2023/01/29 by Jasra, Ajay, Maama, Mohamed, Ombao, Hernando
#Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2301.12371
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 \citemlpf. 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 \citemlanti. 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(ε2) is O(ε-2log(ε)2). In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \citemlpf has a cost of O(ε-2.5) to achieve the same MSE. We support our theory with numerical results in several examples.