Splitting―up approximation for SPDE's and SDE's with application to nonlinear filtering
Résumé
A splitting-up approximation is introduced for diffusion processes, based on the successive composition of two stochastic flows of diffeomorphisms over a given partition of the time interval. In the case where the original diffusion is observed in correlated noise, an equation is derived for the conditional density of the approximating process. This equation is interpreted as a splitting-up approximation of the Zakai equation for the conditional density of the original diffusion process, based on the successive composition of two semigroups, and error estimates are provided using SPDE techniques. The results presented here are of general interest for the approximation of SDE's and SPDE's, independently of the filtering problem. In the context of nonlinear filtering, the main interest of splitting-up approximation is that the original Zakai equation is splitted into a second-order deterministic PDE related with the prediction step, and a degenerate second-order stochastic PDE related with the correction step. This probabilistic interpretation can be used to design further approximation schemes, e.g. in terms of approximating finite-state Markov processes.