Parametric Regularity of the Conditional Expectations via the Malliavin Calculus and Applications

Abstract

Let (W, H, μ) be the classical Wiener space and assume that $U_{\lambda } = I_{W} + u_{\lambda }$ is an adapted perturbation of identity where the perturbation u _λ is an H-valued map, defined up to μ-equivalence classes, such that its Lebesgue density $s \rightarrow \dot{ u}_{\lambda }(s)$ is almost surely adapted to the canonical filtration of the Wiener space and depending measurably on a real parameter λ. Assuming some regularity for u _λ, 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 λ of the estimation $E[\dot{u}_{\lambda }(s)\vert \mathcal{U}_{\lambda }(s)]$ and more generally of the conditional expectations of the type $E[F\mid \mathcal{U}_{\lambda }(s)]$ for nice Wiener functionals, where $(\mathcal{U}_{\lambda }(s),s \in [0,1])$ is the filtration which is generated by U _λ. 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, convexity of the entropy and the quadratic estimation error, and finally to the information theory.

Received3/28/2011; Accepted 11/22/2011; Final 2/11/2012