Nonlinear filtering based on density approximation and deep BSDE prediction

arXiv stat.ML / 4/21/2026

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Key Points

The paper proposes an approximate Bayesian filtering method using backward stochastic differential equations (BSDEs) and a nonlinear Feynman–Kac representation of the filtering problem.
It approximates the unnormalized filtering density with the deep BSDE approach implemented via neural networks, enabling offline training and fast online inference with new observations.
The authors provide a hybrid a priori–a posteriori error bound, assuming a parabolic Hörmander condition.
Numerical experiments are used to confirm the theoretical convergence rate of the proposed method.

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.