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

TL;DR

This paper investigates the regularity of conditional expectations with respect to the signal-to-noise ratio in Wiener space, establishing conditions for their smoothness and applying results to invertibility of perturbations, SDEs, and information theory.

Contribution

It introduces new regularity results for conditional expectations in Wiener space depending on a parameter, with applications to invertibility, SDEs, and entropy.

Findings

Almost sure and L^p-regularity of conditional expectations w.r.t. parameter .

Invertibility of adapted perturbations of identity.

Applications to strong solutions of functional SDEs and entropy convexity.

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.