Deep Learning for the Nonlinear Filtering Problem: Adapting the Deep Backward Stochastic Differential Equation Method to Stochastic Filtering

Abstract

The thesis addresses the filtering problem by estimating the probability density of a latent stochastic process given noisy observations. It does so by analysing the convergence of a novel deep learning based filter, the deep backward stochastic differential equation filter. Rooted in the connection between backward stochastic differential equations and partial differential equations, the new filter is compared against classical filters including the Kalman filter, Extended Kalman filter, and a bootstrap particle filter. While particle filters can handle nonlinear, non Gaussian systems, they suffer from the curse of dimensionality, which motivates the development of new filter approximations for nonlinear and high dimensional settings. The new method is constructed by sequentially applying a deep learning based method for backward stochastic differential equations, while incorporating observations. The thesis studies the empirical strong convergence rate of the method on both the unconditional case without observations, and the conditional case with observations. It demonstrates promising results in three examples, an Ornstein–Uhlenbeck process, a bimodal stochastic differential equation, and a 4-dimensional spring-mass system. The observed convergence rate for the first two cases is approximately O(N^{-1/2}) where N is number of prediction steps between observations, in accordance with the theoretical order. For the final example it is lower and this might be due to a statistical error or an approximation error.