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# lecture-20 - Chapter 20 More on Sto chastic Dierential...

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Chapter 20 More on Stochastic Di ff erential Equations Section 20.1 shows that the solutions of SDEs are di ff usions, and how to find their generators. Our previous work on Feller processes and martingale problems pays o ff here. Some other basic properties of solutions are sketched, too. Section 20.2 explains the “forward” and “backward” equations associated with a di ff usion (or other Feller process). We get our first taste of finding invariant distributions by looking for stationary solutions of the forward equation. Section 20.3 makes sense of the idea of white noise. This topic will be continued in the next lecture, forming one of the bridges to ergodic theory. For the rest of this lecture, whenever I say “an SDE”, I mean “an SDE satisfying the requirements of the existence and uniqueness theorem”, either Theorem 215 (in one dimension) or Theorem 216 (in multiple dimensions). And when I say “a solution”, I mean “a strong solution”. If you are really curious about what has to be changed to accommodate weak solutions, see Rogers and Williams (2000, ch. V, sec. 16–18). 20.1 Solutions of SDEs are Di ff usions Solutions of SDEs are di ff usions: i.e., continuous, homogeneous strong Markov processes. Theorem 217 The solution of an SDE is non-anticipating, and has a version with continuous sample paths. If X (0) = x is fixed, then X ( t ) is F W t -adapted. Proof: Every solution is an Itˆ o process, so it is non-anticipating by Lemma 198. The adaptation for non-random initial conditions follows similarly. (Infor- 123

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CHAPTER 20. MORE ON SDES 124 mally: there’s nothing else for it to depend on.) In the proof of the existence of solutions, each of the successive approximations is continuous, and we bound the maximum deviation over time, so the solution must be continuous too. Theorem 218 Let X x be the process solving a one-dimensional SDE with non- random initial condition X (0) = x . Then X x forms a homogeneous strong Markov family. Proof: By Exercise 19.4, for every C 2 function f , f ( X ( t )) - f ( X (0)) - t 0 a ( X ( s )) f x ( X ( s )) + 1 2 b 2 ( X ( s )) 2 f x 2 ( X ( s )) ds (20.1) is a martingale. Hence, for every x 0 , there is a unique, continuous solution to the martingale problem with operator G = a ( x ) x + 1 2 b 2 ( x ) 2 x 2 and function class D = C 2 . Since the process is continuous, it is also cadlag. Therefore, by Theorem 137, X is a homogeneous strong Markov family, whose generator equals G on C 2 . Similarly, for a multi-dimensional SDE, where a is a vector and b is a matrix, the generator extends 1 a i ( x ) i + 1 2 ( bb T ) ij ( x ) 2 ij . Notice that the coe ffi cients are outside the di ff erential operators. Corollary 219 Solutions of SDEs are di ff usions. Proof: Obvious from Theorem 218, continuity, and Definition 177. Remark: To see what it is like to try to prove this without using our more general approach, read pp. 103–114 in Øksendal (1995).
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lecture-20 - Chapter 20 More on Sto chastic Dierential...

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