Antithetic Multilevel Particle Filters
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In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of <cit.>. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of <cit.>. We show that for a class of diffusion problems, for ϵ>0 given, that the cost to produce a mean square error (MSE) in estimation of the filter, of 𝒪(ϵ^2) is 𝒪(ϵ^-2log(ϵ)^2). In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of <cit.> has a cost of 𝒪(ϵ^-2.5) to achieve the same MSE. We support our theory with numerical results in several examples.
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