I will start the talk by discussing the key subtleties of the theory of weak convergence for stochastic integrals in Skorokhod space, as developed by Jakubowski, Memin & Pages (PTRF, ’89) and Kurtz & Protter (AOP, ’91). While the theory appears both highly elegant and surprisingly powerful in the classical setting of Skorokhod’s J1-topology, things change drastically when looking at coarser topologies. Jakubowski (AOP, ’96) manages to give a strikingly elegant statement for a rather coarse non-Skorokhod topology, but, unlike the J1 setting, it seems that elegance and power no longer go hand-in-hand. Hoping to close part of this gap, I will present some recent results, both positive and negative, on what can be said for Skorokhod’s M1 topology. This topology is coarser than J1 and has recently seen a surge of interest in the applied probability literature. More generally, the aim is twofold: to make the general theory more transparent and to contribute concrete verifiable criteria for convergence in both the J1 and M1 topologies. At its core, we rely on a combination of ideas from Jakubowski (AOP, ’96) and Kurtz \& Protter (AOP, ’91), but we make several new technical contributions and derive conditions that can be of practical interest. In particular, I will briefly discuss some illuminating applications to co-integrating regression in econometrics as well as pricing and hedging for sub-diffusive models in mathematical finance. Both applications are connected to the notion of continuous-time random walks with correlated jump sizes.
The talk is based on joint work with Fabrice Wunderlich (Oxford).