On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

TL;DR

This paper extends Galerkin approximation methods for the Zakai equation in stochastic filtering to models with point-process and mixed observations, providing theoretical convergence results and practical implementation insights.

Contribution

It extends convergence results to models with point-process observations and proposes using Hermite polynomial bases for numerical implementation.

Findings

Extended convergence results to point-process observation models.

Proposed Hermite polynomial basis for numerical Galerkin approximations.

Demonstrated effectiveness through a numerical case study.

Abstract

This paper studies Galerkin approximations applied to the Zakai equation of stochastic filtering. The basic idea of this approach is to project the infinite-dimensional Zakai equation onto some finite-dimensional subspace generated by smooth basis functions; this leads to a finite-dimensional system of stochastic differential equations that can be solved numerically. The contribution of the paper is twofold. On the theoretical side, existing convergence results are extended to filtering models with observations of point-process or mixed type. On the applied side, various issues related to the numerical implementation of the method are considered; in particular, we propose to work with a subspace that is constructed from a basis of Hermite polynomials. The paper closes with a numerical case study.