Variational representations for continuous time processes
type: Item
instance of: scholarly article
title: Variational representations for continuous time processes
zbMATH Open document ID: 1231.60018
DOI: 10.1214/10-AIHP382
author: Amarjit Budhiraja
author: Paul Dupuis
author: Vasileios Maroulas
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2011-10-11T00:00:00Z
full work available at URL: https://eudml.org/doc/242555
review text: The authors prove a variational formula for functionals of a Poisson random measure and of a Brownian motion. Let \(\mathbb X\) be a locally compact metric space and \(T\in (0,\infty)\). For \(\theta>0\), let \(N^{\theta}\) be a Poisson random measure on \(\mathbb X_T=[0,T]\times \mathbb X\) with intensity \(\theta\nu_T\), where \(\nu_T=\lambda_T\otimes \nu\), \(\lambda_T\) is the Lebesgue measure on \([0,T]\) and \(\nu\) is a Radon measure on \(\mathbb X\). Given a bounded measurable functional \(F\) defined on the space of Radon measures on \(\mathbb X_T\), the authors establish a variational formula for \( \log \mathbb E [\exp(-F(N^{\theta}))] \). A similar result is proved for functionals of an infinite-dimensional Brownian motion. As an application, a large deviation principle, as \(\epsilon\downarrow 0\), is established for the solution \(\{Z^{\epsilon}(t): t\in [0,T]\}\) of a small-noise stochastic differential equation \[ dZ^{\epsilon}(t) = b(t,Z^{\epsilon}(t))dt +\sqrt{\epsilon} \sigma(t, Z^{\epsilon}(t))d\beta(t) + \int_{\mathbb X} \gamma(t, Z^{\epsilon}(t-),x) (\epsilon N^{1/\epsilon}(dtdx)-\nu_T(dtdx)). \] Here, \(Z^{\epsilon}(t)\) takes values in \(\mathbb R^d\), \(\beta\) is a \(d\)-dimensional standard Brownian motion and an initial condition \(Z^{\epsilon}(0)=z_0\) is imposed.
reviewed by: Zakhar Kabluchko
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 5956525
zbMATH Keywords: large deviations
zbMATH Keywords: variational representation
zbMATH Keywords: Poisson random measure
zbMATH Keywords: infinite-dimensional Brownian motion
zbMATH Keywords: jump-diffusions
zbMATH Keywords: stochastic differential equations
MaRDI profile type: Publication
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OpenAlex ID: W2024490169
description: scientific article; zbMATH DE number 5956525