A Bismut-Elworthy formula for BSDEs with degenerate noise

TL;DR

This paper extends the Bismut-Elworthy formula to stochastic differential equations with possibly degenerate noise, including infinite-dimensional cases, and explores nonlinear versions relevant to stochastic wave equations.

Contribution

It derives a Bismut-Elworthy formula under weaker assumptions than non-degeneracy, applicable to infinite-dimensional SDEs and nonlinear BSDEs.

Findings

Derived a gradient estimate for transition semigroups with degenerate noise

Extended Bismut-Elworthy formula to nonlinear BSDEs

Applicable to stochastic wave and damped wave equations

Abstract

In this paper we derive a Bismut-Elworthy formula under assumptions weaker than the non degeneracy of the noise. By Bismut-Elworthy formula we mean a gradient type estimate on the transition semigroup of a stochastic differential equation in a possibly infinite dimensional Hilbert space. We also consider a nonlinear version of the Bismut formula for a backward stochastic differential equation, in analogy to what is done in \cite{futeBismut}, where a non-degenerate noise is considered. Our study is motivated by applications to stochastic wave equations and to stochastic damped wave equation.