Conference Agenda

The goal of this presentation is to identify a low-dimensional linear SDE that fits data from a nonlinear stochastic system and can, therefore, reproduce key features of the underlying nonlinear dynamics. Here, we exploit a rough paths perspective on SDEs driven by Brownian motion. The solutions of these equations are continuous functions of the rough path lift corresponding to the driving process. These continuous mappings can be approximated by the (truncated) Stratonovich signature of a Brownian motion using the Universal Approximation Theorem. The truncated Stratonovich signature solves a high-dimensional linear SDE. To reduce the complexity of this problem, we apply dimension reduction techniques, resulting in a reduced-order stochastic linear system.

We introduce the concept of signatures and their application in modeling. Furthermore, we explain the theory behind dimension reduction and present numerical experiments demonstrating the effectiveness of our approach.

Motivated by dynamic risk measures and conditional $g$-expectations, in this work we propose a numerical method to approximate the solution operator given by a Backward Stochastic Differential Equation (BSDE). The main ingredients for this are the Wiener chaos expansion and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it in practice using neural networks, and provide several numerical examples where we can check the accuracy of the method.