Multilevel Particle Filters for the Non-Linear Filtering Problem in Continuous Time
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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 O(ϵ^2), for ϵ>0 arbitrary, such that the associated cost is O(ϵ^-4). We prove, under assumptions, that the multilevel particle filter (MLPF) of Jasra et al (2017) can achieve a mean square error of O(ϵ^2), for cost O(ϵ^-3). This is supported by numerical simulations in several examples.
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