Weak convergence of stochastic integrals

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 ${\int }_0^T{V}_n\mathrm{d}{W}_n\to {\int }_0^{T}V\mathrm{d}W$, 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.

This paper was withdrawn due to an error in the proof of the main theorem