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January 2003 Averaging principle of SDE with small diffusion: Moderate deviations
A. Guillin
Ann. Probab. 31(1): 413-443 (January 2003). DOI: 10.1214/aop/1046294316

Abstract

Consider the following stochastic differential equation in $\mathbb{R}^d$: \begin{eqnarray*} dX^\varepsilon_t &=& b(X_t^\varepsilon,\xi_{t/\varepsilon})\,dt +\sqrt{\varepsilon} a(X_t^\varepsilon,\xi_{t/\varepsilon})\,dW_t,\\ X^\varepsilon_0 &=& x_0, \end{eqnarray*} where the random environment $(\xi_t)$ is an exponentially ergodic Markov process, independent of the Wiener process $(W_t)$, with invariant probability measure $\pi$, and $\varepsilon$ is some small parameter. In this paper we prove the moderate deviations for the averaging principle of $X^\varepsilon$, that is, deviations of $(X^\varepsilon_t)$ around its limit averaging system $(\bar x_t)$ given by %$\bar{d}$. % $d\bar x_t=\bar b(\bar x_t)\,dt$ and $\bar x_0=x_0$ where $\bar b(x)=\mathbb{E}_\pi(b(x,\cdot))$. More precisely we obtain the large deviation estimation about \[ \Bigl(\eta^\varepsilon_t={X^\varepsilon_t-\bar x_t \over \sqrt{\varepsilon} h(\varepsilon)}\Big)_{t\in[0,1]} \] in the space of continuous trajectories, as $\varepsilon$ decreases to 0, where $h(\varepsilon)$ is some deviation scale satisfying $1\ll h(\varepsilon)\ll\varepsilon^{-1/2}$. Our strategy will be first to show the exponential tightness and then the local moderate deviation principle, which relies on some new method involving a conditional Schilder's theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, previously established by the author.

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A. Guillin. "Averaging principle of SDE with small diffusion: Moderate deviations." Ann. Probab. 31 (1) 413 - 443, January 2003. https://doi.org/10.1214/aop/1046294316

Information

Published: January 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1016.60031
MathSciNet: MR1959798
Digital Object Identifier: 10.1214/aop/1046294316

Subjects:
Primary: 60F10 , 60J25 , 60J60

Keywords: averaging principle , exponential ergodicity , Markov process , Moderate deviation , SDE with small diffusion

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 1 • January 2003
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