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February 2016 Rough path recursions and diffusion approximations
David Kelly
Ann. Appl. Probab. 26(1): 425-461 (February 2016). DOI: 10.1214/15-AAP1096

Abstract

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.

Citation

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David Kelly. "Rough path recursions and diffusion approximations." Ann. Appl. Probab. 26 (1) 425 - 461, February 2016. https://doi.org/10.1214/15-AAP1096

Information

Received: 1 February 2014; Revised: 1 December 2014; Published: February 2016
First available in Project Euclid: 5 January 2016

zbMATH: 1335.60093
MathSciNet: MR3449323
Digital Object Identifier: 10.1214/15-AAP1096

Subjects:
Primary: 39A50, 60H05, 60H10, 60H35

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 1 • February 2016
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