Weak Convergence of Stochastic Integrals on Skorokhod Space in Skorokhod's J1 and M1 Topologies

TL;DR

This paper establishes criteria for the continuity of Itô integrals under weak convergence in Skorokhod's J1 and M1 topologies, with new insights into M1 convergence and applications to anomalous diffusion models.

Contribution

It provides the first comprehensive criteria for weak convergence of stochastic integrals in M1 topology and unifies existing J1 results, including counterexamples and applications.

Findings

M1 convergence criteria for stochastic integrals

Unified framework for J1 and M1 topologies

Application to anomalous diffusion models

Abstract

We provide criteria for It\^o integration to behave continuously with respect to Skorokhod's J1 and M1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.