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

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2475-A2502, August 2024.
Abstract. In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes. 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 Lemaire, Thieullen, and Thomas [Adv. Appl. Probab., 52 (2020), pp. 138–172], 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 parameters of the algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.