Antithetic multilevel particle filters

In this paper we consider the filtering of partially observed multidimensional 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 Jasra et al. ( SIAM J. Numer. Anal. 55 (2017), 3068–3096). 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 Giles and Szpruch ( Ann. Appl. Prob. 24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for ε>0 given, the cost to produce a mean squared error (MSE) of O(ε²) in the estimation of the filter is O(ε⁻²log(ε)²) . In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of O(ε-2.5) to achieve the same MSE.