Weighted $L^p$ solutions of scalar BSDEs with general unbounded stochastic coefficients

Abstract

This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs in short) with a general random terminal time $\tau$ taking values in the extended nonnegative real numbers. The generator $g$ of BSDEs satisfies some stochastic growth/continuity conditions in the state variables $(y,z)$, featuring unbounded stochastic coefficients $\mu_\cdot\in\R$ and $\nu_\cdot\in\R_+$ satisfying $\int_0^\tau (|\mu_t|+\nu^2_t) {\rm d}t<+\infty$. For any given real $p>1$, let $\rho_\cdot\geq \mu_\cdot+\frac{\theta}{2(p-1)}\nu_\cdot^2$ (instead of $\rho_\cdot\geq\mu_\cdot+\frac{\theta}{2[1\wedge(p-1)]}\nu_\cdot^2$ used in Zhang, Li, Hu and Fan [2026, arXiv:2603.13873v1]) be a real-valued process for some constant $\theta>1$ such that $\int_0^\tau |\rho_t|{\rm d}t<+\infty$. We work within a weighted $L^p$ space with the weighting factor $e^{\int_0^t \rho_r{\rm d}r}$. Within this framework, we establish several innovative results on the weighted $L^p$ solutions of BSDEs: an existence result, an existence and uniqueness result, an existence and uniqueness result of the minimal (maximal) solution, and two comparison theorems. These findings unify and improve some existing results. Some novel ideas are employed to address the challenges posed by general unbounded stochastic coefficients and general weighted spaces.