Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs

TL;DR

This paper develops a new theory for stochastic integration with cylindrical martingale-valued measures in Hilbert spaces, extending quadratic variation concepts to discontinuous cases, and applies it to solve SPDEs.

Contribution

It introduces a comprehensive stochastic integration framework for cylindrical martingale-valued measures with discontinuous paths, expanding existing quadratic variation theory.

Findings

Established existence and uniqueness of solutions to certain SPDEs.

Extended quadratic variation to discontinuous cylindrical martingale-valued measures.

Provided a new mathematical foundation for stochastic integration in infinite-dimensional spaces.

Abstract

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, introduced by Veraar and Yaroslavtsev (2016), to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths (this is carried out within the context of separable Banach spaces). Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces.