Smoothness of the unnormalized conditional measures of stochastic nonlinear filtering

We prove that the unnormalized conditional measures of stochastic nonlinear filtering have smooth densities when the infinitesimal generator of the state process satisfies H??rmander-like nondegeneracy conditions and when certain LP estimates can be proved. Our results are obtained by application of Malliavin´s calculus. Examples that satisfy the conditions of our main theorem are given.