Itô s Theory of Stochastic Integration

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1 chapter 5 Itô s Theory of Stochastic Integration Up to this point, I have been recognizing but not confronting the challenge posed by integrals of the sort in There are several reasons for my decision to postpone doing so until now, perhaps the most important of which is my belief that, in spite of, or maybe because of, its elegance, Itô s theory of stochastic integration tends to mask the essential simplicity and beauty of his ideas as we have been developing them heretofore. However, it is high time that I explain his theory of integration, and that is what I will be doing in this chapter. However, we will not deal with the theory in full generality and will restrict our attention to the case when the paths are continuous. In terms of equations like , this means that we will not try to rationalize the dp t integral except in the case when p is Brownian motion i.e., M = and p = w, p. The general theory is beautiful and has been fully developed by the French school, particularly by C. Dellacherie and P.A. Meyer who have published a detailed account of their findings in [5]. Because it already contains most of the essential ideas, we will devote this chapter to stochastic integration with respect to Brownian motion. 5.1 Brownian Stochastic Integrals Let Ω, F, P be a complete probability space. Then βt, F t, P will be called an R n -valued Brownian motion if {F t : t } is a non-decreasing family of P-complete sub σ-algebras of the σ-algebra F and β : [, Ω R n is a B [, F-measurable map with the properties that properties that 1 a β = and t βt, ω is continuous for P-almost every ω, b ω βt, ω is F t -measurable for each t [,, c for all s [, and t,, βs + t βs is P-independent of F s and has the distribution of a centered Gaussian with covariance ti R n under P. 1 It should be noticed that our insistence on the completeness of all σ-algebras imposes no restriction. Indeed, if F or the F t s are not complete but a, b, and c hold for some ω β, ω, then they will continue to hold after all σ-algebras have been completed. 125

2 126 5 Itô s Theory of Stochastic Integration Notice that the preceding definition is very close to saying that β, ω is continuous for all ω and that the P-distribution of ω Ω β, ω C [, ; R n is given by Wiener measure, the measure which would have been denoted by P I,, in 2.4 and by P starting in To be more precise, first observe that, without loss in generality, one can assume that β, ω C [, ; R n for all ω, in which case a, b, and c guarantee that P I,, is the P-distribution of ω β, ω. Conversely, if Ω = C [, ; R n, P = P I,,, and F and F t are, respectively, the P-completions of B and B t, then one gets a Brownian motion by taking βt, p = pt. Thus, the essential generalization afforded by the preceding definition is that the σ-algebras need not be inextricably tied to the random variables βt. That is, F t must contain but need not be the completion of σ {βτ : τ [, t]} A Review of Paley Wiener Integral. As an aid to understanding Itô s theory, it may be helpful to recall the theory of stochastic integration which was introduced by Paley and Wiener. Namely, let βt, F t, P be a Brownian motion on the complete probability space Ω, F, P, and assume β, ω C [, ; R n for all ω Ω. Given a Borel measurable function θ : [, R n which has bounded variation on each finite interval, one can use Riemann-Stieltjes theory 2 to define t [, I θ t, ω = θτ, dβτ, ω R n R Because it is given by a Riemann Stieltjes integral, we can say that cf I θ t, ω = lim N m= By c, we know that, for each N N, ω θm2 N, N mβt, ω, R n θm2 N, N mβt, ω R n m= is a centered Gaussian with variance θ[τ]n 2 dτ, and so it is an easy step to the conclusion that ω I θ t, ω is a centered Gaussian with variance equal to θτ 2 dτ. In fact, with only a little more effort, one sees 2 What is needed here is the fact cf. Theorem in [34] that Riemann-Stieltjes theory is completely symmetric: ϕ is Riemann Stieltjes integrable with resect of ψ if and only if ψ is with respect to ϕ. In fact, the integration by parts formula is what allows one to exchange the two.

3 5.1 Brownian Stochastic Integrals. 127 that Is + t Is is P-independent of σ {Iσ : σ [, s]} and that its P-distribution is that of a centered Gaussian with variance s θτ 2 dτ. Moreover, I θ, ω C [, ; R n, and so we now know that I θ t, F t, P is a continuous, square integrable martingale. In particular, by Doob s inequality, θ 2 L 2 [, ;R n = lim EP[ I θ t 2] E P[ I θ 2 ] t [, 4 θ 2 L2 [, ;R n. The relations in can be used as the basis on which to extend the definition of θ I θ to square integrable θ s which do not necessarily possess locally bounded variation. Indeed, says that, as a map taking θ L 2 [, ; R n with locally bounded variation into continuous square integrable martingales on Ω, F t, P, θ I θ is a continuous. Hence, because the smooth elements of L 2 [, ; R n are dense there, this map admits a unique continuous extention. To be precise, define M 2 P; R to be the space of all R-valued, square integrable, P-almost surely continuous P-martingales M relative to {F t : t } such that M M 2 P;R = sup E P[ Mt 2] 1 2 <. t [, Although it may not be apparent, M 2 P; R is actually a Hilbert space. In fact, it can be isometrically embedded as a closed subspace of L 2 P; R. Namely, if M M 2 P; R, then M is an L 2 -bounded martingale and therefore, by the L 2 -martingale convergence theorem cf. Theorem in [36], there exists an M L 2 P; R to which {Mt : t } converges both P- almost surely and in L 2 P; R. In particular, this means that, for each t, Mt = E P[ M F t ] P-almost surely. Moreover, because Mt 2, F t, P is a submartingale, Mt L 2 P;R M M 2 P;R = M L 2 P;R. Hence, the map M M 2 P; R M L 2 P; R is a linear isometry. Finally, to see that {M : M M 2 P; R} is closed in L 2 P; R, suppose that {M k } k=1 M2 P; R and that M k X is L 2 P; R. By Doob s Inequality, sup E P[ M l M k 2 [, l>k ] 4 sup M l M k 2 L 2 P;R l>k as k. Hence, there exists an F-measurable map ω Ω M C [, ; R n such that lim k E P[ M M k [, ] 2 =. But clearly, for each t, Mt = E P [X F t ] P-almost surely, and so not only is M M 2 P; R but also, since X is σ t F t -measurable, X = M. For future reference, we will collect these observations in a lemma.

4 128 5 Itô s Theory of Stochastic Integration Lemma. The space M 2 P; R with the norm given by is a Hilbert space. Moreover, for each M M 2 P; R, M lim t Mt exists both P-almost surely and in L 2 P; R, and the map M M 2 P; R M L 2 P; R is a linear isometry. By combining Lemma with the remarks which precede it, we arrive at the following statement, which summarizes the Paley Wiener theory of stochastic integration Theorem. There is a unique, linear isometry θ L 2 [, ; R n I θ M 2 P; R with the property that I θ t, ω is given by when θ has locally bounded variation. In particular, for each T, I θ T = I 1[,T ] θ P-almost surely. Finally, for each θ L 2 [, ; R n and all s < t, I θ t I θ s is P-independent of F s and its P-distribution is that of a centered Gaussian with variance s θτ 2 dτ Itô s Extension. Itô s extention of the preceding to θ s which may depend on ω as well as t is completely natural if one keeps in mind the reason for his wanting to make such an extention. Namely, he was trying to make sense out integrals which appear in his method of constructing Markov processes. Thus, he wanted to find a notion of integration which would allow him to interpret lim N m= σ X N m2 N, x N mp t as an integral. In particular, he had reason to suppose that it was best to make sure that the integrand is independent of the differential by which it is being multiplied. With this in mind, we say that a map F on [, Ω into a measurable space is progressively measurable if F [, T ] Ω is B [,T ] F T -measurable for each T [,. 3 The following elementary facts about progressive measurability are proved in Lemma of [36] Lemma. Let PM denote the collection of all A [, Ω such that 1 A is an R-valued, progressively function. Then PM is a sub σ-algebra of B [, F, and a function on [, Ω is progressively measurable if and only if it is measurable with respect to PM. Furthermore, if F : 3 So far as I know, the notion of progressive measurability is one of the many contributions which P.A. Meyer made to the subject of stochastic integration. In particular, Itô dealt with adapted functions: F s for which ω F T, ω is F T -measurable for each T. Even though adaptedness is more intuitively appealing, there are compelling technical reasons for preferring progressive measurability. See Remark in [36] for further comments on this matter.

5 5.1 Brownian Stochastic Integrals. 129 [, Ω E, where E is a metric space, with the properties that F, ω is continuous for each ω and F T, is F T for each T, then F is progressively measurable. In fact, if F t is P-complete for each t and F, ω is continuous for P-almost every ω, then F is progressively measurable if F T, is F T -measurable for each T. Now let Θ 2 P; R n denote the space of all progressively measurable θ : [, Ω R n with the property that θ Θ2 P;R n E P [ ] θt 2 dt <. Since, by Lemma 5.1.6, an equivalent way to describe Θ 2 P; R n is as the subspace of progressively measurable elements of L 2 Leb [, P, we know that Θ 2 P; R n is a Hilbert space. Our goal is to show cf. Exercise below that there is a unique linear isometry θ Θ 2 P; R n I θ M 2 P; R with the property that when θ is an element of Θ 2 P; R n such that θ, ω has locally bounded variation for each ω I θ t, ω = = θt, ω, βt, ω R n θτ, ω, dβτ, ω R n βτ, ω, dθτ, ω where the integrals are taken in the sense of Riemann Stieltjes. R n, Lemma. Let SΘ 2 P; R n 4 be the subspace of uniformly bounded θ Θ 2 P; R n for which there exists an N N with the property that θt, ω = θ[t] N, ω for all t, ω [, Ω. Then SΘ 2 P; R n is dense in Θ 2 P; R n. Moreover, if θ SΘ 2 P; R n, I θ is given as in 5.1.7, and A θ t, ω then E θ t, F t, P is a martingale when θτ, ω 2 dτ, E θ t, ω exp I θ t, ω 1 2 A θt. In particular, both I θ t, F t, P and I θ t 2 A θ t, F t, P are martingales. 4 The S here stands for simple.

6 13 5 Itô s Theory of Stochastic Integration Proof: To prove the density statement, first observe that, by any standard truncation procedure, it is easy to check that uniformly bounded elements of Θ 2 P; R n which are supported on [, T ] Ω for some T [, form a dense subspace of Θ 2 P; R n. Thus, suppose that θ Θ 2 P; R n is uniformly bounded and vanishes off of [, T ] Ω. Choose a ψ Cc R; [, so that ψ on, ] and ψt dt = 1, and set R θ k t, ω = k ψ kt τ θτ, ω dτ. Then, for each k Z + and ω Ω, θ k, ω is smooth and vanishes off of [, T +1]. In addition, because ψ is supported on the right half line, it follows from Lemma that θ k is progressively measurable. At the same time, for each ω, θ k, ω L2 P;R n θ, ω L2 P;R n and θ k, ω θ, ω L2 [, ;R n as k. Hence, by Lebesgue s Dominated Convergence Theorem, θ k θ Θ 2 P;R n. In other words, we have now proved the density of the uniformly bounded elements of Θ 2 P; R which are supported on [, T ] Ω for some T and are smooth as functions of t [, for each ω Θ. But clearly, if θ is such an element of Θ 2 P; R and θ N t, ω θ[t] N, ω, then θ N θ in Θ 2 P; R. To prove the second assertion, let θ be a uniformly bounded element of Θ 2 P; R which satisfies θt, ω = θ[t] N, ω. Then, for m2 N s < t m + 12 N, E P[ E θ t ] Fs = E θ se t s 2 θm2 N 2 E P[ exp θm2 N, βt βs ] Fs = E R n θ s since βt βs is P-independent of F s. Clearly, for general s < t, one gets the same conclusion by iterating the preceding result. Hence, we now know that E θ t, F t, P is a martingale. To complete the proof from here, first observe that we can replace θ by λθ for any λ R. In particular, this means that E P[ e I θt ] e 1 2 A2 t E P[ E θ t + E θ t ] = 2e 1 2 A2t, where A is the uniform upper bound on θt, ω. At the same time, by Taylor s Theorem, E λθ t 1 I θ t λ λ 2 e I θt and E λθ t + E λθ t 2 λ 2 I θ t 2 A θ t λ 3 e I θt for < λ 1. Hence, by Lebesgue s Dominated Convergence Theorem, we get the desired conclusion after letting λ.

7 5.1 Brownian Stochastic Integrals Theorem. There is a unique linear, isometric map θ Θ 2 P; R n I θ M 2 P; R with the properties that I θ is given by for each θ SΘ 2 P; R. Moreover, given θ 1, θ 2 Θ 2 P; R, I θ1 ti θ2 t θ1 τ, θ 2 τ, F R n t, P is a martingale. Finally, if θ Θ 2 P; R and E θ t is defined as in 5.1.1, then E θ t, F t, P is always a supermartingale and is a martingale if A θ T is bounded for each T [,. See Exercises and for more refined information. Proof: The existence and uniqueness of θ I θ is immediate from Lemma Indeed, from that lemma, we know that this map is linear and isometric on SΘ 2 P; R n and that SΘ 2 P; R n is dense in Θ 2 P; R n. Furthermore, Lemma says that I θ t 2 A θ t, F t, P is a martingale when θ SΘ 2 P; R, and so the general case follows from the fact that I θk t 2 A θk t I θ t 2 A θ t in L 1 P; R when θ k θ in Θ 2 P; R. Knowing that I θ t 2 A θ t, F t, P is a martingale for each θ Θ 2 P; R n, we get by polarization. That is, one uses the identity I θ1 ti θ2 t = 1 4 θ1 τ, θ 2 τ R n dτ Iθ1+θ 2 t 2 I θ1 θ 2 t 2 A θ1+θ 2 t + A θ1 θ 2 t. Finally, to prove the last assertion, choose {θ k } 1 SΘ 2 P; R so that θ k θ in Θ 2 P; R n. Because E θk t, F t, P is a martingale for each k and E θk t E θ t in P-probability for each t, Fatou s Lemma implies that E θ t, F t, P is a supermartingale. Next suppose that θ is uniformly bounded by a constant Λ <, and choose the θ k s so that they are all uniformly bounded by Λ as well. Then, for each t [, T ], E P[ E θk t 2] e Λt E P[ E 2θk t ] = e Λt, and so E θk t E θ t in L 1 P; R for each t [, T ]. Hence, we now know that E θ t, F t, P is a martingale when θ is bounded. Finally, if A θ t, ω Λt < for each t [, and θ m t, ω 1 [,m] θt, ω θt, ω, then E θm t E θ t in P-probability and E P[ E θm t 2] e Λt for each t, which again is sufficient to show that E θ t, F t, P is a martingale. Remark With Theorem , we have completed the basic construction in Itô s theory of Brownian stochastic integration, and, as time

8 132 5 Itô s Theory of Stochastic Integration goes on, we will increasingly often replace the notation I θ by the more conventional notation θτ, dβτ R n = I θ t. Because it recognizes that Itô theory is very like a classical integration theory, is good notation. On the other hand, it can be misleading. Indeed, one has to keep in mind that, in reality, Itô s integral is, like the Fourier transform on R, defined only up to a set of measure and via an L 2 -completion procedure. In addition, for the cautionary reasons discussed in & 3.3.3, it is a serious mistake to put too much credence in the notion that an Itô integral behaves like a Riemann Stieltjes integral Stopping Stochastic Integrals and a Further Extension. The notation θτ, dβτ for I R n θ t should make one wonder to what extent it is true that, for ζ 1 ζ 2, ζ2 t ζ 1 θτ, dβτ R n I θ t ζ 2 I θ t ζ 1 = 1 [ζ1,ζ 2t θτ, dβτ R n. Of course, in order for the right hand side of preceding to even make sense, it is necessary that 1 [ζ1,ζ 2θ be progressively measurable, which is more or less equivalent to insisting that ζ 1 and ζ 2 be stopping times Lemma. Given θ Θ 2 P; R n and stopping times ζ 1 ζ 2, holds P-almost surely. In fact, if α is a bounded, F ζ1 -measurable function and α1 [ζ1,ζ 2θt, ω equals αωθt, ω or depending on whether t is or is not in [ζ 1 ω, ζ 2 ω, then α1 [ζ1,ζ 2θ Θ 2 P; R n and α I θ t ζ 2 I θ t ζ 1 = α1[ζ1,ζ 2θτ, dβτ R n. Proof: Clearly, to check that θ α1 [ζ1,ζ 2θ is in Θ 2 P; R n, it is enough to check that θ is progressively measurable, and this is an elementary exercise. Next, set t = I θ t ζ 2 I θ t ζ 1 and Ĩt = θτ, dβτ R n.

9 5.1 Brownian Stochastic Integrals. 133 Then, by Doob s Stopping Time Theorem and , E P[ α t Ĩt 2] = E P[ α 2 t 2] 2E P[ α tĩt] + E P[ Ĩt 2] [ ] [ t ζ2 ] t ζ2 = E P α 2 θτ 2 dτ 2E P α θτ, θτ dτ R n t ζ 1 t ζ 1 [ ] + E P θτ 2 dτ =. The preceding makes it possible to introduce the following extension of Itô s theory. Namely, define Θ 2 loc P; Rn to be the space of progressively measurable θ : [, Ω R n with the property that, P-almost surely, A θ t T θt 2 dt < for all T [,. At the same time, define M loc P; R to be the space of continuous local martingales. That is, M M loc P; R if M : [, Ω R is a progressively measurable function, M, ω is continuous for P-almost every ω, and there exists a sequence {ζ k } 1 of stopping times such that ζ k P-almost surely and, for each k Z +, Mt ζ k, F t, P is a martingale Theorem. There is a unique linear map θ Θ 2 loc P; Rn I θ M loc P; R with the property that for any θ Θ 2 loc P; Rn and stopping time ζ: [ ζ E P θτ ] 2 dτ < = I θ t ζ = 1 [,ζ τ θτ, dβτ. R n Because it is completely consistent to do so, we will continue to use the notation θτ, dβτ to denote I R n θ, even when θ Θ 2 loc P; Rn. Another direction in which it is useful to extend Itô s theory is to matrixvalued integrands. Namely, we have the following, which is a more or less trivial corollary of the preceding theorem Corollary. Let βt, F t, P be an R n -valued Brownian motion, and suppose that σ : [, Ω HomR n ; R n is a progressively measurable function with the property that T σt 2 H.S. dt < P-almost surely for all T,. Then there is a P-almost surely unique, R n -valued, progressively measurable function t στ dβτ with the property that, P-almost surely, ξ, στ dβτ στ ξ, dβτ for all t [, & ξ R n. R n R n =

10 134 5 Itô s Theory of Stochastic Integration In particular, if ζ is a stopping time and then [ ζ E P στ ] 2 H.S. dτ <, ζ is an R n -valued martingale, and ζ is an R-valued martingale Exercises. στ dβτ στ dβτ, F t, P 2 ζ στ 2 H.S. dτ, F t, P Exercise We claimed, but did not prove, that Itô s integration theory coincides with Riemann Stieltjes s when the integrand has locally bounded variation. To be more precise, let θ Θ 2 loc P; Rn and assume that θ, ω has locally bounded variation for each ω Ω. Show that, one version of the random variable ω I θ, ω is given by the indefinite, Riemann Stieltjes integral of θ, ω with respect to β, ω. Hint: First show that there is no loss in generality to assume that θ is uniformly bounded and that, for each ω, θ, ω is right continuous at, and left continuous on,. Second, because θ, ω is Riemann Stieltjes integrable with respect to β, ω on each compact interval, verify that the Riemann Stieltjes integral of θ, ω on [, t] with respect to β, ω can be computed as the limit of the Riemann sums θm2 N, ω, N mβt, ω. R n m= Finally, use the boundedness of θ and the left continuity of θ, ω to see that [ θτ θ[τ]n ] 2 dτ =. lim N EP Exercise One of the most important applications of the Paley Wiener integral was made by Cameron and Martin. To explain their application, use, as in 3.1.1, P to denote the measure on C [, ; R n corresponding to the Lévy system I,,. That is, P is the standard

11 5.1 Brownian Stochastic Integrals. 135 Wiener measure on C [, ; R n, and so pt, B t, P is a Brownian motion. Next, given η L 2 [, ; R n, set ht = ητ dτ, and let τ h : C [, ; R n C [, ; R n denote the translation map given by τ h p = p + h. Then the theorem of Cameron and Martin states that the measure τ h P is equivalent of P and that its Radon Nikodym derivative R h is given by the Cameron Martin formula R h = exp I η 1 2 η 2 L 2 [, ;R n. Prove their theorem. Hint: There are several ways in which to prove their theorem. Perhaps the one best suited to the development here is to first prove that P can be characterized as the unique probability measure P on C [, ; R n with the property that E P[ ] e I θ 1 = exp 2 θ 2 L 2 [, ;R n for all piecewise constant, compactly supported θ : [, R n. In this expression, I θ, p denotes the Riemann Stieltjes integral of θ with respect to p over taken over the support of θ. Knowing this, it is easy to check that R 1 h dτ h P = dp for compactly supported, piecewise constant η s. To complete the proof, one need only construct a sequence {η k } 1 of compact supported, piecewise constant functions so that η k η in L 2 [, ; R n. Since the corresponding functions h k tend uniformly on compacts to h while the corresponding R hk s tend in L 1 P ; R to R h, the general case follows. Exercise Because we are dealing with processes which are almost surely continuous, most of the subtlety in the notion of a local martingale is absent. Indeed, given any progressively measurable M : [, Ω R with the property that M, ω is continuous for P-almost every ω, show that Mt, F t, P is a local martingale if and only if, for each k Z +, Mt ζk, F t, P is a martingale, where ζ k ω inf { t : Mt, ω k }. In this connection, show that a local martingale Mt, F t, P is a martingale if M [,T ] L 1 P for each T [,. In particular, use this to see that an Itô stochastic integral I θ t is a P-square integrable martingale if and only if [ T E P θt ] 2 dt < for all T [,. In a slightly different direction, show that a local martingale Mt, F t, P is a supermartingale if it is uniformly bounded from below.

12 136 5 Itô s Theory of Stochastic Integration Exercise Assume that σ : [, Ω HomR n ; R n is a progressively measurable function for which holds, set Xt = στ dβτ and At = στστ dτ, and assume that, for some T, and ΛT,, ξ, AT ξ R n ΛT ξ 2, ξ R n. Show that, for each ɛ, 1, ] ɛ X E [exp P [,T ] e1 ɛ n 2. 2ΛT In particular, conclude that P X [,T ] R e1 ɛ n 2 e ɛr2 2ΛT for all R >, and E P [exp X 2 [,T ] 2nΛT ] e 3 2 for all n 1. Hint: Given ξ R n, set X ξ t = ξ, Xt R n and Et, ξ = exp X ξ t 1 2 ξ, Atξ, R n and show, using the last part of Theorem and Doob s inequality, that, for each q 1,, E P [ sup e q t [,T ] e 1 2 qλt ξ 2 ] Rn ξ,xt e 1 2 qλt ξ 2 E P[ E, ξ q [,T ] q q E P[ ET, ξ q] q 1 q e 1 2 q2 ΛT ξ 2 q E P[ ET, qξ] q 1 ] q = e 1 q 2 qλt ξ 2. q 1 Next, multiply the preceding through by 2πτ n 2 e ξ 2 2τ where τ = ɛ q 2 ΛT, and integrate with respect of ξ over R n. Finally, let q. Exercise In this exercise we will develop some of the intriguing relations between Itô stochastic integrals and Hermite polynomials. These relations reflect the residual Gaussian characteristics which stochastic integrals inherit from their driving Brownian fore-bearers. Deeper examples of these relations will be investigated in Exercise

13 5.2 Itô s Integral Applied to Itô s Construction Method. 137 i Given a, define H m x, a for x R so that the identity λ2 λx e 2 a = m= λ m m! H mx, a, λ R, holds. Show that H m x, = x m and, for a >, H m x, a = a m 2 H m a 1 2 x where H m x H m x, 1. In addition, show that H m x = 1 m e x2 2 m x e x2 2, and conclude that the H m s can be generated inductively from H x = 1 and H m = x x Hm 1. In particular, use this to conclude that H m is an mth order polynomial with the properties that: the coefficient of x m is 1, the coefficient of x l is unless l has the same parity as m, and the constant term in H 2m is 1 m 2m! 2 m m!. ii Assume that θ Θ 2 P; R with A θ T is uniformly bounded, and show that e I θt is P-integrable. Use this along with the last part of Theorem to justify for all m 1. E P[ H m Iθ T, A θ T ] ] = dm dλ m EP[ E λθ T λ= = iii By combining i and ii, show that there exist universal constant B 2m for m Z + such that B 1 2m EP[ I θ T 2m] E P[ A θ T m] B 2m E P[ I θ T 2m], which is a primitive version of Burkholder s inequality. Finally, check that these inequalities continue to hold for any θ Θ 2 loc P; Rn, not just those for which A θ T is uniformly bounded. 5.2 Itô s Integral Applied to Itô s Construction Method Because our motivation for introducing stochastic integration was the desire to understand equations like , it seems reasonable to ask whether the theory developed in 5.2 does in fact do anything to increase our understanding. Of course, because we have been dealing with nothing but Brownian stochastic integrals, we will have to restrict our attention to processes for which the Lévy measure is Basic Existence and Uniqueness Result for S.D.E. s. Let σ : [, Ω R n HomR n ; R n and b : [, Ω R n R n be functions with the properties that, for each x R n, t, ω σt, x, ω

14 138 5 Itô s Theory of Stochastic Integration and t, ω bt, x, ω are progressively measurable, t, ω σt,, ω and t, ω bt,, ω are bounded, and σt, x 2, ω σt, x 1, ω H.S. sup sup x 1 x 2 t,ω x 2 x 1 bt, x 2, ω bt, x 1, ω x 2 x 1 < Theorem. Given functions σ and b satisfying and an R n - valued Brownian motion βt, F t, P, there exists for each x R n a P-almost surely unique R n -valued, progressively measurable function X, x which solves 5 cf. Corollary Xt, x = x + σ τ, Xτ, x dβτ + Moreover, there exist a C,, depending only on sup x R n sup t,ω such that, for all T,, and σt, x, ω H.S. bt, x, ω, 1 + x E P[ X, x 2 ] [,T ] C 1 + x 2 e CT 2 b τ, Xτ, x dτ, t. E P[ Xt, x Xs, x 2 ] C 1 + x 2 e CT 2 t s, s < t T. Proof: Set X t, x x. Assuming that X N, x has been defined and that t σ t, X N t, x satisfies , set cf. Corollary * X N+1 t, x = x + σ τ, X N τ, x dβτ + b τ, X N τ, x dτ, and observe that t σ t, X N+1 t, x will again satisfy Hence, by induction on N, we can produce the sequence {X N, x : N } so 5 Just in case it is not clear, the condition that X, x satisfy the equation implicitly contains the condition that X, x is P -almost surely continuous. In particular, this guarantees that the right hand side of makes sense.

15 5.2 Itô s Integral Applied to Itô s Construction Method. 139 that, for each N N, t σ t, X N t, x satisfies and * holds. Moreover, by the last part of Corollary plus Doob s Inequality, E P[ X 1, x X, x [ ] T 2 8E P σ τ, x ] 2 [,T ] H.S. dτ and, for any N 1: [ ] T + 2T E P b τ, x 2 dτ < E P[ X N+1, x X N, x 2 [,T ] [ T 8E P στ, XN τ, x σ X N 1 τ, x ] 2 H.S. [ T + 2T E P b τ, X N τ, x b X N 1 τ, x ] 2 dτ. Thus, by 5.2.1, there is a K < such that E P[ XN+1, x X N, x ] 2 [,T ] K1 + T T ] E P[ XN, x X N 1, x 2 [,t] ] dt for all N 1; and so, by induction, we see that, for each T,, E P[ XN+1, x X N, x ] 2 [,T ] KN 1 + T N T N E P[ X1, x X, x ] 2. N! [,T ] From the preceding it is clear that, for each T,, lim sup E P[ XN, x X N, x ] 2 =. [,T ] N N >N Hence, we have now verified the existence statement. To prove the uniqueness assertion, suppose that X, x and X, x are two solutions, and note that, for each k 1, ] E P[ X, x X, x 2 [,T ζ k ] K1 + T T E P[ X, x X, x 2 [,t ζ k ] ] dt,

16 14 5 Itô s Theory of Stochastic Integration where ζ k inf{t : Xt, x X t, x k}. But e.g., by Gronwall s inequality this means that E P[ X, x X, x ] 2 =, [,T ζ k ] and so we obtain the desired conclusion after letting k. Turning to the asserted estimates, note that E P[ X, x [ ] T 2 3 x E P σ t, Xt, x ] 2 [,T ] H.S. dt [ ] T + 3T E P b t, Xt, x 2 dt, and so there exists a K, with the required dependence, such that E P[ 1 + X, x 2 [,T ] ] T 3 x 2 + K1 + T E P[ 1 + X, x ] 2 dt. [,t] Hence the first estimate follows from Gronwall s inequality As for the second estimate, assume that s < t T, and use E P[ ] [ ] Xt, x Xs, x 2 2E P σ τ, Xτ, x 2 dτ H.S. s [ + 2T E P b τ, Xτ, x ] 2 dτ together with the first estimate to arrive at the second estimate Corollary. Assume that σ : R n HomR n ; R n and b : R n R n are continuous functions which satisfy σx 2 σx 1 H.S. bx 2 bx 1 sup <. x 1 x 2 x 2 x 1 Refer to the setting in 3.1.2, and take M = and F. Then p t, B t, P is an R n -valued Browning motion and the function p X, x, p C [, ; R n described in G1 is the P -almost surely unique, {B t : t }-progressively measurable solution to Xt, x = x + σ Xτ, x dp τ + s b Xτ, x dτ, t.

17 5.2 Itô s Integral Applied to Itô s Construction Method. 141 Proof: Let X, x be the solution to 5.2.5, and define {X N, x : N } as in H4 of Note that X N t, x, p = x + Thus, σ X N [τ] N, x, p dp τ + ] b X N [τ] N, x, p dτ. [ E P X, x X N, x 2 [,t] [ 16E P σ X[τ] N, x σ X N [τ] N, x ] 2 H.S. dτ [ + 4tE P b X[τ]N, x b X N [τ] N, x ] 2 dτ [ + 16E P σ Xτ, x σ X[τ] N, x ] 2 H.S. dτ [ + 4tE P b Xτ, x b X[τ]N, x ] 2 dτ ; and so, by the last estimates in Theorem and the Lipschitz estimates on σ and b, we see that, for each T, x, R n, there is a KT, x < such that [ E P X, x X N, x ] 2 [,t] [ KT, x2 N + KT, x E P X, x X N, x ] 2 dτ [,τ] for all t [, T ]. Finally, apply Gronwall s inequality to conclude that [ lim X, x X N, x ] 2 =. N EP [,t] Remark In the literature, is called the stochastic integral equation for a diffusion processes with diffusion coefficient σσ and drift coefficient b. Often such equations are written in differential notation: dxt, x = σ Xt, x dp t + b Xt, x dt with X, x = x, in which case they are called a stochastic differential equation. It is important to recognize that the joint distribution of p p, X, x, p under P would be the same were we to replace the canonical Brownian motion p t, B t, P by any other R n -valued Brownian motion βt, F t, P. Indeed, as both the construction in Theorem and the one in make clear, this joint distribution depends only on the distribution of ω β, ω, which is the same no matter which realization of Brownian motion is used.

18 142 5 Itô s Theory of Stochastic Integration Subordination. When solving a system of ordinary differential equations one can sometimes take advantage of an inherent lower triangularity in the system. That is, one may be able to arrange the equations in such a way that the system contains a subsystem which is autonomous onto itself. In this case, the entire system can be solved by first solving the autonomous subsystem, plugging the solution into the remaining equations, and then solving resulting, now time-dependent, equations. The same device applies when solving stochastic differential equations, and is particularly interesting in the following situation. Let σ : R n HomR n ; R n and b : R n R n be as in Corollary Suppose that n = k + l, n = k + l, and αx vx σz = & bz = for z = x, y R k R l, βx, y wx, y where v : R k R k, w : R n R l, α : R k HomR k ; R k and β : R n HomR l ; R l. Next, define Σ : [, R l C [, ; R k HomR l ; R l and B : [, R l C [, ; R l R l so that Σ t, y; p = β pt, y and B t, y; p w pt, y. Finally, write r t = p t, q t, where p t and q t are, respectively, the orthogonal projections of r t onto R k and R l thought of as subspaces of R n = R k R l, and note that, when C [, ; R n is identified with C [, ; R k C [, ; R l, P can be identified with P Q, where P and Q are the standard Wiener measures on C [, ; R k and C [, ; R l, respectively. Under these conditions, we want go about solving by first solving the equation Xt, x, p = x + + α Xτ, x, p dp τ v Xτ, x, p dτ, relative to the Wiener measure P, then, for each p C [, ; R k and y R l, solving Y t, y, q ; p = y + + Σ τ, Y τ, y, q ; p; p dq τ B τ, Y τ, y, q ; p; p dτ

19 5.2 Itô s Integral Applied to Itô s Construction Method. 143 relative to Q, and finally showing that t Z t, z, p, q Xt, x, p = Y t, y, q ; X, x, p is a solution to original stochastic integral equation whose coefficients were σ and b. The following theorem says that this subordination procedure works Theorem. Let r = p, q Z, z, r be defined for z = x, y as in Then r Z, z, r is the P -almost surely unique solution to the stochastic integral equation Zt, z, r = z + σ Zτ, z, r dr τ + b Zτ, z, r dτ, t. Hence, if Φ : C [, ; R k C [, ; R l R is measurable and either bounded or non-negative, then E P[ Φ Z, z, r ] = Φ X, x, p ; Y, y, q ; X, x, p Qdq Pdp. Proof: The proof of this result is really just an exercise in the use of Fubini s Theorem. Namely, let r Z, z, r be the P -almost surely unique solution to the stochastic integral equation Zt, z, r = z + σ Zτ, z, r dr τ + b Zτ, z, r dτ, t. What we have to show is that Z, z, r = Z, z, r for P -almost every r C [, ; R n. Let Xt, z, r and Ỹ t, z, r be the orthogonal projections of Zt, z, r onto R k and R l. By Fubini s Theorem, we know that, for Q-almost every q, the function p X, z, p, q satisfies X t, z, p, q = x + + α X τ, z, p, q dp τ v X τ, z, p, q dτ, t, P-almost surely. Thus, by uniqueness, for Q-almost every q, X, z, p, q = X, x, p for P-almost every p.

20 144 5 Itô s Theory of Stochastic Integration Starting from the preceding and making another application of Fubini s Theorem, we see that, for P-almost every p, the function q Ỹ, z, p, q satisfies Ỹ t, z, p, q = y + + β Xτ, x, p, Ỹ τ, z, p, q dq τ w Xτ, x, p, Ỹ τ, z, p, q dτ, t, Q-almost surely. Hence, by uniqueness, for P-almost every p, Ỹ, z, p, q = Y, y, q ; X, x, p Q-almost surely. After combining these two and making yet another application of Fubini s Theorem, we have now shown that Z, z, r = Z, z, r for P -almost every r Exercises. Exercise Here is an example which indicates how Theorem gets applied. Namely, referring to the setting in that theorem, suppose that β and w are independent of y. That is, βx, y = βx and wx, y = wx. Next, define V : [, C [, ; R k HomR l ; R l and B : [, C [, ; R k R l so that V t, p = ββ pτ dτ and B t, p = w pτ dτ, and let Γ t p, dη denote the Gaussian measure on R l with mean Bt, p and covariance V t, p. Given a stopping time ζ : C [, ; R k [, ] and a bounded measurable f : R n R show that E P[ f Zζp, z, r ], ζp < = E P[ Φ, ζ < ] where Φp = f X, x, p, y + η Γ ζp X, x, p, dη for ζp <. R l 5.3 Itô s Formula The jewel in the crown of Itô s stochastic integration theory is Itô s formula. Depending on ones point of view, his formula can be seen as the solution to any one of a variety of problems. From the point of view which comes out of the considerations in Chapters 3 and 4, especially G2 in 3.1.2, it gives us a representation for the martingales being discussed there. From a more general standpoint, Itô s formula is the fundamental theorem of the calculus for which his integration theory is the integral, and, as such, it is the identity on which nearly everything else relies.

21 5.3 Itô s Formula. 145 In the following statement: βt, F t, P is an R n -valued Brownian motion, and X X : [, Ω R k is a progressively measurable and, for P-almost every ω, X, ω is a continuous function of locally bounded variation Y Y : [, Ω R l is a progressively measurable function and, for each 1 j l, the jth component Y j of Y Y is the dβt-itô stochastic integral I θj of some θ j Θ 2 loc P; Rn. Z Zt = Xt, Y t. With this notation, Itô s Formula is the formula given in the next theorem. Our proof is based on the technique introduced by Kunita and Watanabe in their now famous article [21] Theorem. Given any F C 1,2 R k R l ; R, F Zt F Z = k i= xi F Zτ dx j τ + l j=1 yj F Zτ θ j τ, dβτ R n l θj τ, θ j τ R n yj yj F Zτ dτ, t, j,j =1 P-almost surely. Here, dx i -integrals are taken in the sense of Riemann Stieltjes and dβ-integrals are taken in the sense of Itô. Proof: We begin by making several reductions. In the first place, without loss in generality, we may assume that Z, ω is continuous for all ω Ω. Secondly, we may assume that all the X i s have uniformly bounded variation and that all the θ j s are elements of Θ 2 P; R n. Indeed, if this is not the case already, then we can introduce the stopping times k l ζ R = inf var [,t] X i + θj τ 2 dτ R, i=1 replace Zt by Zt ζ R, and, at the end, let R. Similarly, we may assume that F has compact support. Finally, under these assumptions, a standard mollification procedure makes it clear that we need only handle F s which are both smooth and compactly supported. Thus, from now on, we will be assuming that: Z, ω is continuous for all ω, the X i s have uniformly bounded variation, the θ j s are elements of Θ 2 P; R n, and F is smooth and compactly supported. Finally, by continuity, it suffices to prove that the identity holds P-almost surely for each t. j=1

22 146 5 Itô s Theory of Stochastic Integration Given N N and t,, define stopping times {ζm N : m } so that and { ζm+1 N = inf τ ζm N : Zτ Zζ N m τ } max θ j σ 2 dσ 2 N t. 1 j l ζm N ζ N Next, set Zm N = Xm N, Ym N = Z ζm N, and N m,j = Y j ζm+1 N Y j ζm. N By continuity, we know that, for each ω, ζmω N = t for all but a finite number of m s. Hence, F Zt F Z = F Z N m+1 F Zm N. m= Furthermore, by the Fundamental Theorem of Calculus, F Zm+1 N F Zm N = F Xm N, Ym+1 N F Zm N k ζ N m+1 + xi F Xτ, Ym+1 N dxi τ, and, by Taylor s Theorem, F X N m, Y N m+1 F Z N m = i=1 ζ N m l yj F Zm N N m,j j= l j,j =1 yj yj F Z N m N m,j N m,j + EN m, where there exists a C 3 <, depending only on the bound on the third order derivatives of F, such that l Em N C 3 2 N N m,j 2. Next, we use Lemma to first write yj F Z N m N m,j = and then conclude that F Zt F Z k = i= j=1 yj F Z N m1 [ζ N m.ζn m+1 θ j τ, dβτ R n, F N i τ dx i τ + l m= j,j =1 l j=1 yj θn j τ, dβτ R n yj yj F Zm N N m,j N m,j + Em, N m=

23 5.3 Itô s Formula. 147 where F N i τ xi F Xτ, Y N m+1 & θn j τ yj F Z N mθ j τ for τ [ζ N m, ζ N m+1. Because E N m C 3 2 N l j=1 N m,j 2 and [ ] E P N m,j 2 = E P m= m= N m,j 2 = E P[ Y j t Y j 2], we know that m= EN m tends to in L 2 P; R as N. At the same time, it is clearly that, if C 2 is a bound on the second derivatives of F, then and Fi N τ dx i τ xi F Zτ dx i τ C 22 N var [,t] X i [ E P θn j τ, dβτ R n yj F Zτ θ j τ, dβτ ] 2 R n [ = E P θn j τ yj F Zτ θ j τ ] 2 dτ C24 2 N θ j 2 Θ 2 P;R n. Hence, since it is obvious that ζ N m+1 θj τ, θ j τ R n yj yj F Zζ N m dτ m= ζ N m θj τ, θ j τ R n yj yj F Zτ dτ, all that remains is to show that, for all 1 j, j l, ζ yj yj F Zm N N m,j N m,j m= in P-measure. To this end, remark that ζ N m N m+1 θj τ, θ j τ R n dτ E P[ N m,j N ] [ ] m,j Fζ N t = E P Y j ζm+1y N j ζm+1 N Y j ζmy N j ζm N F ζ N m [ ζ N = E P m+1 θj τ, θ j τ ] dτ R n F ζm N, ζ N m

24 148 5 Itô s Theory of Stochastic Integration and conclude that the terms S N m,j,j y j yj F Z N m N m,j N m,j ζ N m+1 ζ N m are orthogonal for different m s. Hence, E P At the same time, m= 2 Sm,j,j N = θj τ, θ j τ R n yj yj F Z N m dτ E P[ Sm,j,j N 2]. m= Sm,j,j N 2 C24 2 N ζ N N m,j 2 + N m,j 2 + θ j m+1 θj τ θ j τ 2 dτ. Hence, E P m= 2 Sm,j,j N C 2 4 N E P [ Yj t Y j 2 + Yj t Y j 2 + θj τ 2 + θ j τ 2 ] dτ 2C 2 4 N E P[ Y t Y 2]. Remark If one likes to think about the Fundamental Theorem of Calculus in terms of differentials, then the following is an appealing way to think about Itô s formula. In the first place, given a progressively measurable function α : [, Ω R which satisfies P ατ 2 θ j τ 2 dτ < = 1 for all t, one should introduce the notation ατdy j τ ατθj τ, dβτ R n.

25 5.3 Itô s Formula. 149 That is, in terms of differentials, dy j t θ j t, dβt. At the same R N time, one should define ατ dy j τdy j τ ατ θ j τ, θ j τ R n dτ for progressively measurable α s satisfying P ατ θ j τ 2 + θ j τ 2 dτ < = 1 for all t. In terms of differentials, this means that we are taking dy j tdy j t = θj t, θ j t dt. Finally, define dx R n i tdx i t = = dx i tdy j t for all 1 i, i k and 1 j l. With this notation, a differential version of Itô s formula becomes df Zt = grad Zt, dzt + 1 dzt, Hess R k+l Zt F dzt. 2 R k+l Although it looks a little questionable, this way of writing Itô s formula has more than mnemonic value. In fact, it highlights the essential fact on which his formula rests. Namely, dβ m t may be a differential, but it is not, in a classical sense, an infinitesimal. Indeed, there is abundant evidence that dβ m t is of order dt. 6 Thus, it is only reasonable that, when dealing with Brownian paths, one will not get a differential relation in which the left hand side differs from the right hand side by odt unless one uses a two-place Taylor s expansion. Furthermore, it is clear why dβ m t 2 = dt. In order to figure out how dβ m dβ m should be interpreted when m m, remember that βm+β m 2 and βm β m 2 are both Brownian motions. Hence, 2dβ m tdβ m t = d β 2 m + β m t d β 2 m β m t = dt dt = dt. 2 2 In other words, dβ m tdβ m t = δ m,m dt, which now explains why the preceding works. Namely, dx i tdx i t = dt and dx i tdy j t = dx i t θ j t, dβt = dt R n because dx i tdx i t and dβtdy j t are odt, and dy j tdy j t = m,m θ j t m θ j t m dβ m tdβ m t = θ j t, θ j t R n dt because dβ m tdβ m t = δ m,m dt. 6 Lévy understood this point thoroughly and in fact made systematic use of the notation dt.

26 15 5 Itô s Theory of Stochastic Integration Exercises. Exercise Given an R n -valued Brownian motion βt, F t, P and a θ Θ 2 loc Rn, define E θ as in i Show that E θ is P-uniquely determined by the fact that it is progressively measurable and de θ t = E θ t θt, dβt R n and E θ = 1. For this reason, E θ is sometimes called the Itô exponential. Hint: Checking that E θ is a solution to is an elementary application of Itô s formula. To see that it is the only solution, suppose that X is a second solution, and apply Itô s formula to see that d Xt E θ t =. ii Show that E θ t, F t, P is always a supermartingale and that it is a martingale if and only if E P[ E θ t ] = 1 for all t. iii From now on, assume that E P[ e 1 2 A θt ] < for each T. First observe that I θ t, F t, P is a square integrable martingale, and conclude that, for every α, e αi θt, F t, P is a submartingale. In addition, show that E P[ e 1 2 I θt ] E P[ e 1 2 A θt ] 1 2 < for all T. Hint: Write e 1 2 I θ = E 1 2 θ e 1 4 A θ, and remember that E P[ E θ T ] 1. iv Given λ, 1, determine p λ 1, so that λp λp λ λ 1 λ = For 1 λ 2 E λ 2 pθ, and p [1, p λ ], set γλ, p = λpp λ 1 λ 2 conclude that, for any stopping time ζ,, note that E p2 λθ = e γλ,pi θ E P[ E λθ T ζ p2] E P[ e 1 2 I θt ] 2λpp λ. By taking p = p λ and applying part iii, see that this provides sufficient uniform integrability to check that E λθ t, F t, P is a martingale for each λ, 1. v By taking p = 1 in the preceding, justify 1 = E P[ E λθ T ] E P[ e 1 2 I θt ] 2λ1 λ E P [ E θ T ] λ 2 for all λ, 1. After letting λ 1, conclude first that E P[ E θ T ] = 1 and then that E θ t, F t, P is a martingale. The fact that implies Eθ t, F t, P is a martingale is known as Novikov s criterion.