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

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 $\mathcal{O}(\hat{I}{\mu }^2)$, for $\hat{I}\mu >0$ arbitrary, such that the associated cost is $\mathcal{O}(\hat{I}{\mu }^{-4})$. We prove, under assumptions, that the multilevel particle filter (MLPF) of Jasra et al (2017) can achieve a mean square error of $\mathcal{O}(\hat{I}{\mu }^2)$, for cost $\mathcal{O}(\hat{I}{\mu }^{-3})$. This is supported by numerical simulations in several examples.