Limits for parabolic partial differential equations with wide band stochastic coefficients andan application to filtering theory

Abstract

Let first-order differential operator), B\ and BL0 be functions of x and of a “wide band-widthe” random process “band width” Define by {A is an elliptic operator} Under appropriate conditions, converges weakly to a process u which solves a stochastic PDE, driven by a cylindrical Wiener process. The treatment is entirely probabilistic. Methods which the authors applied previously to the finite dimensional cases are modified and extended for the current class of problems. Anonlinear filtering problem with “wide bandwidth” observation noise is treated, and it is shown that the weakconvergence point of view provides a natural robust approximate filter with a clear statistical interpretation.