Backward stochastic partial differential equations with quadratic growth

TL;DR

This paper investigates the existence and uniqueness of weak solutions to backward stochastic partial differential equations with quadratic growth, relevant for non-Markovian stochastic control problems with quadratic cost functionals.

Contribution

It establishes theoretical results for quadratic BSPDEs with nonhomogeneous terms, expanding understanding of their solutions in complex stochastic control contexts.

Findings

Proves existence and uniqueness of weak solutions for quadratic BSPDEs.

Connects quadratic BSPDEs to non-Markovian stochastic control problems.

Provides a framework for analyzing value functions in quadratic cost settings.

Abstract

This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of the first unknown and the second unknown. As an example, we consider a non-Markovian stochastic optimal control problem with cost functional formulated by a quadratic BSDE, where the corresponding value function satisfies the above quadratic BSPDE.