Multilevel Particle Filters for the Non-Linear Filtering Problem in Continuous Time

TL;DR

This paper develops a multilevel particle filter approach for continuous-time non-linear filtering, achieving better computational efficiency with comparable accuracy compared to traditional particle filters.

Contribution

It introduces a multilevel particle filter method that reduces computational cost while maintaining accuracy in continuous-time non-linear filtering problems.

Findings

MLPF achieves mean square error of O(ε^2) with cost O(ε^{-3})

Numerical simulations confirm improved efficiency over standard particle filters

Theoretical analysis supports the convergence and cost reduction claims

Abstract

In the following article we consider the numerical approximation of the non-linear filter in continuous-time, where the observations and signal follow diffusion processes. Given access to high-frequency, but discrete-time observations, we resort to a first order time discretization of the non-linear filter, followed by an Euler discretization of the signal dynamics. In order to approximate the associated discretized non-linear filter, one can use a particle filter (PF). Under assumptions, this can achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ arbitrary, such that the associated cost is $\mathcal{O}(\epsilon^{-4})$. We prove, under assumptions, that the multilevel particle filter (MLPF) of Jasra et al (2017) can achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for cost $\mathcal{O}(\epsilon^{-3})$. This is supported by numerical simulations in several examples.