# A nonlinear Bismut-Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces

## TL;DR

This paper develops a nonlinear Bismut-Elworthy formula for quadratic growth BSDEs in Banach spaces, enabling solutions to complex HJB equations and stochastic control problems in high-dimensional settings.

## Contribution

It extends the Bismut-Elworthy formula to cases with unbounded pseudo-inverse diffusion operators and quadratic growth generators in Banach spaces.

## Key findings

- Derived a Bismut-Elworthy formula for quadratic BSDEs in Banach spaces.
- Applied the formula to solve semilinear Kolmogorov equations with quadratic growth.
- Addressed stochastic control problems with quadratic cost functions.

## Abstract

We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut-Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to $Z$. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations for the unknown $v$, with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth.