Lecture 15: Ito Сonstruction

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Lecture 15 Fall 2013 10/30/2013 Ito integral for simple processes Content. 1. Simple processes. Ito isometry 2. First 3 steps in constructing Ito integral for general processes 1 Ito integral for simple processes. Ito isometry Consider a Brownian motion Bt adopted to some filtration Ft such that (Bt , Ft ) is a strong Markov process. As an example we can take filtration generated by the Brownian motion o o itself. Our goal is to give meaning to expressions of the form Xt dBt = Xt (ω)dBt (ω), where Xt is some stochastic process which is adapted to the same filtration as Bt . We will primarily deal with the case X ∈ L2 , although it is possible to extend definitions to more general pro­ cesses using the notion o of local martingales. As in the case of usual integration, the idea is to define Xt (ω)dBt (ω) as some kind of a limit of (random) sums j Xtj (ω)(Btj+1 (ω)−Btj (ω)) and show that the limit exists in some appropri­ ate sense. As Xt we can otake all kinds of processes, including Bt itself. For exT ample we will show that 0 Bt dBt makes sense and equals (1/2)BT2 − (1/2)T . Definition 1. A process X ∈ L2 = L2 (Ω, F, (Ft ), P) is called simple if there exists a countable partition Π : 0 = t0 < · · · < tn < · · · with limn tn = ∞ such that Xt (ω) = Xtj (ω) for all t ∈ [tj , tj+1 ), j = 0, 1, 2, . . . for all ω ∈ Ω. The subspace of simple processes is denoted by L02 We assume that partition is such that tj → ∞ as j → ∞. It is important to note that we assume that the partition Π does not depend on ω. Thus not every piece-wise constant process is a simple process. Give an example of a piecewise constant process which is not simple. Note that since Xt ∈ Ft we have Xtj ∈ Ftj for each j. As an example of simple process, fix any partition Π and a process Xt ∈ L2 and consider the process X̂t (ω) defined by X̂t (ω) = 1 Xtj (ω), where tj is defined by t ∈ [tj , tj+1 ). In the definition it is important that X̂t = Xtj and not Xtj+1 . Observe that the latter is not necessarily adopted to (Ft )t∈R+ . Given a simple process X and t, define its integral by Xtj (ω)(Btj+1 (ω) − Btj (ω)) + Xtn (ω)(Bt (ω) − Btn (ω)), It (X(ω)) = 0≤j≤n−1 where n = max{j : tj ≤ t}. Observe that It (X) is an a.s. continuous function (as Bt is a.s. continuous). Theorem 1. The following properties hold for It (X) It (αX + βY ) = αIt (X) + βIt (Y ). (1) t E[It2 (X)] = E[ Xs2 ds] [Ito isometry], (2) 0 It (X) ∈ M2,c , (3) t E[(It (X) − Is (X))2 |Fs ] = E[ Xu2 du], ∀ 0 ≤ s < t ≤ T. (4) s Notice that (4) is a generalization of Ito isometry. We only prove Ito isome­ try, the proof of (4) follows along the same lines. Proof. Define tn = t for convenience. We begin with (1). Let {t1j } and {tj2 } be partitions corresponding to simple processes X and Y . Consider a partition {tj } obtained as a union of these two partitions. For each tj which belongs to the second partition but not the first define Xtj = Xt1 , where t1i is the largest i point not exceeding tj . Do a similar thing for Y . Observe that now Xt = Xtj for t ∈ [tj , tj+1 ). The linearity of Ito integral then follows straight from the definition. Now for (2) we have E[It2 (X)] = E[Xtj1 Xtj2 (Btj1 +1 − Btj1 )(Btj2 +1 − Btj2 )]. 0≤j1 ,j2 ≤n−1 When j1 < j2 we have E[Xtj1 Xtj2 (Btj1 +1 − Btj1 )(Btj2 +1 − Btj2 )] = 0 which we obtain by conditioning on Ftj2 , using the tower property and observ­ ing that all of the random variables involved except for Btj2 +1 are measurable with respect to Ftj2 (recall that Ftj1 ⊂ Ftj2 ). 2 Now when j1 = j2 = j we have E[Xt2j (Btj+1 − Btj )2 ] = E[Xt2j E[(Btj+1 − Btj )2 |Ftj ]] = E[Xt2j (tj+1 − tj )]. Combining, we obtain E[It2 (X)] = X Z t X E[Xt2j (tj+1 − tj )] = E[ Xt2j (tj+1 − tj )] = E[ Xs2 ds]. j 0 j Let us show (3). We already know that the process It (X) is continuous. From Ito isometry it follows that E[It2 (X)] < ∞. It remains to show that it is a martingale. Thus fix s < t. Define tn = t and define j0 = max{j : tj ≤ s}. X Xtj (Btj+1 − Btj )|Fs ] E[It (X)|Fs ] = E[ j≤n−1 = E[ X Xtj (Btj+1 − Btj )|Fs ] + E[Xtj0 (Bs − Btj0 )|Fs ] j≤j0 −1 + E[Xtj0 (Btj0 +1 − Bs )|Fs ] + E[ X Xtj (Btj+1 − Btj )|Fs ] j>j0 = E[ X Xtj (Btj+1 − Btj )|Fs ] + E[Xtj0 (Bs − Btj0 )|Fs ] j≤j0 −1 = Is (X). (think about justifying last two equalities). 2 Constructing Ito integral for general square integrable processes o The idea for defining Ito integral XdB for general o processes in L2o is to ap­ proximate X by simple processes X (n) and define XdB as a limit of X (n) dB, which we have already defined. For this purpose we need to show that we can indeed approximate X with simple processes appropriately. We do this in 3 steps. Step 1. Proposition 1. Suppose X ∈ L2 is an a.s. bounded continuous process in the sense ∃M s.t. P(ω : supt≥0 |Xt (ω)| ≤ M ) = 1. Then for every T > 0 there 3 exists a sequence of simple processes X n ∈ L02 such that T Z lim E[ n (Xtn − Xt )2 dt] = 0. (5) 0 Proof. Fix a sequence of partitions Πn = {tnj } of [0, T ] such that Δn = max(tnj+1 − tnj ) → 0 as n → ∞. Given process X, consider the modified process Xtn = Xtnj for all t ∈ [tnj , tnj+1 ). This process is simple and is adapted to Ft . Since X is a.s. continuous, then a.s. Xt (ω) = limn→∞ Xtn (ω) (notice that we are using leftcontinuity part of continuity). We conclude that a sequence of measurable func­ tions X n : Ω × [0, T ] → R a.s. converges to X : Ω × [0, T ] → R. On the other hand P(ω : supt≤T |Xtn (ω)| ≤ M ) = 1. Using Bounded Convergence Theo­ oT rem, the a.s. convergence extends to integrals: E[ 0 (Xtn − Xt )2 dt] → 0. Step 2. Proposition 2. Suppose X ∈ L2 is a bounded, but not necessarily continuous process: |X| ≤ M a.s. For every T > 0, there exists a sequence of a.s. bounded continuous processes Xn such that Z T lim E[ (Xtn − Xt )2 dt] = 0. (6) n 0 Proof. We use a certain ”regularization” trick to oturn a bounded process into a t bounded continuous approximation. Let Xtn = n t−1/n Xs ds. We have |X n | ≤ n(1/n)M = M and |Xtn; − Xtn | ≤ 2n|t' − t|M (verify this), implying that Xtn is a.s. bounded continuous. Since Xt is a.s. Riemann integrable, then for almost all ω, the set of discontinuity points of of Xt (ω) has measure zero and for all continuity points t by Fundamental Theorem of Calculus, we have limn→∞ Xtn (ω) = Xt (ω). We conclude that X n : Ω × [0, T ] → R converges a.s. to X on the same domain. Applying the Bounded Convergence Theorem we obtain the result. Step 3. Proposition 3. Suppose X ∈ L2 . For every T > 0 there exists a sequence of a.s. bounded processes Xn ∈ L2 such that Z T lim E[ (Xtn − Xt )2 dt] = 0. (7) n 0 4 Proof. Define X n by Xtn = Xt when −n ≤ Xt ≤ n, Xtn = −n, when Xt < −n and Xtn = n, when Xt > n. We have X n → X a.s. w.r.t both ω and t ∈ [0, T ]. Also |Xtn | ≤ |Xt | implying Z T (Xtn − Xt )2 dt ≤ 2 T Z 0 (Xtn )2 dt + 2 0 Z T Z Xt2 dt 0 T ≤4 Xt2 dt. 0 oT Since E[ 0 Xt2 dt] < ∞, then applying Dominated Convergence Theorem, we obtain the result. Exercise 1. Establish (7) by applying instead Monotone Convergence Theorem. 3 Additional reading materials • Karatzas and Shreve [1]. • Øksendal [2], Chapter III. References [1] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Springer, 1991. [2] B. Øksendal, Stochastic differential equations, Springer, 1991. 5