A small time approximation for the solution to the Zakai Equation

TL;DR

This paper introduces a new small time approximation method for the Zakai equation in nonlinear filtering, using deterministic PDEs and Wiener chaos techniques, with proven convergence rates and explicit bounds.

Contribution

It presents a novel approximation approach for the Zakai equation over short intervals, combining PDE analysis with Wiener chaos and explicit error bounds.

Findings

Approximation accuracy of order one in interval length

Explicit bounds for the approximation error

Integration of PDE and Wiener chaos methods

Abstract

We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds.