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Unformatted text preview: Chapter 20 More on Stochastic Differential Equations Section 20.1 shows that the solutions of SDEs are diffusions, and how to find their generators. Our previous work on Feller processes and martingale problems pays off here. Some other basic properties of solutions are sketched, too. Section 20.2 explains the forward and backward equations associated with a diffusion (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. 1618). 20.1 Solutions of SDEs are Diffusions Solutions of SDEs are diffusions: 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 It o process, so it is non-anticipating by Lemma 198. The adaptation for non-random initial conditions follows similarly. (Infor- 123 CHAPTER 20. MORE ON SDES 124 mally: theres 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 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 , 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 coefficients are outside the differential operators....
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06