Regularity of the conditional expectations with respect to signal to noise ratio

Abstract

Let $(W,H,\mu)$ be the classical Wiener space, assume that $U_\la=I_W+u_\la$ is an adapted perturbation of identity where the perturbation $u_\la$ is an $H$-valued map, defined up to $\mu$-equivalence classes, such that its Lebesgue density $s\to \dot{u}_\la(s)$ is almost surely adapted to the canonical filtration of the Wiener space and depending measurably on a real parameter $\la$. Assuming some regularity for $u_\la$, its Sobolev derivative and integrability of the divergence of the resolvent operator of its Sobolev derivative, we prove the almost sure and $L^p$-regularity w.r. to $\la$ of the estimation $E[\dot{u}_\la(s)|\calU_\la(s)]$ and more generally of the conditional expectations of the type $E[F\mid\calU_\la(s)]$ for nice Wiener functionals, where $(\calU_\la(s),s\in [0,1])$ is the the filtration which is generated by $U_\la$. These results are applied to prove the invertibility of the adapted perturbations of identity, hence to prove the strong existence and uniqueness of functional SDE's; convexity of the entropy and the quadratic estimation error and finally to the information theory.