lecture-15 - Chapter 15 Convergence of Feller Processes...

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Unformatted text preview: Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller pro- cesses to a limiting process. Section 15.1 lays some ground work concerning weak convergence of processes with cadlag sample paths. Section 15.2 states and proves the central theorem about the convergence of sequences of Feller processes. Section 15.3 examines a particularly important special case, the approximation of ordinary differential equations by pure-jump Markov processes. 15.1 Weak Convergence of Processes with Cad- lag Paths (The Skorokhod Topology) Recall that a sequence of random variables X 1 , X 2 , . . . converges in distribution on X , or weakly converges on X , X n d X , if and only if E [ f ( X n )] E [ f ( X )], for all bounded, continuous functions f . This is still true when X n are ran- dom functions, i.e., stochastic processes, only now the relevant functions f are functionals of the sample paths. Definition 161 (Convergence in Finite-Dimensional Distribution) Random processes X n on T converge in finite-dimensional distribution on X , X n fd X , when, J Fin( T ) , X n ( J ) d X ( J ) . Proposition 162 Convergence in finite-dimensional distribution is necessary but not sufficient for convergence in distribution. 80 CHAPTER 15. CONVERGENCE OF FELLER PROCESSES 81 Proof: Necessity is obvious: the coordinate projections t are continuous func- tionals of the sample path, so they must converge if the distributions converge. Insufficiency stems from the problem that, even if a sequence of X n all have sample paths in some set U , the limiting process might not: recall our example (78) of the version of the Wiener process with unmeasurable suprema. Definition 163 (The Space D) By D ( T, ) we denote the space of all cadlag functions from T to . By default, D will mean D ( R + , ) . D admits of multiple topologies. For most purposes, the most convenient one is the Skorokhod topology , a.k.a. the J 1 topology or the Skorokhod J 1 topology , which makes D () a complete separable metric space when is itself complete and separable. (See Appendix A2 of Kallenberg.) For our purposes, we need only the following notion and theorem....
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lecture-15 - Chapter 15 Convergence of Feller Processes...

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