Time-discretization of the zakai equation for diffusion processes observed in correlated noise

A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional density of a diffusion process observed in white-noise. The case where the observation noise and the state noise are correlated, is considered. The numerical scheme is based on a Trotter-like product formula, which exhibits prediction and correction steps, and for which an error estimate of order δ is proved, where δ is the time discretization step. The correction step is associated with a degenerate second-order stochastic PDE, for which a representation result in terms of stochastic characteristics has been proved by Krylov-Rozovskii [13] and Kunita [15,17]. A discretization scheme is then provided to approximate these stochastic characteristics. Under the additional assumption that the correlation coefficient is constant, an error estimate of order is proved for the overall numerical scheme. This has been proved to be the best possible error estimate by Elliott-Glowinski [7].