Summary
A convenient method for proving weak convergence of a sequence of non-Markovian processesx ε(·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.
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Brown University, Divisions of Applied Mathematics and Engineering and Lefschetz Center for Dynamical Systems. Research supported in part by the Air Force Office of Scientific Research under AFOSR AF-76-3063, by the National Science Foundation under NSF Eng. 73-03846A03 and in part by the Office of Naval Research ONR N00014-76-C-0279 P003
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Kushner, H.J. A martingale method for the convergence of a sequence of processes to a jump-diffusion process. Z. Wahrscheinlichkeitstheorie verw Gebiete 53, 207–219 (1980). https://doi.org/10.1007/BF01013317
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DOI: https://doi.org/10.1007/BF01013317