Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes

TL;DR

This paper introduces a multilevel particle filter for efficiently approximating the filtering problem of partially observed piecewise deterministic Markov processes driven by ODEs, demonstrating computational advantages through theoretical bounds and numerical tests.

Contribution

It develops a novel multilevel particle filter tailored for PDMPs with ODE-driven dynamics, providing error bounds and showing computational improvements over traditional particle filters.

Findings

The MLPF achieves lower mean square error for a given computational cost.

Theoretical bounds guide parameter selection for the MLPF.

Numerical examples confirm the efficiency of the proposed method.

Abstract

In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes (PDMPs). In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in [20], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameter of that algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.