Numerical analysis of a time discretized method for nonlinear filtering problem with L\'evy process observations

TL;DR

This paper develops and analyzes a numerical method for solving nonlinear filtering problems with Le9vy process observations, demonstrating convergence and accuracy through theoretical proofs and numerical experiments.

Contribution

It introduces a splitting-up technique for the Zakai equation with Le9vy observations and proves its convergence, providing a new numerical approach for nonlinear filtering.

Findings

Half-order convergence of the splitting-up method

Finite difference scheme achieves half-order convergence

Numerical experiments confirm theoretical results

Abstract

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter solution. Then we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.