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Multilevel particle filters for the non-linear filtering problem in continuous time. (English) Zbl 1452.62696

Summary: 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 A. Jasra et al. [SIAM J. Numer. Anal. 55, No. 6, 3068–3096 (2017; Zbl 1477.62260)] 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.

MSC:

62M20 Inference from stochastic processes and prediction
60J60 Diffusion processes
62-08 Computational methods for problems pertaining to statistics
65C05 Monte Carlo methods

Citations:

Zbl 1477.62260
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References:

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