗R). Sets in ∗V are known as internal sets. One way5 to construct ∗V is by means of an ultrapower VN /U although there is a little more work to do (compared to the corresponding construction of ∗R). The set membership relation ∈ that gives the structure (V, ∈), when extended pointwise to the ultrapower VN /U, gives a “pseudomembership” relation E, say, resulting in the structure (VN /U, E). It is then necessary to take the “Mostowski collapse” of this structure, which constructs simultaneously the collection ∗V and an injection i : (∗V, ∈) → (VN /U, E). Although i is not surjective, its range includes the equivalence class of each constant sequence, and then ∗M is defined by ∗

M = i−1 ((M, M, M, . . . M)/U).

The key property of the nonstandard universe ∗V is a Transfer Principle which again indicates precisely which properties of the superstructure V are inherited by ∗V. Theorem 1.8 (The Transfer Principle) Suppose that ϕ is a bounded quantifier statement. Then ϕ holds in V if and only if ∗ϕ holds in ∗V. A bounded quantifier statement (bqs) is simply a statement of mathematics that can be written in such a way that all quantifiers range over a prescribed set. That is, we have subclauses such as ∀x ∈ A and ∃y ∈ B but not unbounded quantifiers such as ∀x and ∃y. Most quantifiers in mathematical practice are bounded (often only implicitly in exposition). A bqs ϕ may also contain fixed sets M from V, which will be replaced in ∗ϕ by ∗M. Members of internal sets are internal (this follows easily from the construction) and since the sets ∗M are also internal, it follows that the information we obtain from the Transfer Principle is entirely about internal sets. To illustrate, the Transfer Principle tells us that any internal bounded subset of ∗R 5

This sketch of a construction of ∗V can be skipped without any loss – it is included to show that a nonstandard universe is a very down-to-earth and non-mysterious mathematical construct.

10

1 Loeb Measures

has a least upper bound, whereas this can fail for external6 sets. For example, the set N is a subset of ∗R that is bounded (by any infinite hyperreal) but has no least upper bound – from which we deduce that N is external. Incidentally this demonstrates that there actually are external sets – i.e. V(∗R) \ ∗V = Ω. An easy application of the Transfer Principle gives the following very useful properties. Proposition 1.9 Let A ⊆ ∗R be an internal set. (a) (Overflow) If A contains arbitrarily large finite numbers then it also contains an infinite number; (b) (Underflow) If A contains arbitrarily small positive infinite7 numbers then it contains a positive finite number. Taking reciprocals gives a corresponding pair of principles for the set of infinitesimals. As with ∗R it is possible (and quite convenient) to take an axiomatic approach to ∗V, which simply postulates the existence of a set ∗V and a mapping ∗ : V → ∗V that obeys the Transfer Principle. For most purposes (and certainly the construction of Loeb measures) one further assumption is needed, which we now discuss. 1.2.3 ℵ1 -saturation A nonstandard universe constructed as a countable ultrapower has an additional property called ℵ1 -saturation, which we highlight here because of its importance. Definition 1.10 A nonstandard universe ∗V is said to be ℵ1 -saturated if the following holds: if (Am )m∈N is a countable decreasing sequence of internal sets with each Am = Ω, then m∈N Am = Ω. Theorem 1.11 A nonstandard universe ∗V constructed as a countable ultrapower is ℵ1 -saturated. Proof (Sketch) Each set Am is represented by a sequence of standard sets (Xm,n )n∈N . Since each Am is nonempty and the sequence is decreasing, then for U-almost all8 n we have Xm+1,n ⊆ Xm,n and Xm,n = Ω. By a systematic modification of the sets Xm,n on a U-small9 set of indices n we may assume that Xm+1,n ⊆ Xm,n and Xm,n = Ω for all n and m. Now pick xn ∈ Xn,n 6 7 8 9

an external set is one that is not internal that is, for every positive infinite x ∈ ∗R there is an element a ∈ A with a infinite and a < x that is, the set {n : Xm+1,n ⊆ Xm,n } belongs to U. i.e. a set that is not in the ultrafilter U .


11

and let y = (xn )U be the element represented by this sequence. Then y ∈ Am for every m since xn ∈ Xm,n for n ≥ m. ℵ1 -saturation is a kind of compactness property that is essential for the Loeb measure construction. For the rest of these lectures we assume that ∗V is a nonstandard universe that is ℵ1 -saturated. It is possible to build nonstandard universes with stronger saturation type properties, by an extension of the techniques discussed above. These are needed in some applications of Loeb measures involving topological spaces that do not have a countable sub-base, and in other “non-separable” mathematical applications. An equivalent and very useful formulation of ℵ1 -saturation, known as countable comprehension, goes as follows. Countable comprehension Given any sequence (An )n∈N of internal subsets of an internal set A, there is an internal sequence10 (An )n∈∗N of subsets of A that extends the original sequence. To see that ℵ1 -saturation implies countable comprehension, apply ℵ1 saturation to the sets Bm consisting of internal sequences (Cn )n∈∗N with Cn = An for n ≤ m. The reverse implication is proved using the overflow principle. (The reader may like to try proving this as an exercise.) 1.2.4 Nonstandard topology We gather together here some of the basic nonstandard topological notions that will be referred to later. First, we note that the idea of being infinitely close generalises to any topological space, extending the idea of a monad. Recall that for a ∈ R the monad of a is the set monad(a) = {x ∈ ∗R : x ≈ a}. More generally we have: Definition 1.12 Let (X, T ) be a topological space. (i) For a ∈ X the monad of a is ∗ monad(a) = U. a∈U ∈T

(ii) If x ∈ ∗X, we write x ≈ a to mean x ∈ monad(a). (Note that in general this is not a symmetric relationship.) (iii) x ∈ ∗X is nearstandard if x ≈ a for some a ∈ X. (iv) ns(Y ) is the set of nearstandard points in Y , for any Y ⊆ ∗X. (v) st(Y ) = {a ∈ X : x ≈ a for some x ∈ Y }; this is called the standard part of Y . 10

that is, an internal function with domain ∗N.

12

1 Loeb Measures

The idea of the pointwise standard part mapping for ∗R generalizes to Hausdorff spaces because of the next result. Proposition 1.13 A topological space X is Hausdorff if and only if monad(a) ∩ monad(b) = Ω Proof An easy exercise.

for

a = b,

a, b ∈ X.

This means that for Hausdorff spaces we can define the standard part mapping st : ns(∗X) → X by st(x) = the unique a ∈ X with a ≈ x. The following notation is often used: ◦

x = st(x).

If necessary we write stX or stT to denote the space or topology concerned. The following is another important notion that plays a key rˆ ole in constructing solutions to differential equations of all kinds. Definition 1.14 Suppose that Y is a subset of ∗ X for some topological space X, and F : ∗X → ∗R is internal. Then F is said to be S-continuous on Y if for all x, y ∈ Y we have x ≈ y =⇒ F (x) ≈ F (y). The importance of this notion is seen in the following result. Theorem 1.15 If F : ∗R → ∗R is S-continuous on an interval ∗[a, b] for real a, b, and F (x) is finite for some x ∈ ∗[a, b], then the standard function defined on [a, b] by f (t) = ◦F (t) is continuous, and ∗f (τ ) ≈ F (τ ) for all τ ∈ ∗[a, b]. Remark This theorem shows that S-continuous functions in ∗C[a, b]are precisely those that are nearstandard in the uniform topology on C[a, b], and the function f defined above is the standard part ◦F for this topology. One final result from general nonstandard topology that we will need is: Proposition 1.16 Let (X, T ) be separable, Hausdorff. Suppose that Y ⊆ ∗X is internal, and A ⊆ X. Then (a) st(Y ) is closed, (b) if X is regular and Y ⊆ ns(∗X) then st(Y ) is compact, (c) st(∗A) = A (the closure of A), (d) if X is regular, then A is relatively compact iff ∗A ⊆ ns(∗X).

1.3 Construction of Loeb Measures

13

Remark The condition that X should be separable in Proposition 1.16 can be omitted if the nonstandard model has more saturation – namely κsaturation (see the Remark at the end of the previous section), where the topology on X has a base of cardinality κ. However, in all our applications the relevant spaces X are separable, and so ℵ1 -saturation (which we have in our model) is sufficient.

1.3 Construction of Loeb Measures A Loeb measure is a measure constructed from a nonstandard measure by the following construction of Peter Loeb [71]. We confine our attention in these lectures mainly to finite (or bounded) Loeb measures. Suppose that an internal set Ω and an internal algebra A of subsets of Ω are given, and suppose further that µ is a finite internal finitely additive measure on A. This means that µ is an internal mapping µ : A → ∗[0, ∞) with µ(A∪B) = µ(A)+µ(B) for disjoint A, B ∈ A, and that µ(Ω) is finite.11 Thus µ(A) is finite for each A ∈ A, so we may define the mapping ◦

µ : A → [0, ∞)

by ◦µ(A) = ◦(µ(A)). Clearly ◦µ is finitely additive, so that (Ω, A, ◦µ) is a standard finitely additive measure space. In general this is not a measure space, because A is not σ-additive unless A is finite. Nevertheless, if (An )n∈N is a family of sets from A, then the set n∈N An is almost in A. It differs from a set in A by a null set (a notion to be defined shortly); see the Key Lemma (Lemma 1.19) and its corollary below. This is what lies at the heart of the following fundamental result proved by Loeb. Theorem 1.17 There is a unique σ-additive extension of ◦µ to the σ-algebra σ(A) generated by A. The completion of this measure is the Loeb measure corresponding to µ, denoted µL and the completion of σ(A) is the Loeb σalgebra, denoted by L(A). Proof For a quick proof we can apply Caratheodory’s extension theorem. It is only necessary to check σ-additivity of ◦µ on A. Suppose that (An )n∈N is a sequence of pairwise disjoint sets from A such that An ∈ A. A= n∈N 11

It also means of course that all sets in A are internal

14

1 Loeb Measures

By ℵ1 -saturation (applied to the decreasing sequence of sets A \ there is m ∈ N such that m An = An . n∈N

m n=1

An )

n=1

So Ak = Ω for k > m, and m m ◦ ◦ ◦ ◦ µ An = µ An = µ(An ) = µ(An ), n∈N

n=1

n=1

n∈N

using finite additivity. Caratheodory’s theorem (see [88] for example) now gives the result. It is quite straightforward and rather more illuminating to prove Loeb’s theorem from “first principles” and here is one way to proceed – based around the idea of a Loeb null set. (See [29] for full details of this approach.) Definition 1.18 Let B ⊆ Ω (not necessarily internal). We say that B is a Loeb null set if for each real ε > 0 there is a set A ∈ A with B ⊆ A and µ(A) < ε. The following result makes it clear that A is almost a σ-algebra. Lemma 1.19 (Key Lemma) Let (An )n∈N be an increasing family of sets, with each An in A, and let B = n∈N An . Then there is a set A ∈ A such that (a) B ⊆ A; (b) ◦µ(A) = limn→∞ ◦µ(An ); (c) A \ B is null. Proof

Let α = limn→∞ ◦µ(An ). For each finite n, µ(An ) ≤ ◦µ(An ) +

1 1 ≤ α+ . n n

Now, using ℵ1 -saturation, take an increasing internal sequence (An )n∈∗N of sets in A extending the sequence (An )n∈N . Overflow gives an infinite N such that 1 µ(AN ) ≤ α + . N Let A = AN . Then (a) holds because A ⊇ An for each finite n. Moreover, µ(An ) ≤ µ(A) for each finite n, so ◦µ(An ) ≤ ◦µ(A) ≤ α, giving ◦µ(A) = α, which is (b). Moreover, ◦µ(A\An ) = ◦µ(A)− ◦µ(An ) → 0. Now A\B ⊆ A\An so A \ B is null. From this Key Lemma, it is clear that A is almost a σ-algebra – and in fact it is a σ-algebra modulo null sets. The following makes this precise.


15

Definition 1.20 (i) Let B ⊆ Ω. We say that B is Loeb measurable if there is a set A ∈ A such that A∆B 12 is Loeb null. Denote the collection of all Loeb measurable sets by L(A). (ii) For B ∈ L(A) define µL (B) = ◦µ(A) for any A ∈ A with A∆B null, and call µL (B) the Loeb measure of B. It is then quite straightforward to prove: Theorem 1.21 L(A) is a σ-algebra, and µL is a complete (σ-additive) measure on L(A). The measure space Ω = (Ω, L(A), µL ) is called the Loeb space given by (Ω, A, µ), and L(A) is called the Loeb algebra. Of course L(A) depends on both A and µ, so strictly we should write L(A, µ), but usually it is clear which measure is intended. If µ(Ω) = 1 then Ω is a Loeb probability space and µL is the Loeb probability measure given by µ. The following are alternative characterisations of Loeb measurable sets, and are often taken as the fundamental definition (see [3], [20] or [69] for example). First some definitions are required. Definition 1.22 Let B ⊆ Ω (not necessarily internal). (i) B is µ-approximable if for every real ε > 0 there are sets A, C ∈ A with A ⊆ B ⊆ C and µ(C \ A) < ε. (ii) The inner and outer Loeb measure of B, µ(B) and µ(B) are given by µ(B) = sup{◦µ(A) : A ⊆ B, A ∈ A} µ(B) = inf{◦µ(A) : A ⊇ B, A ∈ A} Then we have Theorem 1.23 The following are equivalent: (a) B is Loeb measurable. (b) B is µ-approximable. (c) µ(B) = µ(B).

12

A∆B is the symmetric difference (A \ B) ∪ (B \ A)

16

1 Loeb Measures

Loeb counting measure For a simple illustration of the Loeb construction (but one which has far reaching applications) consider the Loeb counting measure, as follows. Let Ω = {1, 2, . . . , N } where N ∈ ∗N \ N, so that Ω is a infinite hyperfinite set (necessarily internal), and let ν be the counting probability measure on Ω, defined by |A| |A| = ν(A) = |Ω| N for A ∈ ∗P(Ω) = A, say.13 Here |A| denotes the number14 of elements in A. Note that ∗P(Ω) is a proper subset of P(Ω), since, for example, the set N ∈ P(Ω) \ ∗P(Ω), which in turn shows that A is not a σ-algebra. The Loeb counting measure νL is the completion of the extension to σ(A) of the finitely additive measure ◦ν. 1.3.1 Example: Lebesgue measure A first simple application of Loeb measure is an intuitive construction of Lebesgue measure. First we define the hyperfinite (time)15 line T corresponding to the interval [0, 1]. Definition 1.24 Fix N ∈ ∗N \ N and let ∆t = N −1 . The hyperfinite time line (based on ∆t, for the interval [0, 1]) is the set T = {0, ∆t, 2∆t, 3∆t, . . . , 1 − ∆t}. (In applications hyperfinite time lines may be taken with different end points, according to need.) We will use sanserif symbols t, s for elements of T to distinguish them from those in [0, 1]. Theorem 1.25 Let νL be the Loeb counting measure on the hyperfinite time line T. Define −1 (i) M = {B ⊆ [0, 1] : st−1 T (B) is Loeb measurable}, where stT (B) = ◦ {t ∈ T : t ∈ B}. (ii) λ(B) = νL (st−1 T (B)) for B ∈ M. Then ([0, 1], M, λ) is Lebesgue measure (i.e. M is the Lebesgue completion of the Borel sets B[0, 1], and λ(B) is the Lebesgue measure of B ∈ M.) 13 14

15

An application of the Transfer Principle tells us that this is the collection of all internal subsets of Ω. The Transfer Principle tells us that for internal subsets A of Ω there is a unique M ∈ ∗N, M ≤ N , such that there is an internal bijection F : {1, 2, . . . , M } → A – and this M is what is meant by |A|. Equivalently, | · | is the extension to ∗V of the standard function | · | that gives the cardinality of finite sets. This has become the conventional terminology for this discrete representation of the interval [0,1] when it is used to represent time.


17

Proof (Sketch) It is routine to check that M that contains each ∗ is a σ-algebra 1 1 ([a, b]) = [a− , b+ ]∩T , which is standard interval [a, b] (since st−1 T n∈N n n a countable intersection of internal sets), and that λ is a complete probability measure on M. Showing that λ is translation invariant and λ([a, b]) = b − a is straightforward, so that ([0, 1], M, λ) is an extension of Lebesgue measure. Now take B ∈ M, and an inner approximation A ⊆ st−1 T (B) with A internal. Then the set st(A) is a closed inner approximation of B, and this suffices to show that B is Lebesgue measurable. This result is a particular case of a general theorem of Anderson [6] that shows how any Radon measure on a Hausdorff space can be represented by a hyperfinite Loeb counting measure. A famous example of this is Anderson’s representation of Wiener measure, below. A less well known but very pleasant example is David Ross’ very intuitive construction of Haar measure16 , as follows (taken from [85, 87]). 1.3.2 Example: Haar measure Let G be a compact group, and take an internal infinitesimal neighbourhood17 V of 1. Take a minimal ∗open cover Ω of ∗G consisting of sets that are translates of V . So Ω = {V1 , . . . , VN } say with each Vi = gi V for some gi ∈ ∗G. Let νL be the Loeb counting probability measure on Ω. For Borel sets B ⊆ G define −1

m(B) = νL (stΩ (B)) where stΩ : Ω → G is the generalisation of the standard part mapping18 to this context. Then m is Haar measure on G. To see this, first it routine to show that m is a Borel probability measure on G; the other required property is that m should be translation invariant – that is, m(B) = m(gB) for each B ∈ B and g ∈ G. It is sufficient to show that m(B) ≤ m(gB), and for this take an internal set A ⊆ st−1 Ω (B). Let C = {Vj : Vj ∩ gVk = Ω for some Vk ∈ A} and note that C ⊆ st−1 Ω (gB). It is easy to check that the collection (Ω \ A) ∪ g −1 C is a cover of ∗G by sets that are translates of V , so by minimality of the collection Ω this gives |C| ≥ |A|. Thus m(gB) ≥ m(B) as required. 1.3.3 Example: Wiener measure Perhaps the best known measure construction using Loeb measure theory is Anderson’s construction [5] of Wiener measure, which we now describe. Recall 16 17 18

Haar measure on a compact group is the unique probability measure that is invariant under multiplication by group elements. This means that V ⊂ ∗U for each open neighbourhood U of 1. Actually we have the mapping stG : ∗G → G, but since V is an infinitesimal neighbourhood, the set stG (Vi ) is a singleton for each i, so it makes sense to define stΩ : Ω → G.

18

1 Loeb Measures

that Wiener measure W on C = C0 [0, 1] (the set of continuous functions x with x0 = 0) is the unique Borel probability on C such that

−y 2 1 dy exp W ({x : xt − xs ∈ B}) = 1 2(t − s) (2π(t − s)) 2 B for s < t and Borel B ⊂ R, and such that disjoint increments xt − xs of paths x ∈ C are independently distributed under W . Take the hyperfinite time line T = T ∪ {1}, where T is as above and let CN be the set of all polygonal paths B(t)t∈T filled in linearly between the time points t ∈ T, with B(0) = 0 and √ B(t + ∆t) − B(t) = ∆B(t) = ± ∆t. Let WN =counting probability on CN , giving the internal probability space (CN , AN , WN ) where AN = ∗P(CN ). This gives the corresponding Loeb space Ω = (CN , L(AN ), PN ) where PN = (WN )L .

√ 3 ∆t √ 2 ∆t √ ∆t 0

∆t

5∆t

10∆t

15∆t

√ − ∆t √ −2 ∆t √ −3 ∆t

An infinitesimal random walk


19

Theorem 1.26 (Anderson) (a) For a.a.19 B ∈ CN , B is S-continuous, and gives a continuous path b = ◦B ∈ C. (b) For Borel D ⊆ C W (D) = PN (st−1 (D)) is Wiener measure.20 (c) Writing Ω = CN and ω instead of B for a generic point in Ω, the process b : [0, 1] × Ω → R defined by b(t, ω) = ◦ω(t) is Brownian motion on the probability space Ω. This is arguably the most intuitive of all the many constructions of Brownian motion/Wiener measure, and captures precisely the stochastic analyst’s rule of thumb “db2 = dt”, since we really do have ∆B 2 = ∆t. Anderson [5] used it to give an elementary proof of Donsker’s invariance principle, together with a pathwise construction of the Itˆ o integral and an intuitive proof of Itˆ o’s Lemma. His construction opened the way for a large number of important applications in stochastic analysis and related fields, either directly or as an inspiration in more general situations. One of the first and most important of these, due to Keisler [61] is the idea of solving stochastic differential equations by means of hyperfinite difference equations. The paper [63] indicates some of the more recent developments in this area. We will discuss these ideas later (see section 1.5.3 below, and also Lecture 2), after we have outlined the basics of Loeb integration theory. First it is necessary to consider Loeb measurable functions. 1.3.4 Loeb measurable functions Suppose we have a Loeb space Ω = (Ω, L(A), µL ) constructed from the internal space (Ω, A, µ). A Loeb measurable function f : Ω → R is simply a function that is measurable in the conventional sense with respect to the Loeb algebra L(A). That is, f −1 (] − ∞, a]) ∈ L(A) for every real interval [a, b]. There is of course another concept of measurable function, given by the transfer of the standard definition. A function F : Ω → ∗R is ∗measurable if F is internal and F −1 ([α, β]) ∈ A for every hyperreal interval [α, β] (with α, β ∈ ∗R). The fundamental connection between these two notions is as follows. 19 20

with respect to the Loeb measure PN of course. The standard part mapping here is the restriction to CN of the mapping st : ∗ C → C for the uniform topology – see section 1.2.4 above.

20

1 Loeb Measures

Theorem 1.27 Let f : Ω → R. Then the following are equivalent. (a) f is Loeb measurable; (b) there is a ∗measurable function F : Ω → ∗R such that f (ω) ≈ F (ω) for almost all ω ∈ Ω (with respect to the Loeb measure µL ).21 For a proof see [29], [87], or the original paper of Anderson [6], who proved the result for measurable functions into a second-countable Hausdorff space. David Ross has extended this further to include all metric spaces [86]. Definition 1.28 A function F as given by Theorem 1.27 is called a lifting of f ; that is, a lifting (with respect to µL ) of a function f : Ω → R is an internal ∗measurable function F : Ω → ∗R such that f (ω) ≈ F (ω) for almost all ω ∈ Ω (with respect to the Loeb measure µL ). A general lifting result that is very useful is Anderson’s ‘Luzin’ theorem [6]. Theorem 1.29 Let (X, C, µ) be a complete Radon space and suppose that f : X → R is measurable. Then ∗f is a lifting of f with respect to µL . That is, ∗ f (x) ≈ f (◦x) for (∗µ)L almost all x ∈ ∗X. Remarks 1. The kind of lifting given by this theorem is known as a two-legged lifting, to distinguish it from the kind of lifting in Definition 1.28. 2. Anderson actually established this result for the situation where the range of f is any Hausdorff space with a countable base of open sets.

1.4 Loeb Integration Theory Given a Loeb space Ω = (Ω, L(A), µL ) and its originating internal space (Ω, A, µ), there are two integrals to consider. First, there is the internal integral

F dµ Ω 21

This result also holds for extended real valued functions f : Ω → R provided we adopt the terminology ◦x = ±∞ and hence x ≈ ±∞ if x ∈ ∗R is positive (resp. negative) infinite.

1.4 Loeb Integration Theory

21

for any (internal) ∗integrable function F : Ω → ∗R. The value of this integral is a hyperreal that is given by the transfer of the construction of the integral on a standard space. Secondly there is the classical Lebesgue integral

f dµL Ω

defined in the usual way for a Loeb integrable function f : Ω → R: the term Loeb integrable function f means simply that f is integrable (in the conventional sense) with respect to the Loeb measure µL on Ω. Loeb integration theory gives the connection between these two

integrals. Its importance stems from the fact that the internal integral F dµ may be quite simple (for example a hyperfinite sum) while a closely related Loeb integral can represent a general standard integral (such as a Lebesgue integral on the real line or a Wiener integral). Here are the details. Theorem 1.30 If F is a finitely bounded internal measurable function then

◦ F dµ = ◦F dµL .

Corollary 1.31 If F is a (finitely) bounded lifting of a Loeb measurable f , then

◦ f dµL = F dµ.

We cannot in general expect equality of ◦ F dµ and ◦F dµL since F may be large on a set of infinitesimal measure, as in the following example. Example Consider Ω = ∗[0, 1] and define F : ∗[0, 1] → ∗R by 1 K for τ ≤ K F (τ ) = 0 otherwise. Let Λ denote ∗Lebesgue measure. Then ◦F (τ

◦

) = 0 almost everywhere with respect to ΛL , and hence F dΛL = 0. But F dΛ = 1. We always have Theorem 1.32 For any internal A-measurable F with F ≥ 0

◦ ◦ F dµL ≤ F dµ, where we allow the value ∞ on either side.

22

1 Loeb Measures

To obtain equality of ◦ F dµ and ◦F dµL it is necessary to have some condition on F akin to standard integrability — roughly, so that F is not too big on small sets. The following is the appropriate condition. Definition 1.33 Let a function F : Ω → ∗R be A-measurable and internal and µ an

internal finite measure. Then F is S-integrable if (i) Ω |F |dµ is finite,

(ii) if A ∈ A and µ(A) ≈ 0, then A |F |dµ ≈ 0. Note If µ is not finite an extra condition

has to be added: (iii) if A ∈ A and F ≈ 0 on A, then A |F |dµ ≈ 0. This is always satisfied for a finite measure µ. If F ≈ 0 on A and µ(A) = 0, then for any 0 < ε ∈ R we have |F | < ε. Hence A |F |dµ < εµ(A), which is enough since µ(A) is finite. 1 The function in the example above is not S-integrable because A = [0, K ] has Λ(A) ≈ 0 but A F dΛ = 1.

Note that F is S-integrable if and only if its positive and negative parts F + and F − are S-integrable, and equivalently if |F | is S-integrable. The next result shows the importance of S-integrability. Theorem 1.34 Let F : Ω → ∗R be A-measurable with F ≥ 0. Then the following conditions are equivalent: (a) F is S-integrable, (b) ◦F is Loeb integrable and

◦ F dµ = ◦F dµL .

The following is an equivalent formulation of S-integrability (the proof is left as an exercise). Proposition 1.35 An internal function F is S-integrable if and only if for all infinite K

|F |dµ ≈ 0. |F |>K

To complete the basic theory of Loeb integration we have: Theorem 1.36 Let f : Ω → R be Loeb measurable. Then f is µL -integrable if and only if it has an S-integrable lifting F : Ω → ∗R. Definition 1.37 We say that F : Ω → ∗R is SLp (p > 0) if |F |p is Sintegrable (so SL1 means S-integrable). Here is a very useful test for S-integrability isolated by Lindstrøm [68] and frequently applied in the case p = 2.

1.5 Elementary Applications

Theorem 1.38 Suppose µ(Ω) < ∞. If F : Ω → measurable, and

|F |p dµ < ∞

∗

23

R is internal, A-

Ω

for some p > 1, p ∈ R, then F is S-integrable.

1.5 Elementary Applications As a warm up for the more substantial applications of Loeb measures in later lectures, we present here a few simple illustrations of their power. 1.5.1 Lebesgue integration Recall the hyperfinite time set T defined above (Definition 1.24), which carries the counting Loeb measure νL . For any function f : [0, 1] → R we may define a corresponding function fˆ : T → ∗R by fˆ(t) = f (◦t). The characterisation (or definition) of Lebesgue measure given by Theorem 1.25, combined with Theorem 1.27 yields immediately: Theorem 1.39 The following are equivalent: (a) f is Lebesgue measurable; (b) fˆ is Loeb measurable (wrt νL ); (c) there is an internal function F : T → ∗R (a lifting of fˆ) such that for a.a. t ∈ T f (◦t) = ◦F (t) The lifting F of fˆ is a two-legged lifting in the sense described earlier. Now apply Theorem 1.36 to give the following pleasant characterisation of the Lebesgue integral. Theorem 1.40 Suppose that f, fˆ are as above. Then the following are equivalent: (a) f is Lebesgue integrable; (b) fˆ is Loeb integrable; (c) there is an S-integrable function F : T → ∗R that is a lifting of f (and fˆ). If any of (a)–(c) holds then

1

f dλ = F (t)∆t, fˆdνL = ◦ T 0 t∈T

the summation term t∈T F (t)∆t being another way of writing T F dν.

24

1 Loeb Measures

1.5.2 Peano’s Existence Theorem The above characterisation of the Lebesgue integral as a hyperfinite sum leads naturally to the method of hyperfinite difference equations for solving ODEs – an appealing technique pioneered by Keisler and extended to great effect especially for stochastic differential equations – see [61]. Here is an outline of a proof of Peano’s fundamental existence theorem using this technique. Theorem 1.41 (Peano) Suppose that f : [0, 1] × R → R is bounded, measurable, and continuous in the second variable, and let x0 ∈ R. Then there is a solution to the differential equation dx(t) = f (t, x(t))dt x(0) = x0

(1.1)

(Of course, what is meant is really the corresponding integral equation.) Proof Without any loss of generality we may assume that x0 = 0 (otherwise consider the equation for x(t) − x0 ). Suppose that |f | ≤ c. An extension of the Lifting Theorem 1.27 is used (see below for details) to obtain an internal function F : T × ∗[−c, c] → ∗R such that |F | ≤ c and for almost all t∈T F (t, X) ≈ f (◦t, ◦X) (1.2) for all |X| ≤ c. The hyperfinite difference equation corresponding to (1.1) is now ∆X(t) = F (t, X(t))∆t, where ∆X(t) = X(t + ∆t) − X(t), together with the initial condition X(0) = 0. This is an internal equation for an internal function X : T → ∗R, with solution X(t) defined recursively by X(0) = 0 X(t + ∆t) = X(t) + F (t, X(t))∆t. Then X is S-continuous, and |X(t)| ≤ ct ≤ c for all t ∈ T. So we may define a continuous function x : [0, 1] → R by x(t) = ◦X(t) for any t ≈ t. Clearly |x(t)| ≤ c. To see that x(t) is a solution, observe that, by (1.2) and the definition of x, for almost all t ∈ T F (t, X(t)) ≈ f (◦t, ◦X(t)) = f (◦t, x(◦t)) which means that the function G(t) = F (t, X(t)) is a lifting of the function g(t) = f (t, x(t)). So, applying Theorem 1.40 to g(t) and its lifting G(t) we have (putting t = ◦t)

1.5 Elementary Applications

x(t) = ◦X(t) =

◦

25

F (s, X(s))∆t

s 0 such that St B ⊆ E whenever t ≥ t0 . If there is a bounded absorbing set E then a simple way to construct an attractor is as follows. First note that ∗E is S-absorbing – meaning that for any finitely bounded set B ⊂ ∗X there is a finite time τ0 such that ∗Sτ B ⊆ ∗E for all τ ≥ τ0 . This follows by transfer of the absorbing property of E. Now write Tτ for ∗Sτ and define the internal set C by C= Tτ ∗E = Tτ ∗E τ −infinite

n∈N τ ≥n

(the equivalence follows by an application of ℵ1 -saturation). Then it follows easily that C is a global S-attractor , by which we mean that C has the three properties noted in the following theorem.

2.6 Attractors for Navier–Stokes Equations

49

Theorem 2.11 (a) C is a countable intersection of internal sets; in fact C= Cn n∈N

where Cn =

Tτ ∗E.

τ ≥n

(b) Tτ C = C for all finite τ ; (c) For each n ∈ N and finitely bounded set B ⊂ ∗X there is t0 ∈ [0, ∞) such that T (τ )B ⊆ Cn for all τ ≥ t0 . The set C is bounded and hence weakly nearstandard in ∗X, so we may take the standard part in the weak topology A = w-stC = {w-stx : x ∈ C} and this is a weakly compact subset of X. It is now quite straightforward to see that A is a global attractor for the semigroup St , given certain natural continuity assumptions. The proof draws on the fact that C is an S-attractor. This idea for constructing attractors is adapted to give the particular applications to the Navier–Stokes equations that are described in the following sections. 2.6.3 Attractors for 3-dimensional Navier–Stokes equations In three dimensions, because of possible non-uniqueness, some alternative ideas are needed to discuss attractors for Navier–Stokes equations. The radical approach adopted by Sell [92] was to work with the entire set of solutions as phase space, and then define the semigroup St as translation of the solution in time by t. In the paper [14] the above nonstandard ideas were used, for 3-dimensional Navier–Stokes equations, but starting from the internal semigroup Tτ on HN which gives solutions to the Galerkin equation (2.4) on HN (this was mentioned in 2.5.2) – which exists because of uniqueness at this level. For simplicity the force f was taken to be constant. Briefly, the idea is as follows. The techniques used to give an attractor in dimension 2 can be applied to the Galerkin approximation on HN in dimension 3 to give a ball B(ρ) of finite radius in HN that is a global Sattractor for the semigroup Tτ . Then define C ⊂ HN and A = w-stC (the weak standard part) as above, and we have a weakly compact subset of H, which is an attractor in some sense for the 3-dimensional Navier–Stokes equations. To be more precise, we can define two multi-valued semiflows

50

2 Stochastic Fluid Mechanics

St u = {w-st(Tt U ) : w-stU = u, |U | < ∞}, if u ∈ /A w-st{Tt U : st(U ) = u} ˆ St u = w-st{Tt U : w-st(U ) = u, U ∈ C} if u ∈ A A sample of the results obtained for these semiflows is as follows, where An = w-stCn : ˆ Theorem 2.12 (a) St1 +t2 u ⊆ St1 (St2 u), and similarly for S; (b) A ⊆ St A for all t; (c) A = Sˆt A for all t; (d) for each n and bounded set B ⊂ H there is t0 such that for t ≥ t0 Sˆt B ⊆ An ; (e) for each weakly open set O ⊇A and bounded set B ⊂ H there is t0 such that for t ≥ t0 Sˆt B ⊆ O. As is to be expected, the results are less pleasing than in dimension 2. Rather more satisfactory results were obtained for an approach that uses small initial pieces of trajectories of solutions as phase space – this is an idea that is intermediate between the above and that of Sell. For full details, and further variations on this theme, see [14].

2.7 Measure Attractors for Stochastic Navier–Stokes Equations As noted above, the notion of measure attractor is the most appropriate for the general time-homogeneous stochastic Navier–Stokes equations in two dimensions du = (ν∆u − u, ∇u + f (u))dt + g(u)dwt . (2.24) A measure attractor is, roughly speaking, an attractor for the Foias equation. The precise definition is given below. Measure attractors for the Navier–Stokes equations were first studied by B.Schmalfuß [89], [90] (see also [75], which deals with a general class of equations that does not include the Navier–Stokes equations). A measure attractor for statistical solutions will be a subset of the set M1 (H) of Borel probabilities on H, viewed as a subset of the space of Borel measures M(H) on H (which we equip with the topology of weak convergence). The paper [15], on which the present section is based, establishes results on existence of measure attractors for Navier–Stokes equations that are more general than those obtained by Schmalfuß.

2.7 Measure Attractors for Stochastic Navier–Stokes Equations

51

Here are brief details. First, the phase space (where all the activity takes place) is the subset of M1 (H) given by

X = {µ ∈ M1 (H) : |u|2 dµ(u) < ∞}.

Write B = {µ ∈ X : r

|u|2 dµ(u) ≤ r}.

These sets play the rôle of bounded sets in the discussion of attractors in the introduction above. The semigroup St on X is defined as the evolution of the (unique) statistical solution to the Navier–Stokes equations in two dimensions. That is, if (µt )t≥0 is the statistical solution with µt = µ then St µ = µt . An alternative description is that St µ = µt is defined by

ϑ(u)dµt (u) = Eϑ(v(t, u))dµ(u) H

H

for any continuous bounded ϑ, where v(t, u) is the solution to the stochastic Navier–Stokes equations with initial condition u. In other words, drawing on the uniqueness property we transport the initial measure along the trajectories of the solution, to give the semigroup St . Loeb space methods are used to give a simple proof of continuity of the function u → Eϑ(v(t, u)) (using the nonstandard construction of solution to the stochastic Navier–Stokes equations, as in Section 2.3). This is needed to show that for any r > 0 and t ≥ 0 the set St B r is compact. The next step is to show that there is ρ > 0 such that B ρ absorbs the sets B r . Having done this, a measure attractor is obtained as described in Section 2.6.2. Theorem 2.13 Write Tτ for ∗Sτ and define the internal set C ⊂ ∗X by C= Tτ ∗B ρ = Tτ ∗B ρ τ −infinite

n∈N τ ≥n

and let A = stC. Then A is a measure attractor for the stochastic Navier– Stokes equations (2.24). That is (a) A is weakly compact; (b) St A = A for all t; (c) for each open set O ⊇ A, and for each r > 0 St B r ⊆ O for all sufficiently large t.

52


Remarks The above is a slightly simplified account of the main results of the paper [15]. There are two natural weak topologies that may be considered on M1 (H) and the above holds for each – but with slightly differing requirements on the growth and continuity of the coefficients f, g.

2.8 Stochastic Attractors for Navier–Stokes Equations Consider the stochastic Navier–Stokes equations in 2-dimensions

t

t [−νAu(s) − B(u(s)) + f ]ds + g(u(s))dws u(t) = u0 + 0

(2.25)

0

where w is a 1-dimensional Wiener process and g has the special property that (g(u), v) = −(u, g(v)) (which is equivalent to (g(u), u) = 0 if g is linear). As noted in Section 2.3.1, this allows the construction of a stochastic flow of solutions to (2.25). This in turn makes it possible to investigate the possibility of a stochastic attractor for this particular case. We first explain what is meant by this. 2.8.1 Stochastic attractors For a stochastic system such as (2.25), the introduction of new noise as time evolves means that there cannot be an attractor in the sense considered so far for a deterministic system. The idea developed by Crauel & Flandoli [19] is to introduce a system of shifts of the noise in time, and then consider a stochastic attractor to be a set which, at time 0, attracts trajectories “starting at −∞” (compared to the usual idea of an attractor being a set “at time ∞” that attracts trajectories starting at time 0). This idea is spelled out below. In proving the existence of a stochastic attractor for the system (2.25) the nonstandard framework makes it particularly easy to consider −∞. In a general setting, let ϕ be a stochastic flow of solutions to a system such as (2.25), as discussed in section 2.3.1. That is, ϕ is a measurable function ϕ : [0, ∞) × H × Ω → H such that ϕ(·, ·, ω) is continuous for a.a. ω, and for each fixed initial condition u0 the process u(t, ω) = ϕ(t, u0 , ω) is a solution to (2.25). Suppose that in addition there is given a one parameter group θt : Ω → Ω of measure preserving maps, which should be thought of as a shift of the noise to the left by t. The notion of a semigroup of operators in the definition of a deterministic attractor, along with the notion of an attractor itself is now replaced by the following.

2.8 Stochastic Attractors for Navier–Stokes Equations

53

Definition 2.14 (i) The flow ϕ is a crude cocycle if for each s ∈ R there is a full set Ωs such that for all ω ∈ Ωs ϕ(s + t, x, ω) = ϕ(t, ϕ(s, x, ω), θs ω) holds for each x ∈ H and t ∈ R. (ii) A cocycle is perfect if Ωs does not depend on s. (iii) Given a perfect cocycle ϕ, a random global attractor is a random compact subset A(ω) of H such that for almost all ω ϕ(t, A(ω), ω) = A(θt ω),

t ≥ 0,

lim dist(ϕ(t, B, θ−t ω), A(ω)) = 0

t→∞

for each bounded set B ⊂ H. Note that the existence of a perfect cocycle is necessary for the possibility of having a stochastic attractor. Constructing a perfect cocycle is difficult, particularly for infinite dimensional systems that are truly stochastic (as compared to random dynamical systems where paths may be treated individually). The papers [2], [19] for example, discuss this in a slightly more general setting. Under the condition of invertibility of ϕ(t, ·, ω) in finite dimensional spaces, it is shown in [2] that a crude cocycle can be made perfect, i.e. it may be modified to a stochastically indistinguishable perfect one. In the setting described below we can improve on this using Loeb space machinery: we construct a perfect cocycle in the infinite dimensional space H for a flow ϕ that is not invertible. Although the equation considered is a special case of the stochastic Navier–Stokes equations, the technique is applicable to a more general class of stochastic flows. 2.8.2 Existence of a stochastic attractor for the Navier–Stokes equations Turning now to the particular system (2.25) above in dimension 2, with the special form of the noise g as described, we assume in addition that we have periodic boundary conditions. The main result of the paper [17] is the following. Theorem 2.15 (a) With appropriate growth and continuity conditions on g, there is an adapted Loeb space carrying a stochastic flow of solutions to (2.25) that is a perfect cocycle, and there is a stochastic attractor A(ω) (compact in the strong topology of H) for this system. (b) If g has the additional property that ((g(v), v)) = 0 for v ∈ V the stochastic attractor is bounded and weakly compact in V.

54


The proof of this result is quite long and complicated, so here we only sketch the basic ideas. Step 1 We begin with the Galerkin approximation (2.12) on HN with a one dimensional Wiener process Wτ defined for all time τ ∈ ∗(−∞, ∞) – so that the paths of this process all lie in ∗C0 (R) (∗continuous functions that are 0 at 0.) As underlying nonstandard probability space we take Ω = ∗C0 (R) with the Wiener measure. On Ω take the group of measure preserving shifts Θτ given by (Θτ ω)(σ) = ω(τ + σ) − ω(τ ). Step 2 On Ω construct a flow of solutions Φ(τ, V, ω) – which is a crude cocycle by the transfer of a result of [2] for finite dimensional stochastic systems. Step 3 From Φ define a superflow Ψ of solutions to (2.12) by Ψ (τ, s, V, ω) = Φ(τ − s, V, Θs ω). for all τ ≥ s, for s ∈ ∗Q, for all V ∈ HN , for all ω in a ∗full subset Ω1 of Ω. Ψ has the following properties (i) for each s the process U (τ, ω) = Ψ (τ, s, V, ω) is a solution to (2.12) on ∗ [s, ∞) with initial condition U (0) = V (ii) for all r ≥ s ∈ ∗Q, all τ ≥ r, all V ∈ HN and all ω ∈ Ω1 Ψ (τ, s, V, ω) = Ψ (τ, r, Ψ (r, s, V, ω), ω) Step 4 Show that there is a Loeb full subset of Ω1 such that (i) Ω1 is invariant under Θt for all finite t ∈ ∗Q; (ii) for all ω ∈ Ω1 the superflow Ψ (τ, s, V, ω) is S-continuous in (τ, s, V ) for finite τ and s and (strongly) nearstandard V ∈ HN . It is this step that is the crucial one, and requires a special adaptation of the Lindstrøm-Kolmogorov Continuity Theorem in [3]. In order to carry it out, it is necessary to establish some new strong regularity properties of the solutions to (2.12) in the particular case under consideration. Step 5 Show that there is a fixed finite radius ρ such that B(ρ) absorbs the paths of the superflow Ψ . This is where the special form of the noise is used – since it gives a deterministic equation for the evolution of the energy |U (τ )|2 of solutions to (2.12). Step 6 Define a stochastic S-attractor C(ω) ⊆ HN by

2.9 Attractors for 3-dimensional Stochastic Navier–Stokes Equations

C(ω) =

55

Ψ (0, −s, B(ρ), ω)

0<s∈∗Q, infinite

=

Ψ (0, −s, B(ρ), ω)

n∈N n≤s∈∗Q

=

Cn (ω)

n∈N

say. This has the following properties: (i) for finite s ∈ ∗Q, s ≥ 0 Φ(s, C(ω), ω) = C(Θs ω) (ii) for finite r and n there is finite t0 (n, r) ∈ Q such that Φ(t, BN (r), Θ−t ω) ⊂ Cn (ω) for all t with t0 (n, r) ≤ t ∈ ∗Q Step 7 Take standard parts! The natural probability space to take is the quotient of Ω1 under the equivalence ω ∼ ω iff ω = Θt ω for some hyperrational t ≈ 0. Then, abusing notation somewhat, on this quotient space define ϕ(◦τ, ◦V, ω) = ◦Φ(τ, V, ω) for nearstandard τ, V . Then this is a perfect cocycle, and the random set A(ω) = ◦C(ω) is a stochastic attractor, as required by Theorem 2.15.

2.9 Attractors for 3-dimensional Stochastic Navier–Stokes Equations The difficulties encountered in considering attractors for 3-dimensional Navier– Stokes equations noted above (Section 2.6.3) are compounded when considering the corresponding stochastic equations. Recently Keisler and the author [34] have used Loeb space methods to develop an extension of Sell’s approach [92] (see Section 2.6.3 above) to these equations. The idea is to take as phase space the set X of all stochastic processes on a given Loeb space that are solutions to the stochastic Navier–Stokes equations satisfying some natural conditions that can be derived heuristically. The equations considered take the form

t

t [−νAu(s) − B(u(s)) + f (u(s)]ds + g(u(s))dws (2.26) u(t) = u0 + 0

0

56


These are similar to the equations discussed in Sections 2.7 and 2.8 but in dimension 3, with a general time-homogeneous multiplicative noise g and taking w to be a 1-dimensional Wiener process. (The only restrictions on f, g are that they are continuous and obey a certain growth condition which is certainly satisfied if they are bounded.) For the above equations, Sell’s idea was used by Flandoli & Schmalfuss in the paper [51] for the Navier–Stokes equations with a special form of multiplicative noise g which allowed essentially a pathwise solution. At the same time the notion of stochastic attractor had to be the one developed by Crauel & Flandoli [19], as in the previous Section 2.8 – involving running time backwards to −∞. In a later paper [52] Flandoli & Schmalfuss consider in the same framework 3D-Navier–Stokes equations with an irregular forcing term in place of gdw, having no feedback. The idea in the paper [34], outlined here, is to use Sell’s approach at the level of processes rather than paths. In this way the idea of an attractor is formulated in the conventional sense, examining the long term behaviour of solutions as t → ∞. To do this, it is necessary to have a single underlying probability space that is rich enough to carry a supply of solutions to the 3D stochastic Navier–Stokes equations that is sufficient for the concepts to make sense. For this we need the adapted Loeb space described later. But first we describe the basic idea in a little more detail. Suppose we are given a 1-dimensional Wiener process wt on a filtered6 probability space Ω = (Ω, F, (Ft )t≥0 , P ). It is necessary to assume that the space Ω is equipped with some additional structure – namely a family of measure preserving maps θt : Ω → Ω for t ≥ 0 with the following properties: (i) θ0 =identity and θt ◦ θs = θt+s ; (ii) θt Fs = Ft+s for all s, t ≥ 0; (iii) w(t + s, θt ω) − w(t, θt ω) = w(s, ω) for all s ≥ 0. Note that the property (iii) tells us that for a fixed t the increments of the process w(t + s, θt ω) are the same as those of the process w(s, ω). Thus θt can be thought of as a shift of the noise to the right by t. (This is in contrast to the shifts discussed in Section 2.8, which by convention are essentially shifts to the left. Here it is simply more convenient and natural to consider shifts to the right.) The family (θt ) allows the following definition of a semiflow Sr of stochastic processes. Definition 2.16 (Semiflow of Processes) Suppose that u = u(t, ω) is a stochastic process defined for t > 0. Then for any r ≥ 0 the process v = Sr u is defined by v(t, ω) = u(r + t, θr ω) 6

For the reader unfamiliar with stochastic analysis, the σ-algebras Ft are an increasing sequence of sub-algebras of F that represent the information available at each time t.


57

Note that this ensures that if u(t, ·) is Ft -measurable, then so is v(t, ·).7 Suppose now that X is a class of solutions to the stochastic Navier–Stokes equations (2.26) on Ω (so that u ∈ X will be a stochastic process) with the property that St X ⊆ X for all t ≥ 0. We can make a preliminary definition of an attractor for the semiflow St as follows. Definition 2.17 (Provisional) An attractor for the semiflow St on X is a set A ⊆ X such that (i) St A = A for all t ≥ 0; (ii) A attracts bounded subsets of X, in some sense; that is, if B is a bounded subset of X then eventually St B gets “close” to A. (iii) A is “compact” in some sense to be made precise. Since existence results for the stochastic Navier–Stokes equations require a rather large probability space, a fortiori any single space Ω carrying an entire class of solutions X with the required closure properties is likely to be too big to allow a compact attractor in the usual sense. A suitable notion however is that of neocompactness, to be explained later. We can now state (modulo the definitions being made precise) the main theorem of the paper [34]. Theorem 2.18 On a suitable space Ω (which carries solutions to the stochastic Navier–Stokes equations for all L2 (F0 )-measurable initial conditions), there is a neocompact attractor for the class of solutions X described below. Here are a few more details showing how this result is established and fulfilling on the promises made earlier to explain the undefined notions. For Ω take Ω = ∗C0 (R), the internal space of ∗continuous functions ω : ∗ R → ∗R with ω(0) = 0, and let Q be the internal ∗Wiener measure on Ω. Thus the canonical process W (t, ω) = ω(t) is a two-sided ∗Wiener process under Q. This gives the internal filtered space Ω = (Ω, A, (Aτ )τ ≥0 , Q) where Aτ = ∗σ({W (τ ) : τ ≤ τ }) and A = τ ∈∗R Aτ . A family of internal measure preserving maps Θτ : Ω → Ω is defined for τ ≥ 0 by (Θτ (ω))(σ) = ω(σ − τ ) − ω(−τ ). That is, Θτ is a shift of the path ω to the right by τ . (Again note the contrast with the shifts Θτ defined in Section 2.8.2.) 7

Hence, if the process u is adapted then so is v.

58


Now let P = QL be the Loeb measure obtained from Q with the corresponding Loeb σ-algebra F = L(A), giving the Loeb probability space Ω = (Ω, L(A), QL ) = (Ω, F, P ). The filtration (Ft )t≥0 is obtained from (Aτ )τ ≥0 in a routine way (see [61] or [13] for example). The family θt is simply the restriction of Θτ to standard times. The appropriate class X of solutions to (2.26) is described by a number of energy inequalities which can be derived heuristically from the equations themselves using an informal version of Itˆ o’s lemma. These take the following form. For a.a. t0 > 0 and all t1 ≥ t0 E(|u(t1 )|2 ) ≤ E(|u(t0 )|2 ) exp(−k(t1 − t0 )) + k

(2.27)

and

E supt0 ≤s≤t1 |u(s)|2 + ν

t1

u(s)2 ds

≤ αE(|u(t0 )|2 ) + β(t1 − t0 ) + γ

t0

(2.28) and various generalisations involving powers |u(t)|q for 1 ≤ q ≤ 2. The paths of solutions in X are also required to have certain natural regularity properties – such as weak continuity and various forms of Lp boundedness. Denoting by 8 Xk the solutions in X that are bounded in the appropriate norm by k, we have that X = k∈N Xk . The notion of neocompactness is that developed by Keisler & Fajardo in a series of papers (see [49, 50, 62]), and is a weakening of the standard notion of compactness that is appropriate for highly non-separable metric spaces. Although the general definition of neocompact requires considerable elaboration, in the context of a metric space obtained from an internal space, a neocompact set is one that is obtained as the standard part of a set C = n∈N Cn with each Cn internal and with all members of the set C nearstandard. The power of the notion of neocompactness derives from the connection with a related notion of neocontinuity which is stronger than the classical notion of continuity. Neocompactness and neocontinuity can be used together in much the same way that compactness and continuity are often combined to good effect. The papers noted above develop a powerful approach to existence theorems in analysis and probability theory, using the fact that many functions involved are actually neocontinuous, and frequently sets and spaces constructed are neocompact. Keisler and the author used these ideas in [33] to provide an alternative proof of existence of solutions to the general 3dimensional stochastic Navier–Stokes equations together with some new optimality results for such solutions. With almost all notions now explained, here is an outline of the procedure involved in the proof of Theorem 2.18. 8

∞ 1 The norm is given by |u| = E 0 |u(t, ω)|2 exp(−t)dt 2 .


59

First, using the method of construction described in Section 2.3, show that the set X of solutions is non-empty. For the required energy inequalities, use the internal Itˆ o lemma, which is valid because internally we are essentially working in ∗RN . Next, formulate the idea of an internal approximate solution to the equations (2.26). The approximation extends to every aspect of being a solution, so that in addition to the equation itself, the energy inequalities are also only approximately satisfied - that is, to within an infinitesimal. The class of all approximate solutions is denoted X , and a crucial result shows that X = ◦X - where ◦U denotes the solution process obtained from an internal process U as in Section 2.3. Moreover, the set X = k∈N Xk where Xk denotes those approximate solutions that are bounded by k, and Xk = ◦Xk for each k. The natural internal counterpart of the semiflow St is denoted Tτ and it is easy to see from the definition that Tτ X ⊂ X for finite τ . Having shown that there is an S-absorbing set B ⊂ X , continue by defining the set Cn C= n∈N

where for each finite n Cn =

Tτ B.

τ ≥n

τ finite

Now show that C is an S-attractor ; that is Tτ C = C

(2.29)

for all finite τ , and for every finite n and k there is finite t(n, k) such that Tτ Xk ⊆ Cn for all finite τ ≥ t(n, k). Finally, the required attractor is defined to be A = ◦C having shown that all members of C are nearstandard. The neocompactness of A follows by showing that C is a countable intersection of internal sets, and the invariance property St A = A is a direct consequence of (2.29). The ◦ sense in which A attracts bounded sets is as follows. Putting An = Cn we have A = n∈N An . Then for every finite n and k there is finite t(n, k) such that St Xk ⊂ An for all t ≥ t(n, k). The above is necessarily an abbreviated account of the results and methods of [34] – and it is also slightly simplified in parts. Nevertheless it outlines

60


the main ideas. The need (or so it seems) for the use of a Loeb space is that only such a space would be rich enough to carry a class of solutions to the equations (2.26) that is closed under the semiflow St and contains the “limit” processes that must appear in an attractor. As a consequence of the main result above, we can extend Sell’s approach to measure attractors for the 3-dimensional stochastic Navier–Stokes equations – simply by taking the laws of all the processes involved. For details consult [34].

3. Stochastic Calculus of Variations

3.1 Introduction This Lecture is a story in four acts. The common thread is provided by some intuitions about Brownian motion that can be made precise using the technology of nonstandard analysis, particularly Loeb measures. Recall that a one-dimensional Brownian motion (also called a Wiener process) on the interval [0, 1] is a stochastic process b : [0, 1] × Ω →R (where Ω is some probability space) with the following properties. 1. b(0, ω) = 0 2. For a.a. ω the path b(·, ω) is continuous; 3. The increments b(t, ·) − b(s, ·) are Gaussian distributed1 with mean 0 and variance |t − s|. 4. Disjoint increments are independent (for example if s1 < t1 ≤ s2 < t2 then the random variables b(ti , ·) − b(si , ·) (i = 1, 2) are independent). Wiener measure W on C = C0 [0, 1] (the set of continuous functions that are zero at zero) is the probability measure induced by Brownian motion. That is, for Borel A ⊆ C W (A) = P (b(·, ω) ∈ A). Recall Anderson’s famous construction of Brownian motion that we described in Section 1.3.3: the process b is obtained√as the standard part of an infinitesimal random walk B(t) with step sizes ± ∆t, where ∆t ≈ 0 (in fact we take ∆t = N −1 for a fixed, but arbitrary, infinite natural number N ). Since ∆B(t)2 = ∆t this construction captures the stochastic analyst’s rule of thumb (or intuition) that “db2t = dt” (Note that we use interchangeably the notation b(t) = bt ). Another intuitive formula – this time for Wiener measure – that is often employed by physicists, is Donsker’s “flat integral”

1 W (A) = κ exp − 12 0 x˙ 2t dt dλ(x) (3.1) 1

A

We will write N (µ, σ 2 ) for the Gaussian (normal) distribution with mean µ and variance σ 2 .

N.J. Cutland: LNM 1751, pp. 61–84, 2000. c Springer-Verlag Berlin Heidelberg 2000

62

3 Stochastic Calculus of Variations

for the Wiener measure W (A) of a Borel subset A ⊆ C. Here λ is ‘Lebesgue’ measure on C (i.e. a hypothetical translation invariant measure) and κ is a constant; x˙ denotes the derivative of a path in C. This formula fails to make sense for many reasons. First, there is no ‘Lebesgue’ measure on C (or C[0,1]). Second, the only way to make sense of

1 the term 0 x˙ 2t dt, which is the action (i.e. the integral of the kinetic energy along a path) is to define it to be infinite unless x is absolutely continuous with x˙ ∈ L2 [0, 1]. This means that the integrand in (3.1) is almost surely zero; thus, finally, the constant κ would need to be infinite to have W (A) finite. Donsker’s flat integral is the precursor of many other intuitive flat integral formulae that occur in physics, and can be used to give heuristic arguments for correct results – such as the Cameron-Martin formula for the translation of Wiener measure (see Theorem 3.2 below). It is based on the intuitive idea that the increments dbt of Brownian motion are Gaussian distributed with mean 0 and variance dt. Thus for each fixed t the random variable xt = dbt has “density” 1

(2πdt)− 2 exp(− 12

x 2 1 x2t t ) = (2πdt)− 2 exp(− 12 dt) dt dt

against Lebesgue measure. The informal product of these terms over t ∈ [0, 1] then gives (3.1). In Section 3.2 (the first Act) we will show how this can be made precise by a modification of Anderson’s construction, taking ∆B(t) to be N (0, ∆t) distributed. Although we lose the fact that ∆B(t)2 = ∆t we do have E(∆B(t)2 ) = ∆t which is quite sufficient. Act 2 (that is, Section 3.3) begins with another intuition about Wiener measure – this time due to Wiener himself [98]. He originally thought in terms of the differential b˙ t = dbt /dt of a path of Brownian motion, and informally ˙ thus, using db2 = dt: calculated its L2 [0, 1] norm b t

˙ 2= b 0

1

b˙ 2t dt =

0

1

(db2t /dt2 )dt

=

1

(1/dt)dt = 1/dt = ∞

0

˙ = √∞, and b˙ lives in the space S ∞ (√∞), which Wiener called differSo b ential space. He then thought of Wiener measure as the uniform probability on differential space. The apparatus of nonstandard analysis allows us to make sense of this; the appropriate space that√gives precision to the above ideas turns out to be the infinite sphere S N −1 ( N ) in ∗RN for an infinite integer N . For convenience only2 , in discussing this in Section 3.3 we rescale and work with the unit sphere S N −1 (1) which we called the Wiener sphere in [41]. 2

because it is more natural work with the Euclidean norm in ∗RN than to use to 2 2 2 the L norm X = t Xt ∆t

3.1 Introduction

63

In this framework it is shown that Wiener measure “is” the uniform measure on the Wiener sphere, and a pleasant geometrical derivation of the Cameron-Martin formula emerges. There is a discussion of differential space (which we can think of as the Wiener sphere) and related issues in the brilliant little paper [72]. In the third Act (Section 3.4) we pick up a remark of David Williams concerning the infinite dimensional Ornstein–Uhlenbeck process v, which plays an important rˆ ole in the stochastic calculus of variations (the ‘Malliavin Calculus’). This process is fundamental to infinite dimensional stochastic calculus, and it has been suggested that v should be regarded as the infinitedimensional counterpart of Brownian motion. Williams [99] remarks that the “correct” way to think of the infinite dimensional Ornstein–Uhlenbeck √ process is as “Brownian motion on the infinite dimensional sphere S ∞ ∞ ”. Given the Wiener sphere, it is possible to make sense of this – by √ considering ∗ Brownian motion on S N −1 (1), which is a scaled version of S N −1 ( N ) (again with N infinite). This, and some consequences, is spelled out in Section 3.4. The final part of the story (Act 4, Section 3.5) returns to the idea that the increments bt of Brownian motion are N (0, ∆t), and discusses the so called Malliavin calculus on the space L2 (W ) of functions ϕ : C → R that are L2 with respect to the Wiener measure. Informally, we can think of a sample point b ∈ C as completely determined by the ‘vector’ of increments x = (dbt )t∈[0,1] . Writing xt = dbt and x = (xt )t∈[0,1] we can therefore think of ϕ ∈ L2 (W ) as ϕ = ϕ(x) = ϕ((xt )t∈[0,1] ). At the heart of the Malliavin calculus is the idea of differentiation with respect to the variables xt . Thus, intuitively, the gradient or derivation operator D is given by ∂ϕ Dϕ(b, t) = (b). ∂xt There is a similar intuitive definition of the other basic operators of this calculus. In Section 3.5 we show how the construction of Brownian motion from infinitesimal increments ∆B(t) that are N (0, t) distributed allows the above ideas to be made precise using the transfer of classical calculus on ∗RN , and the basic results of the Malliavin calculus are derived as simple applications. 3.1.1 Notation Throughout this Lecture we will work with the following notation. Fix N ∈ ∗N \ N and let ∆t = N −1 . As in Section 1.3.1 we take the hyperfinite time line T = {0, ∆t, 2∆t, 3∆t, . . . , 1 − ∆t}. On occasions we need to include the end point, so write

64

3 Stochastic Calculus of Variations

T = T ∪ {1}. We use sanserif letters s, t, u as variables from T. For any internal function F : T → ∗R write ∆F (t) = F (t + ∆) − F (t) and

∆F (t) F˙ (t) = ∆t

so that ∆F, F˙ ∈ ∗RT . Write C = {Y ∈ ∗RT : Y (0) = 0}, which we identify with the space of polygonal paths obtained by joining the points (t, Y (t)). Then C is the internal path space that we will be working with throughout. The standard part mapping st(Y ) = ◦Y ∈ C is defined for S-continuous paths in C (see Section 1.2.4). A vector X in the internal space ∗RT will usually be regarded as a vector of increments of a path Y = ΣX ∈ C, where Σ is the following mapping Σ : ∗RT → C (ΣX)(0) = 0 (ΣX)(t) = X(s)

if t > 0.

s 0 the price process is defined by Sn (t + ∆n , ω) = Sn (t, ω)(1 + µ∆n + σ∆B(t, ω))

(4.1)

or ∆Sn (t, ω) = Sn (t, ω)(µ∆n + σ∆B(t, ω)) (4.2) √ giving Sn : Tn × Ωn → R>0 provided that 1 + µ∆n ± σ ∆n > 0, which is assumed. In this definition the parameters µ and σ of the price process are the drift and volatility repectively. The drift µ gives the general trend of the price and the volatilty σ > 0 is a measure of the random fluctuation.3 The other component of the CRR model is the price of a bond, which is deemed to be a safe investment at a fixed interest rate r. By discounting to current prices, and by changing the unit of currency, we may without loss of generality take r = 0 and the initial price of the bond as 1. Within the CRR model a variety of financial activities can be represented. First, consider an investor who at any given time t ∈ Tn holds a portfolio consisting of a number of units each of the stock and the bond. Denote this as follows: Θb (t) = the number of units of the bond held at time t Θs (t) = the number of units of the stock held at time t Θ(t) = (Θb (t), Θs (t)) The investor may well, at time t, take account of the performance of the stock up to the time t, and so Θ can depend on ω up to time t. The nonanticipating4 function Θ : Tn × Ωn → R2 is the trading strategy of the investor - that is, it is a function that describes the changes he makes to his portfolio over time. The idea is that he holds the portfolio Θ(t) on the time interval [t, t + ∆n ) and at time t+∆n adjusts it to become Θ(t+∆n ). The value of this portfolio at time t is clearly given by the random process V (t, ω) = Θb (t, ω) + Θs (t, ω)Sn (t, ω).

(4.3)

If the portfolio is not changed at time t+∆n then at that time the investment will be worth Θb (t, ω) + Θs (t, ω)Sn (t + ∆n , ω) = V − (t + ∆n , ω) 3

4

(4.4)

If σ = 0 the price would behave like a fixed interest rate investment giving Sn (t) = Sn (0)(1 + µ∆n )nt , whereas if µ = 0 the price fluctuates but the mean remains at the original price; in fact the average change in price at each time step is zero. Recall that in this context this means that at time t the dependence in ω is only on ω(s) for s ≤ t.

88

4 Mathematical Finance Theory

say, and a gain5 of Θs (t, ω)(Sn (t + ∆n , ω) − Sn (t, ω)) = Θs (t, ω)∆Sn (t, ω) will have been made. The gains process for the trading strategy Θ is thus defined by G(t, ω) = Θs (s, ω)∆Sn (s, ω). s 0 for some ω. Informally an arbitrage opportunity is known as a free lunch. It turns out that in the models Mn there are no free lunches – and the models are said to be viable.

4.3 Options and Contingent Claims Financial options constitute one of the main kinds of derivative that are currently (and increasingly) being traded in financial markets, with levels of activity exceeding the trading of the underlying stocks on which options are based. 5 6

This could of course be negative - in which case it would be a loss. In this simple financial model it is assumed that the cost of making changes is zero - i.e. there are no transaction costs - which is of course unrealistic in practice.

4.3 Options and Contingent Claims

89

A call option on a stock7 is a right (in the form of a contract that has been purchased by the owner) to buy a certain quantity of shares at a fixed price K per share (written into the contract now) at some future date T . The owner of the option is not obliged to buy those shares, so he may or may not exercise the option when the time T comes. Whether or not he does will depend on the price of the shares at that time T , and how it compares with the strike price K. The fundamental question of option pricing is to determine the fair price of an option. Clearly the owner of the option cannot lose at time T : he will buy (and make a gain) if the market price of the shares at that time is more than the predetermined purchase price K guaranteed to him in his contract; otherwise he will not exercise his option. Plainly, an option of this kind is a valuable right, and so there will be a price to pay for it now. A simple example is the European call option, which we can describe in the above CRR model taking T = 1, and thinking of the present as time 0. A European call option with exercise time 1 and strike price K gives its owner the right to buy one unit of stock at time 1 for the price K. At time 1 it makes sense to exercise the option if and only if Sn (1, ω) ≥ K. This gives a guaranteed immediate profit of Sn (1, ω) − K. In the other situation, if Sn (1, ω) < K, to purchase the stock at price K would be absurd – if the stock were desired it could be obtained more cheaply directly from the market. To summarise, the owner of the European call option with strike price K and exercise time T = 1 owns an asset that will be worth (Sn (1, ω) − K)+ at time 1. A slightly more sophisticated option is an American call option, which gives the owner the right to buy at any time of his choosing, up to and including a set expiry time. There are many other so called ‘exotic’ options that are more complex and require more sophistication on the part of the trader. Abstractly an option is an example of a contingent claim -- which is, in the CRR model, simply a non-negative random variable C(ω). This represents the value at the time t = 1 of an asset that can be purchased now. In the case of the European call, we have C(ω) = (Sn (1, ω) − K)+ and in the case of the American option there is a more complicated formula involving stopping times. One of the first goals of the theory is to discover (if possible) the fair price at which such a claim should be traded now (i.e. at time t = 0). In the next section we will outline the general approach to pricing a claim C(ω) in the CRR model, and mention briefly the price so obtained for the European call. 7

The complementary option – the right to sell a quantity of stock - is called a put option. This is not, of course, the same as selling a call option.

90


4.3.1 Pricing a claim The idea for determining the ‘fair’ price Π(C) of an option C(ω) is the following. Suppose I (the person thinking of buying the option now, at time t = 0) knew that there was a self-financing trading strategy Θ such that at time t = 1 its value V (1, ω) was exactly C(ω) for all possible ω. Then in order to embark on that strategy now, I would have to invest a sum of V (0) = Θb (0) + Θs (0)s0 to buy the portfolio of bonds and stocks indicated by the strategy (note that at time t = 0 a strategy is non-random). Thus, assuming that there are no arbitrage opportunities, V (0) would be a fair price to buy (or sell) the option C(ω). This is because buyer or seller could equally well, for that price, purchase a random asset which would be worth C(ω) at time t = 1 by investing in stocks and bonds using the strategy Θ. This argument for fixing the fair price hinges crucially on the existence of a hedging or replicating strategy Θ, so the following result is fundamental. Theorem 4.2 Let C(ω) ≥ 0 be a contingent claim in the CRR model Mn . Then there is a unique self-financing trading strategy Θ such that C(ω) = V (1, ω)

(4.5)

for all ω. Proof (Sketch) This is actually simple linear algebra rather than probability theory, and is best illustrated by the case n = 1 and thinking about how it generalises. In the case n = 1 we have ∆n = 1 and C1 = Ω1 consists of two points (paths) ω = X + and ω = X − say for which X + (1) = 1 and X − (1) = −1. The claim C is given by its two values C(X + ) and C(X − ) and the desired trading strategy Θ has two unknown values Θb (0) and Θs (0). The equation V (1, ω) = C(ω) together with the requirement that Θ be self-financing (so that V (1, ω) = V − (1, ω)) gives the following equations for Θ C(X + ) = Θb (0) + Θs (0)S1 (1, X + ) C(X − ) = Θb (0) + Θs (0)S1 (1, X − ) which have a unique solution for Θ. The general case n > 1 is similar.

Remark There are alternative slightly more sophisticated ways to formulate the proof of the above result in the general case. One is to solve for the 2n values8 V (0) and Θs (t, ω) for t ∈ Tn using the 2n linear equations Θs (t, ω)∆Sn (t, ω) (4.6) C(ω) = V (1, ω) = V (0) + t 0 and (long-term) drift µ ∈ R. By Itˆ o’s formula the solution on [0, 1] is given by 11

We could also include the gains process G(t, ω) = V (t, ω) − V (0) and make this a quadruple, provided V (0) is specified along with G. See the discussion in [37].

4.4 The Black-Scholes Model

93

st = s0 exp(σbt + (µ − 12 σ 2 )t). A process of this kind is a geometric Brownian motion (with drift12 ). The model also has a bond which grows at a fixed interest rate r. As in the discrete case, we may assume without loss of generality that r = 0 and, moreover, that the fixed price of a unit of the bond is 1. The continuous time counterpart of the first part of Theorem 4.4 is: Theorem 4.6 There is a unique probability measure Q on Ω, called the equivalent martingale measure, that makes the price process s(t, ω) a martingale. In this model a trading strategy θ = (θs , θb ) is a pair of adapted processes θ , θb : [0, 1] × Ω → R giving the portfolio of number of units of stock and bond respectively held at time t. The value of this holding at time t is then s

v(t, ω) = θb (t, ω) + θs (t, ω)s(t, ω). The strategy θ is self-financing if for all t we have

t θs (t, ω)ds(t, ω), v(t, ω) = v(0) + 0

which is the counterpart of Definition 4.1(iii). It is a consequence of Theorem 4.6 that, as with the CRR models, there are no arbitrage opportunities in the BS model. A contingent claim in this model is an L2 (Q) random variable C(ω) ≥ 0. The fair price π(c) of such a claim is defined to be π(c) = v(0) where v is the value process for any self-financing strategy θ for which v(1, ω) = c(ω). The validity of this definition follows from the following, which is a disguised version of the martingale representation theorem for the Itˆ o calculus. (See Theorem 3.25, Section 3.5.3.) Theorem 4.7 Let c(ω) ≥ 0 be a contingent claim in the Black-Scholes pricing model. Then there is a unique self-financing trading strategy θ such that c(ω) = v(1, ω)

(4.8)

for a.a. ω. The continuous time counterpart of the second part of Theorem 4.4 is: 12

A (pure) geometric Brownian motion would have µ = 0.

94


Theorem 4.8 For any claim c(ω) ∈ L2 (Q) the fair price π(c) = v(0) derived from the unique replicating strategy θ is also given by π(c) = EQ (c(·)) The famous Black-Scholes formula gives explicitly the fair price π(c) of a European call c(ω) = (s(1, ω) − K)+ according to the preceding theory. Black and Scholes argued that the unique value of a European call option with strike K and expiry at time t = 1 is given by 13 s0 s0 π(c) = s0 ψ σ −1 log( ) + 12 σ − Kψ σ −1 log( ) − 21 σ K K where ψ denotes the normal cumulative density function.

(4.9)

4.5 The Black-Scholes Model and Hyperfinite CRR Models The first connection between the Black-Scholes (BS) model of the previous section and the CRR models Mn is that the former is in some sense the limit of the latter as n → ∞. One way to make this precise is to consider a hyperfinite (infinite) CRR model MN and “take standard parts”. Here are the details. Fix an infinite natural number N and consider the CRR model MN based on the internal probability space Ω N = (CN , AN , WN ). Recall first from Lecture 1 (Section 1.3.3) Anderson’s construction of Wiener measure/Brownian motion via the standard part mapping st : CN → C defined on the set of S-continuous paths in CN . On the Loeb space Ω = (CN , L(AN ), (WN )L ) Brownian motion is given by b(t, ω) = ◦ω(t). The space Ω carries the internal price process SN for the risky stock, given by the formula (4.1). Then we have, writing P = PN for the Loeb measure (WN )L : Theorem 4.9 With respect to P , almost all paths of SN are S-continuous and ◦ SN (t, ω) = st = s0 exp σbt + (µ − 12 σ 2 )t which is the Black-Scholes price model with s0 = ◦SN (0). 13

The usual Black-Scholes formula is a little more complex since it considers any future time T as expiry date, and also allows the bond interest rate to be r > 0. In full generality it gives the fair price at any time t < T in the future that is prior to the expiry date T .

4.5 The Black-Scholes Model and Hyperfinite CRR Models

95

4.5.1 The Black-Scholes formula One of the main results of [35] shows that the Black-Scholes formula is the standard part of the formula for the price of a European option (4.7) in the CRR model MN . Theorem 4.10 Let C(ω) = (SN (1, ω) − K)+ and c(ω) = (s(1, ω) − K)+ be European calls in the CRR and Black-Scholes market models respectively. Then π(c) = ◦Π(C) where the option prices π(c) and Π(C) are given by (4.9) and (4.7). The proof, which can be found in [35], is a straightforward application of the Central Limit Theorem, which links the complementary binomial function Φ with the normal cdf ψ. These results show that in some sense “the Black-Scholes market model contains an built-in version of the CRR model” – which is the economists’ intuition. Further elaboration of this point is provided in the following sections. 4.5.2 General claims The link between the Black-Scholes formula and the corresponding CRR formula is a particular case of a general relationship between entities in the BS and hyperfinite CRR models. Recall first the idea of liftings (see Section 1.3.4 of Lecture 1). In the present context we are concerned with two-legged liftings as follows. Definition 4.11 (i) Let f : C → R be an L2 (Q) random variable and F : CN → ∗R. Then F is an SL2 lifting of f if F is SL2 (with respect to QN ) and for a.a. ω ∈ CN F (ω) ≈ f (◦ω) (ii) Let g : [0, 1]×C → R be an adapted L2 function and G : TN ×CN → ∗R be nonanticipating. Then G is an SL2 lifting of g if G is SL2 (QN ) and for a.a. (t, ω) ∈ TN × CN G(t, ω) ≈ g(◦t, ◦ω) (iii) Let h : [0, 1] × C → R be an adapted L2 function with h(·, ω) a.s. continuous and suppose that H : TN ×CN → ∗R is nonanticipating. Then H is an S-continuous SL2 lifting of h if H is an SL2 (QN ) lifting and for a.a. ω ∈ C N the function H(·, ω) is S-continuous, so in fact for a.a. ω ∈ CN H(t, ω) ≈ h(◦t, ◦ω) for all t ∈ TN .

96


The following result is implicit in [35] and explicit in [37]. Theorem 4.12 Let (C, Θ, V ) be a triple in the hyperfinite CRR model Ω N consisting of a claim C(ω), the self-financing strategy Θ that generates it, and the corresponding value process V (t, ω) as in Section 4.3.1. Suppose further that (c, θ, v) is a triple of the same kind in the BS model, with the selffinancing strategy that generates the claim c and the corresponding value process v as given in Section 4.4. Then the following are equivalent: (a) C is an SL2 lifting of c; (b) Θ is an SL2 nonanticipating lifting of θ; (c) V is an S-continuous SL2 lifting of v.

4.6 Convergence of Market Models When considering the issue of convergence of entities (for example claims, value processes, trading strategies) in the discrete CRR market models Mn to similar entities in the BS model M it must be be recognised that the underlying probability spaces Ω n are changing as well as the entities themselves. In this sort of situation the conventional mode of convergence is weak convergence, but it is easy to see that this does not preserve the intrinsic structure and relationships such as that between a claim, the generating strategy and the corresponding value process that was described above14 . The nonstandard approach suggests an alternative mode of convergence, picking up the fundamental idea from basic calculus that a real sequence sn converges to s ∈ R if and only if ∗sN ≈ s for all infinite N . So we can tentatively make the following definitions. Definition 4.13 Suppose that (Cn )n∈N , (Θn )n∈N and (Vn )n∈N are sequences of claims, self-financing strategies and value processes in the CRR models Mn , and c is a claim, θ a self-financing strategy and v a value process in the BS model on Ω. Then (i) Cn D2 -converges to c if CN is an SL2 (QN ) lifting of c for all infinite N; (ii) Θn D2 -converges to θ if ΘN is a nonanticipating SL2 (QN ) lifting of θ for all infinite N ; (iii) Vn D2 -converges to v if VN is an S-continuous SL2 (QN ) lifting of v for all infinite N . 14

It is easy to construct an example of claims Cn in Mn that converge weakly whose generating trading strategies do not converge weakly.

4.6 Convergence of Market Models

97

D2

When convenient we will write −→ to denote D2 -convergence. The reason for the choice of the name D2 -convergence will be clear a little later on. First there are a number of issues to discuss. Since this notion depends on nonstandard analysis, it could conceivably be a notion whose meaning depends on the particular nonstandard universe being used. A related question is whether there is a standard convergence notion that is equivalent to the above definitions; if so then it would be independent of the nonstandard model. There is next the question of finding D2 -convergent sequences for a given entity in the BS model, and the relationship with other modes of convergence. Finally, there is the question as to whether D2 -convergence is either useful or natural. Taking the last point first, as motivation for the effort involved in answering the others, we have: Theorem 4.14 Suppose that (Cn , Θn , Vn )n∈N is a sequence of claims, together with their generating strategies and value processes, in CRR models Mn , and a similar triple (c, θ, v) in the BS model M. Then the following are equivalent: D2

(a) Cn −→ c; D2

(b) Θn −→ θ; D2

(c) Vn −→ v. Proof This is an immediate application of Theorem 4.12.

This result can be described by saying that D2 -convergence preserves the basic constructions of option pricing – unlike weak convergence – and this suggests that it an interesting notion. In answer to the other issues raised, although the nonstandard characterisation is perhaps the most natural one for D2 -convergence (given the Loeb space machinery), there are two equivalent standard definitions, so this is a genuine down-to-earth notion of the theory. For simplicity we will restrict our remarks here to the case of claims. It is necessary to recall the equivalent martingale measures Qn and Q for the CRR and BS models. Then the first standard characterisation of D2 -convergence is given by the following result. Theorem 4.15 Let (Cn ) be a sequence of contingent claims in the CRR models Mn and let c ∈ L2 (Q). The following are equivalent: D2

(a) Cn −→ c; (b) (Bn , Cn (Bn )) → (b, c(b)) weakly15 and EQn (Cn2 ) → EQ (c2 ). For details of the proof see [37], where a similar characterisation of D2 convergence of trading strategies and value processes may be found. 15

By this we mean convergence of the distribution of (Bn , Cn (Bn )) in Cn ×R under Qn to that of (b, c(b)) in C × R under Q.

98


4.7 Discretisation Schemes The second standard characterisation of D2 -convergence uses the idea of discretisation schemes to relate the continuous space C to the discrete approximations Cn . Here is the definition. Definition 4.16 A family (dn ) of measurable maps C → Cn is an adapted Q-discretisation scheme if for each n: (i) dn is adapted (i.e. (dn (b))(t) depends only on the values (bs )s≤t ); (ii) dn is measure-preserving (with respect to Q and Qn ); (iii) dn (b) → b in Q-probability, i.e. ∀ > 0 : Q(|dn (b) − b| < ) → 1 as n → ∞. (Here |.| denotes the sup norm in C.) The existence of such a scheme can be established by modifying a construction given by Frank Knight in 1962, using polygonal paths approximating Brownian motion; the proof is a little technical, so we refer the reader to [37] for details. The results below indicate that for the present purposes any two discretisation schemes are equivalent. The rôle of such schemes in characterising D2 -convergence is as follows. Theorem 4.17 Let (Cn ) be a sequence of contingent claims in the CRR models Mn and let c ∈ L2 (Q). Suppose that an adapted Q-discretisation scheme (dn ) is given. The following are equivalent: (a) Cn is D2 -convergent to c; (b) Cn (dn (·)) converges to c(·) in L2 (Q)-norm. Remark It is from this characterisation of D2 -convergence that the name is derived – it is L2 -convergence with respect to a discretisation scheme. Note that this theorem shows that as far as D2 -convergence is concerned any two discretisation schemes are equivalent. There is a similar characterisation of D2 -convergence of self-financing strategies and value processes; since there is an extra time parameter it involves discretisation of time also, but this is quite straightforward – see [37] for details. The discretisation scheme characterisation of D2 -convergence may be of value in devising numerical approximation schemes for the calculation of prices and the like. At the least it provides the information that calculations using such schemes give approximations that converge. The results described above regarding convergence of discrete models to the continuous BS one are really a special case of a more abstract formulation of the convergence of discrete stochastic calculus to the continuous counterpart on the classical Wiener space. As has already been mentioned, obtaining the self-financing strategy that replicates a given contingent claim is an application of the martingale representation theorem – which can be thought of as stochastic differentiation. Going in the direction from a trading

4.8 Further Developments

99

strategy to the value process is, abstractly, simply stochastic integration. So D2 -convergence is a mode of convergence that preserves these basic operations of the stochastic calculus. These more abstract (but financial-jargonfree) formulations of the results described here are discussed in the paper [38], in the slightly simpler context of ordinary Brownian motion (as opposed to geometric Brownian motion as here). That paper shows that there are other operations that are stable under D2 -convergence – for example the Wiener-Itˆ o chaos representation of L2 functionals.

4.8 Further Developments The ideas outlined in this lecture so far have been extended in a number of different directions more recently, as follows. 4.8.1 Poisson pricing models An alternative discrete market model was suggested by Cox & Ross [18], based on jump processes modelling the stock prices. This model converges to a continuous model where prices are modelled by a geometric Poisson process. An early result of Loeb in his original paper [71] shows how a Poisson process is obtained as the standard part of a discrete jump process with infinitesimal time steps, using Loeb measure and the standard part mapping, in the same way that Brownian motion is obtained from an infinitesimal random walk. The paper [36] shows how this can be used to give results parallel to those linking the CRR discrete market model to the BS continuous time model discussed earlier. In particular the theory of D2 -convergence applies to this setting also and preserves the operations of finding self-financing trading strategies and value processes. Abstractly this is again a preservation result concerning stochastic integration and differentiation, this time with respect to a driving martingale that is a (compensated) Poisson process. 4.8.2 American options An American option is one where the exercise time16 is at the owner’s discretion at any time up to a fixed expiry time. Such options, in particular an American put option, are more complicated than European options both to deal in and also to investigate theoretically. The time to exercise is obviously a stopping time – that is, a random time that depends only on what has happened so far and doesn’t require any foreknowledge. The question arises as to what is the best or optimal time to exercise the option – naturally defined in terms of the stopping time that gives the best return. In the Black–Scholes 16

that is, the time when the owner can exercise his right to buy (for a call option) or sell (for a put option)

100


model the value vt of an American put option to its owner at time t is therefore defined as the supremum over stopping times τ ≥ t of the expected return if the option is exercised at the time τ . From an abstract point of view this is an optimal stopping problem, with the value process given by the Snell envelope of the return process (K − su )+ , where K is the strike price for this option. It turns out that there is a unique optimal stopping time that gives the first occurrence of the critical price (or optimal stopping boundary) sc at which it is optimal to exercise the option. The critical price is a deterministic function sc (u) that depends on the time u ≥ 0 at which the option is being valued. There is a discrete version of this theory for the CRR models Mn , and it is shown in [40] that all the entities mentioned above D2 -converge to the continuous version. In the case of the critical price, which is non-random, the results show that the discrete critical price functions Snc converge uniformly to sc in the BS model. Thus, in addition to providing a more transparent link between the theory of American options in the discrete setting and those in the continuous context, the machinery of D2 -convergence, coupled with the hyperfinite Loeb space machinery, again gives new stronger convergence information. For a complete exposition of this see the paper [40]. 4.8.3 Incomplete markets A model of a financial market is called complete if every claim can be uniquely replicated by a self-financing strategy as in Theorems 4.2 and 4.7. In such models the fair price for an option is the same whether calculated from the self-financing strategy or using the equivalent martingale idea. In incomplete markets the question of a fair price for an option is still a topic of discussion and research. A number of different but related approaches have been suggested. In his recent thesis [97] Wellmann suggests that stability under convergence – in particular D2 -convergence – should be one basic requirement for a satisfactory pricing methodology. He considers the mean-variance hedging and the variance-optimal pricing methodologies, and using an extension of the methods described above shows that both are stable under D2 -convergence. The thesis also examines practical aspects of these models in terms of their utility for numerical approximations. 4.8.4 Fractional Brownian motion Self-similar processes such as fractional Brownian motion (FBM) have been proposed as providing more realistic models for stock prices than those such as geometric Brownian motion that form the basis for the BS model. From observing the stock market it is clear that the time series in actual stock price data do not conform to the BS hypotheses, but appear to display some evidence of long-term dependence (though the latter is still hotly debated).

4.8 Further Developments

101

Fractional Brownian motion (FBM) models are good limiting models for longterm dependence, and thus have been proposed as alternative pricing models. However, FBM is not a semimartingale, which makes it impossible to use the apparatus of stochastic calculus in its current form. More seriously, a host of other issues is raised by the fact that the absence of the semimartingale property means that there is no equivalent martingale measure for FBM, and the rationale for pricing of options that gave rise to the theory outlined in the CRR and BS models is no longer valid. A nonstandard definition of FBM, based on a fractional version of the Anderson random walk, was constructed in [39] and arbitrage opportunities were identified in the hyperfinite model. How these might be adjusted to yield a set of fractional Brownian paths of positive Loeb measure along which arbitrage is possible, remains an open question. Other (standard) discussions of FBM as a pricing model have not so far displayed an explicit set of ‘arbitrage paths’. 4.8.5 Interest rates So called term structure models for interest rates form a major pre-occupation of finance theorists at present. There is no consensus on the ‘correct’ model to use and competing alternatives abound, based mostly on the insights gained from the Black-Scholes model and its generalisations. Work in this field using nonstandard methods was initiated by Wellmann [96], who discusses a hyperfinite version of the Heath-Jarrow-Morton model. This area seems to offer much scope for the development of nonstandard approaches.

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Index

absorbing set, 48 action, 62 adapted discretisation scheme, 98 adapted function, 27 ℵ1 -saturation, 10 American call option, 89 American option, 99 Anderson’s Brownian motion, 18 Anderson’s Luzin theorem, 20 Anderson’s random walk, 70 arbitrage, 88 attractor, 47 – for Navier–Stokes equations, 46 – for stochastic Navier–Stokes equations, 55 – global, 47 – measure, 50 – random global, 53 – S-attractor, stochastic, 54 – stochastic, 47, 52 Black-Scholes formula, 94, 95 Black-Scholes model, 92 Bochner integral, 31 bounded quantifier statement, 9 Brownian motion, 19, 61 – fractional, 100 – geometric, 93 BS model, 94 call option, 89 Cameron–Martin formula, 62, 65, 68 Cameron–Martin subspace, 65 claim, contingent, 89, 90, 93 cocycle, 47 – crude, 53 – perfect, 53 complete market, 100

comprehension, countable, 11 contingent claim, 89, 90, 93 convergence – D2 -convergence, 97 convergence of market models, 96 countable comprehension, 11 Cox-Ross-Rubinstein model, 86 CRR model, 86, 94 – hyperfinite, 94 D2 -convergence, 97 delayed equation, 25 derivation operator, 63 discretisation scheme, 98 divergence, 79 Donsker’s flat integral, 61 Donsker’s invariance principle, 19 drift, 87 dynamical system, random, 53 equivalent martingale measure, 91, 93 Euler equations, stochastic, 40 European call option, 89, 92 external set, 10 fair price, 89 finance theory, 85 finite hyperreal, 2 flat integral, 61, 64 Foias equation, 42 fractional Brownian motion, 100 gains process, 88 Galerkin approximation, 32 geometric Brownian motion, 93 global attractor, 47 – random, 53 gradient operator, 63, 77

110

Index

Haar measure, 17 Hausdorff space, 12 hedging – mean variance, 100 – variance-optimal, 100 hedging strategy, 90 Hermite polynomial, 76 Hilbert spaces, 30 hyperfinite CRR model, 94 hyperfinite difference equation, 19, 24 hyperfinite set, 16 hyperfinite time line, 16, 63 hyperreals, 2 Ichikawa integral, 31 incomplete market, 100 infinite hyperreal, 2 infinitely close, 2 infinitesimal, 2 infinitesimal delayed equation, 25 infinitesimal random walk, 18 inner Loeb measure, 15 integration by parts, 81 interest rate model, 101 internal set, 9 Itˆ o integral, 19, 27, 74 Itˆ o’s lemma, 19 Key lemma, 14 Lebesgue measure, 16 lifting, 20, 25 – monomial, 76 lifting, two legged, 20 Loeb algebra, 13, 15 Loeb counting measure, 16 Loeb differential equation, 26 Loeb integrable function, 21 Loeb integration, 20 Loeb measurable, 15 Loeb measurable function, 19 Loeb measure, 13 Loeb null set, 14 Loeb probability, 15 Malliavin calculus, 63, 70, 72 – gradient operator, 73, 77 – Malliavin covariance, 81 – Malliavin operator, 73, 83

– Skorohod integral, 73, 79 Malliavin operator, 83 martingale representation theorem, 78, 98 measure attractor, 50 monad, 2, 11 monomial lifting, 76 µ-approximable, 15 multiple Wiener integral, 74 Navier–Stokes equations, 29 – nonhomogeneous, 40 – solution of deterministic, 34 – solution of stochastic, 37 – stochastic, 30, 31 – uniqueness of solution, 34 nearstandard, 11 neocompact, 57, 58 nonstandard densities, 44 nonstandard reals, 2 nonstandard topology, 11 – in Hilbert spaces, 33 nonstandard universe, 7, 8 number operator, 83 Ocone’s formula, 78 optimal stopping, 100 option – American, 89 – American put, 99 – call, 89 – European, 89, 92 – put, 89 option pricing, 89 Ornstein–Uhlenbeck process, 84 – infinite dimensional, 63, 69 outer Loeb measure, 15 overflow, 10 Peano’s existence theorem, 24 Poisson pricing model, 99 portfolio, 87 price process, 87 put option, 89 Q-discretisation scheme, 98 random attractor, 53 random dynamical system, 53 replicating strategy, 90

Index S-absorbing set, 48 S-continuous, 12 S-integrable, 22 saturated, ℵ1 -, 10 self-financing trading strategy, 88, 90, 93 semigroup, 46, 52 Skorohod integral, 79 Snell envelope, 100 standard part, 5, 12 statistical solution, 42, 43 – for stochastic Navier–Stokes equations, 46 – using Loeb measures, 43 – using nonstandard densities, 45 stochastic attractor, 47, 52 – for Navier–Stokes equations, 52, 53 stochastic differential equation, 19, 28 stochastic differentiation, 98, 99 stochastic Euler equations, 40 stochastic flow, 39, 53 stochastic Navier–Stokes equations, 30, 31 – attractor, 55 – definition of solution, 32 – nonhomogeneous, 40 – solution of, 37 stopping time, 99

111

strategy – hedging, 90 – replicating, 90 strike price, 89 strong solution, 32 superflow, 54 superstructure, 8 symmetrisation, 79 term structure for interest rates, 101 term structure models, 101 trading strategy, 87, 93 – self-financing, 88, 90, 93 Transfer Principle, 4, 9 ultrapower, 2, 9 underflow, 10 universe, nonstandard, 7 value process, 87 viable market model, 88 volatility, 87 weak solution, 32 Wiener measure, 17, 61, 65, 67 Wiener process, 61 Wiener sphere, 62, 66 – Brownian motion on, 69 Wiener-Itˆ o chaos decomposition, 75

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