Weak convergence of stochastic integrals

TL;DR

This paper establishes weak convergence results for stochastic integrals driven by Wiener processes, especially when integrand sequences only converge weakly in time, which is important for analyzing SPDEs with singularities.

Contribution

It provides new weak convergence theorems for stochastic integrals with integrands converging weakly in time, extending standard methods for SPDE analysis.

Findings

Proves weak convergence of stochastic integrals under weak integrand convergence.

Addresses convergence issues in SPDEs with singular behavior.

Extends existing stochastic integral convergence theory.

Abstract

The convergence of stochastic integrals driven by a sequence of Wiener processes $W_n\to W$ (with convergence in $C_t$) is crucial in the analysis of stochastic partial differential equations (SPDEs). The convergence we focus on in this paper is of the form $\int_0^T V_n\, {\rm d} W_n \to \int_0^T V\,{\rm d} W$, where $V_n$ takes values in $L^p([0,T];X)$ for some finite $p\ge 2$ and a Banach space $X$. Standard methods do not directly apply when $V_n$ only converges weakly in the temporal variable to $V$. We provide (weak) convergence results that address the need to take limits of stochastic integrals when only weak temporal convergence is available. This is particularly relevant for SPDEs with singular behaviour.