Solvability of BSDEs with possibly unbounded stochastic coefficients on a general weighted $L^p$ space

TL;DR

This paper establishes existence and uniqueness of solutions for multidimensional BSDEs with unbounded stochastic coefficients on a broad weighted $L^p$ space, extending previous results and applying to PDEs and utility theory.

Contribution

It introduces new methods for solving BSDEs with very general growth and unbounded coefficients in a weighted $L^p$ space, unifying and strengthening existing theories.

Findings

Proved existence and uniqueness of solutions under broad conditions.

Connected BSDE solutions to viscosity solutions of PDEs.

Provided dual representation of unbounded utility functions.

Abstract

This paper is devoted to solving a multidimensional backward stochastic differential equation (BSDE for short) with a general random terminal time $\tau$ taking values in $[0,+\infty]$. The generator $g$ of such BSDE satisfies a stochastic monotonicity condition in the state variable $y$ and a stochastic Lipschitz condition in the state variable $z$ with possibly unbounded stochastic coefficients $\mu_\cdot\in\R$ and $\nu_\cdot\in\R_+$ satisfying $\int_0^\tau (|\mu_t|+\nu^2_t) {\rm d}t<+\infty$, along with a very general growth in $y$ that is more easily verified and weaker than existing ones. Let $p>1$ be a given constant and $\rho_\cdot\geq \mu_\cdot+\frac{\theta}{2[1\wedge(p-1)]}\nu_\cdot^2$ be a given real-valued process for some constant $\theta>1$ such that $\int_0^\tau |\rho_t|{\rm d}t<+\infty$. In a general weighted $L^p$ space with a weighted factor $e^{\int_0^t \rho_r{\rm d}r}$, we establish an existence and uniqueness result for the adapted solution of previous BSDE when the terminal value satisfies an associated weighted integrability condition, broadening the scope of the process $\rho_\cdot$ in the weighted factor and thereby unifying and strengthening some corresponding existing results obtained in \citet{DarlingandPardoux1997}, \citet{Briand2003}, \citet{LiFan2024SD} and \citet{Li2025}. Some innovative ideas are presented in order to address the general weighted space and the very general growth condition. As applications, we prove the existence of viscosity solutions for parabolic and elliptic PDEs linked with previous BSDEs under some general assumptions on their nonlinear terms, and establish a dual representation of an unbounded dynamic concave utility defined on a general weighted $L^p$ space via the weighted $L^p$ solutions of previous BSDEs.