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

Abstract: We revisit the theory of stochastic integral convergence on Skorokhod space, providing a new comprehensive treatment under Skorokhod's J1 and M1 topologies based on a simple and tractable notion of good decompositions for the integrators. Our analysis yields new insights even in the classical J1 setting and addresses the sharpness of various key results. As regards convergence of stochastic integrals in the M1 setting, our contribution is twofold. Firstly, we derive useful sufficient conditions. Secondly, we demonstrate the limitations of general weak continuity properties for strictly M1 convergent integrators. Moreover, we provide an important structural result for local martingales on Skorokhod space, showing that their M1 tightness generally implies their J1 tightness. The practicality of our results is illustrated through novel contributions to applications from econometric theory, statistical mechanics, mathematical finance, and insurance mathematics.