A Bismut Elworthy formula for quadratic BSDEs

TL;DR

This paper extends the Bismut-Elworthy formula to quadratic BSDEs in a Markovian setting, enabling new analytical tools for nonlinear PDEs and stochastic control problems with quadratic growth.

Contribution

It introduces a Bismut-Elworthy formula for quadratic BSDEs, broadening the scope of stochastic analysis in non-degenerate Markovian frameworks.

Findings

Derived a Bismut-Elworthy formula for quadratic growth BSDEs.

Applied the formula to semilinear Kolmogorov equations with quadratic nonlinearity.

Provided insights into stochastic control problems with quadratic costs.

Abstract

We consider a backward stochastic differential equation in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. Under non-degeneracy assumptions, we prove an analogue of the well-known Bismut-Elworty formula when the generator has quadratic growth with respect to $Z$. Applications to the solution of a semilinear Kolmogorov equation for the unknown $v$ with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous are given, as well as applications to stochastic optimal control problems with quadratic growth.