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.
JASRA, A., YU, F. et HENG, J. (2020). Multilevel Particle Filters for the Non-Linear Filtering Problem in Continuous Time. Statistics and Computing, 30, pp. 1381-1402.