Malliavin calculus with time dependent coefficients and application to nonlinear filtering (Q1123483)

We prove, using Malliavin calculus, that under a local Hörmander condition the solution of a stochastic differential equation with time- depending coefficients admits a \(C^{\infty}\) density with respect to Lebesgue measure. An application of this result to nonlinear filtering is developed to prove the existence of a \(C^{\infty}\) density for the filter associated with a correlated system whose observation is one dimensional with unbounded coefficients.