https://hal.inria.fr/hal-00912048Florchinger, PatrickPatrickFlorchingerLMAM - Laboratoire de Mathématiques et Applications de Metz - UPVM - Université Paul Verlaine - Metz - CNRS - Centre National de la Recherche ScientifiqueLe Gland, FrançoisFrançoisLe GlandMEFISTO - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en AutomatiqueTime discretization of the Zakai equation for diffusion processes observed in correlated noiseHAL CCSD1991[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Le Gland, Francois2013-12-02 00:59:202022-02-04 03:34:322013-12-02 00:59:20enJournal articles10.1080/174425091088337041A 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 and Rozovskii and by Kunita. 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 √δ is proved for the overall numerical scheme. This has been proved to be the best possible error estimate by Elliott and Glowinski.