Weak convergence of stochastic integrals on Skorokhod space in Skorokhod's J 1 and M 1 topologies.

We provide criteria for Itô integration to behave continuously with respect to Skorokhod's J 1 and M 1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M 1 setting and unify existing theories in the J 1 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, M 1 tightness in fact implies J 1 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 M 1 and J 1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.