Andreas Søjmark - Convergence of stochastic integrals in Skorokhod's J1 and M1 topologies (in-person)
|Dates:||26 October 2022|
|Times:||15:00 - 16:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
Andreas Søjmark (LSE) will speak in the Probability seminar.
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 (1989) and Kurtz & Protter (1991). While the theory appears both highly elegant and surprisingly powerful in the setting of Skorokhod's J1-topology, things change drastically when looking at coarser topologies. Jakubowski (1996) 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 a part of this gap, I will present some recent results, both positive and negative, on what can be said for Skorokhod's M1 topology, which is coarser than J1 and has recently seen a surge of interest in the applied probability literature. The aim is both to make the general theory more transparent and to contribute concrete verifiable criteria for convergence in the M1 setting. At its core, we rely on a combination of ideas from Jakubowski (1996) and Kurtz & Protter (1991), 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, ruin theory with investment in insurance mathematics, and pricing and hedging for sub-diffusive models in mathematical finance. The common attribute will be the presence of stochastic integrals against so-called continuous-time random walks with correlated jump sizes. The talk is based on joint work with Fabrice Wunderlich (Oxford).
Travel and Contact Information
Frank Adams Room 2
Alan Turing Building