Uniform large deviations for multivalued stochastic differential equations with Poisson jumps
type: Item
instance of: scholarly article
title: Uniform large deviations for multivalued stochastic differential equations with Poisson jumps
zbMATH Open document ID: 1230.60062
DOI: 10.1215/21562261-1299891
author: Jing Wu
published in: Kyoto Journal of Mathematics
publication date: 2011-10-21T00:00:00Z
review text: The paper deals with multivalued stochastic differential equations driven by countably many Wiener processes and a Poisson random measure of the form \[ \begin{multlined} dX^\epsilon(t)\in -A(X^\epsilon(t))dt+b(X^\epsilon(t))dt+\sqrt{\epsilon}\sigma(X^\epsilon(t))dW(t)\\ +\int_\mathbb{Y}\gamma(X^\epsilon(t-),y)\bigl(\epsilon N^{\epsilon^{-1}}(dy,dt)-\nu(dy)dt\bigr)\end{multlined} \] with initial condition \(X(0)=x_0\in \overline{D(A)}\subset\mathbb{R}^d\). Here, \(\epsilon\) is a small parameter; \(A:\mathbb{R}^d\to 2^{\mathbb{R}^d}\) is a multivalued maximal monotone operator with domain \(D(A)\); the coefficients \(b,\,\sigma,\,\gamma\) are Lipschitz continuous vector fields; \(W\) is an sequence of independent Brownian motions; and \(N\) is a Poisson random measure with compensator \(\nu(dy)dt\), independent of \(W\) and a locally compact Polish space \(\mathbb{Y}\) for the marks. In the first part of the paper, necessary technical aspects and notations regarding multivalued stochastic differential equations and the Laplace principle are presented. Then, in the main part of the paper the uniform large deviation principle is proved for perturbed equations of the above form employing a variational representation of bounded measurable functionals of a Poisson random measure and an infinite-dimensional Brownian motion. The example of a reflected stochastic differential equation with Poisson jumps concludes the paper.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 5960763
zbMATH Keywords: multivalued stochastic differential equations
zbMATH Keywords: Poisson random measure
zbMATH Keywords: large deviation principle
zbMATH Keywords: stochastic differential equations with reflection
reviewed by: Martin Georg Riedler
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/21562261-1299891
OpenAlex ID: W2077100773
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description: scientific article; zbMATH DE number 5960763