Computational and Applied Mathematics seminar

Abstract: In this talk, I present a new approach to the nonlinear filtering problem based on the deep Backward SDE (BSDE) method. We begin by formulating a system of equations that captures the classical prediction–update steps in Bayesian filtering. While the update step is tractable up to a normalising constant, the focus is on approximating the prediction step, which involves evolving the prediction density over time. When the hidden state follows a Stochastic Differential Equation (SDE), the prediction density satisfies the Fokker—Planck equation, a Partial Differential Equation (PDE). We solve this PDE using a probabilistic BSDE representation, which we approximate through an optimisation scheme involving neural networks, stochastic gradient descent, and the Euler—Maruyama method. The approach is demonstrated on numerical examples, and we numerically examine its strong convergence rate.