Nonlinear filtering for stochastic systems with fixed delay: approximation by a modified Milstein scheme

In this paper we study approximation schemes for a nonlinear filtering problem of a partially observed diffusive system when the state process is the solution of a stochastic delay diffusion equation with a constant time lag τ and the observation process is a noisy function of the state. The approximating state is the linear interpolation of a modified Milstein scheme, which is asymptotically optimal with respect to the mean square -error within the class of all pathwise approximations based on equidistant observations of the driving Brownian motion. Upper bounds for the error of the filter approximations are computed. Some other discretization schemes for the state process are also considered.