December 2023 CDT Mathematics of Random Systems Workshop

14:00    Fabrice Wunderlich, CDT student at University of Oxford

Skorokhod meets Itô. For the second time.
On the Weak Convergence of Stochastic Integrals on Skorokhod Space under Skorokhod's J1 and M1 Topologies

Abstract: I will talk about 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) and 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. Moreover, I hope to convince you that our results lead to a theory of stochastic integral convergence that is much more readily accessible for many applications. As a motivation for the presented results, I will consider a Lévy process framework used in statistical mechanics, mathematical finance and econometrics. Armed with our results, it becomes a straightforward matter to derive functional limit theorems when the approximating integrals are driven by continuous-time random walks. The talk is based on joint work with Andreas Søjmark (LSE).

14:45     Bernat Bassols Cornudella, CDT student at Imperial College London

Conditioned Random Dynamics: noise-induced chaos, equilibrium measures and repellers

Abstract: The theory of conditioned random dynamics is an emerging field that provides a natural mathematical framework to study transient and absorbed (or killed) processes. In this talk we will give an overview of the tools used to analyse such systems and demonstrate the relevance of conditioned random dynamics with two examples. First, we apply this theory to understand noise-induced transition to chaos in a random logistic map (https://doi.org/10.48550/arXiv.2308.07116), describing the dynamics via a two-state compartmental model. Second, we show conditioned stochastic stability for uniformly expanding systems and characterise the so-called natural measure on repellers, key in the study of chaotic transients.

This is ongoing work with Jeroen S.W. Lamb and Matheus M. Castro at Imperial College London. 

Speaker talk:

16:00   Dr Emilio Rossi Ferrucci (University of Oxford)

Branched Itô formula and natural Itô-Stratonovich isomorphism