Multilevel particle filters for the non-linear filtering problem in continuous time

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. 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 of Jasra et al. (SIAM J Numer Anal 55:3068–3096, 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.