Convergence of Stochastic Integrals on Skorokhod Space: New Results and an Application to Ruin Theory with Risky
Seminar in Insurance and Economics
SPEAKER: Andreas Søjmark (LSE).
TITLE: Convergence of Stochastic Integrals on Skorokhod Space: New Results and an Application to Ruin Theory with Risky Investments.
ABSTRACT: I will start the talk by briefly encapsulating key aspects of the theory of weak convergence for stochastic integrals on Skorokhod space, as pioneered by Jakubowski, Memin & Pages (PTRF ’89) and Kurtz & Protter (AOP ’91). I will then proceed to discuss new results that develop this theory in two important directions. Firstly, we seek a simpler set of conditions that are easier to work with, yet apply just as broadly, and which may serve to make the subtleties of the theory more transparent. Secondly, we are interested in understanding exactly what happens when transitioning from the classical J1 topology to Skorokhod's weaker M1 topology, which has recently seen a surge in interest. As we will see, by exploring these directions, we are able to clarify some apparent confusion in the literature when it comes to applications. Moreover, I hope to convince you that our results lead to a theory of stochastic integral convergence that is much more readily accessible for future applications. As an illustration of this latter point, I will consider a Lévy process framework for ruin theory with risky investments as in Paulsen (AAP ’02). Armed with our results, it becomes a straightforward matter to derive functional limit theorems when the insurance risk and the financial returns are modelled by renewal-reward processes. The talk is based on joint work with Fabrice Wunderlich (Oxford).