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.