On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces

TL;DR

This paper extends the characterization of the path-independence of Girsanov transformations from finite-dimensional SDEs with jumps to infinite-dimensional stochastic evolution equations in Hilbert spaces, linking them to partial integro-differential equations.

Contribution

It introduces a method using Galerkin approximations to analyze path-independence in infinite-dimensional stochastic evolution equations with jumps.

Findings

Established a link between infinite-dimensional stochastic equations and partial integro-differential equations.

Extended finite-dimensional results to infinite-dimensional Hilbert space settings.

Provided a framework for analyzing Girsanov transformations in complex stochastic systems.

Abstract

Based on a recent result on characterising the path-independence of the Girsanov transformation for non-Lipschnitz stochastic differential equations (SDEs) with jumps on $R^d$, in this paper, we extend our consideration of characterising the path-indpendent property from finite-dimensional SDEs with jumps to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in the manner that one could then link the characterisation of the path-independence for finite-dimensional jump type SDEs to that for the infinite-dimensional settings. Our result provides an intrinsic link of infinite-dimensional stochastic evolution equations with jumps to infinite-dimensional partial integro-differential equations.