A new numerical scheme for the Zaka\"i equation

TL;DR

This paper introduces a novel numerical scheme for approximating solutions to the Zakai equation, combining stochastic representation and quantization methods to improve non-linear filtering computations.

Contribution

It presents a new approximation method for the Zakai equation using a stochastic representation and a quantization approach based on the diffusion process.

Findings

Effective approximation of the Zakai equation demonstrated

Utilizes a stochastic representation involving observation process

Employs a quantization method based on the diffusion process

Abstract

The aim of this paper is to propose a new method for numerical approximations of the solution of the linear stochastic partial differential equation arising in non-linear filtering problems: the Zaka\"i equation. The approximation scheme is based on a representation of the solution of the Zaka\"i equation involving a stochastic part arising from the observation process and a deterministic partial differential equation in which are involved only the parameters of the signal process. We may then employ a dynamic programming principle in order to write down an approximation of this partial differential equation. A quantization method based on the underlying diffusion process (which is a not the signal itself) is used.