Nonlinear filtering based on density approximation and deep BSDE prediction

TL;DR

This paper introduces a new Bayesian filtering method using deep BSDEs and neural networks, enabling efficient online application with theoretical error bounds and confirmed convergence.

Contribution

It presents a novel nonlinear filtering approach based on density approximation and deep BSDEs, with proven error bounds and convergence analysis.

Findings

Method trained offline for online use with new data

Theoretical error bounds under parabolic Hörmander condition

Confirmed convergence rate through numerical examples

Abstract

A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using the well-known deep BSDE method and neural networks. The method is trained offline, which means that it can be applied online with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic H\"ormander condition. The theoretical convergence rate is confirmed in two numerical examples.