Small time asymptotics for stochastic evolution equations

TL;DR

This paper establishes a large deviation principle for the small time behavior of solutions to stochastic evolution equations with multiplicative noise, under certain linear and Lipschitz conditions.

Contribution

It adapts existing methods for small noise asymptotics to analyze small time asymptotics of stochastic evolution equations with multiplicative noise.

Findings

Large deviation principle derived for small time asymptotics

Applicable to equations with generators of analytic semigroups

Methodology extends Peszat's techniques to small time analysis

Abstract

We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic semigroups and Lipschitz continuity of the nonlinear drift and dispersion coefficients. Methods originally used by Szymon Peszat for the small noise asymptotics problem are adapted to solve the small time asymptotics problem.