Weak convergence of stochastic integrals

2023/01/15 by Kenneth H. Karlsen, Peter H. C. Pang, Karlsen, Kenneth H. +1
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Advanced Banach Space Theory #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2301.06096

Abstract

The convergence of stochastic integrals driven by a sequence of Wiener processes W_n→ 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 ∫₀^T V_n \rm d W_n → ∫₀^T V \rm d W, where V_n takes values in L^p([0,T];X) for some finite p≥ 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.

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