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lecture-18 - Chapter 18 Stochastic Integrals with the...

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Chapter 18 Stochastic Integrals with the Wiener Process Section 18.1 addresses an issue which came up in the last lecture, namely the martingale characterization of the Wiener process. Section 18.2 gives a heuristic introduction to stochastic integrals, via Euler’s method for approximating ordinary integrals. Section 18.3 gives a rigorous construction for the integral of a function with respect to a Wiener process. 18.1 Martingale Characterization of the Wiener Process Last time in lecture, I mentioned (without remembering much of the details) that there is a way of characterizing the Wiener process in terms of some mar- tingale properties. Here it is. Theorem 183 If M ( t ) is a continuous martingale, and M 2 ( t ) - t is also a martingale, then M ( t ) is a Wiener process. There are some very clean proofs of this theorem 1 — but they require us to use stochastic calculus! Doob (1953, pp. 384 ff ) gives a proof which does not, however. The details of his proof are messy, but the basic idea is to get the central limit theorem to apply, using the martingale property of M 2 ( t ) - t to get the variance to grow linearly with time and to get independent increments, and then seeing that between any two times t 1 and t 2 , we can fit arbitrarily many little increments so we can use the CLT. We will return to this result as an illustration of the stochastic calculus. 1 See especially Ethier and Kurtz (1986, Theorem 5.2.12, p. 290). 97
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CHAPTER 18. STOCHASTIC INTEGRALS 98 18.2 A Heuristic Introduction to Stochastic In- tegrals Euler’s method is perhaps the most basic method for numerically approximating integrals. If we want to evaluate I ( x ) b a x ( t ) dt , then we pick n intervals of time, with boundaries a = t 0 < t 1 < . . . t n = b , and set I n ( x ) = n i =1 x ( t i - 1 ) ( t i - t i - 1 ) Then I n ( x ) I ( x ), if x is well-behaved and the length of the largest interval 0. If we want to evaluate t = b t = a x ( t ) dw , where w is another function of t , the natural thing to do is to get the derivative of w , w , replace the integrand by x ( t ) w ( t ), and perform the integral with respect to t . The approximating sums are then n i =1 x ( t i - 1 ) w ( t i - 1 ) ( t i - t i - 1 ) (18.1) Alternately, we could, if w ( t ) is nice enough, approximate the integral by n i =1 x ( t i - 1 ) ( w ( t i ) - w ( t i - 1 )) (18.2) (You may be more familiar with using Euler’s method to solve ODEs, dx/dt = f ( x ). Then one generally picks a Δ t , and iterates x ( t + Δ t ) = x ( t ) + f ( x ) Δ t (18.3) from the initial condition x ( t 0 ) = x 0 , and uses linear interpolation to get a continuous, almost-everywhere-di ff erentiable curve. Remarkably enough, this converges on the actual solution as Δ t shrinks (Arnol’d, 1973).) Let’s try to carry all this over to random functions of time X ( t ) and W ( t ).
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