Averaging principle of SDE with small diffusion: Moderate deviations
type: Item
instance of: scholarly article
title: Averaging principle of SDE with small diffusion: Moderate deviations
zbMATH Open document ID: 1016.60031
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The main subject of the paper is the formulation of moderate deviations for the averaging principle of a stochastic differential equation (SDE) with small diffusion, in the case when the fast random environment is an exponentially ergodic Markov process and does not depend on the Wiener process driving the SDE. The original SDE has the form \[ dX_t^{\varepsilon}=b(X_t^{\varepsilon},\xi_{t/\varepsilon})+\sqrt{\varepsilon}a(X_t^{\varepsilon},\xi_{t/\varepsilon})dW_t,\quad X_0^{\varepsilon}=x_0, \] and the large deviation estimation is obtained for the process \(\eta_t^{\varepsilon}=(X_t^{\varepsilon}-\overline x_t)/\sqrt \varepsilon h(\varepsilon)\) in the space of continuous trajectories, as \(\varepsilon\) decreases to \(0\), where \(h(\varepsilon)\) is some deviation scale, and \(d\bar{x}(t)=\bar{b}(\bar{x}(t))dt\), \(\bar{b}(x)=\mathbb{E}_{\pi}(b(x,\cdot))\). The author's strategy is to show the exponential tightness and then the local moderate deviation principle, which relies on the new method involving a conditional Schilder's theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, established by the author [Stochastic Processes Appl. 92, 287-313 (2001)].
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1906127
zbMATH Keywords: moderate deviations
zbMATH Keywords: averaging principle
zbMATH Keywords: stochastic differential equations with small diffusion
zbMATH Keywords: Markov process
zbMATH Keywords: exponential ergodicity
author: Arnaud Guillin
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1046294316
OpenAlex ID: W2054806859
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DOI: 10.1214/AOP/1046294316
description: scientific article; zbMATH DE number 1906127