带一般无界随机系数的标量倒向随机微分方程的加权$L^p$解
Weighted $L^p$ solutions of scalar BSDEs with general unbounded stochastic coefficients
AI总结：
本文针对带一般无界随机系数和随机终端时间的一维BSDEs，在更宽松的加权$L^p$空间框架下，建立了解的存在性、唯一性、极值解及比较定理等新结果，统一并改进了已有结论。
AI中文摘要：
本文致力于求解取值于扩展非负实数集的一般随机终端时间$\tau$下的一维倒向随机微分方程（简称BSDEs）。BSDEs的生成元$g$关于状态变量$(y,z)$满足若干随机增长/连续性条件，其特征为无界随机系数$\mu_\cdot\in\R$与$\nu_\cdot\in\R_+$满足$\int_0^\tau(|\mu_t|+\nu^2_t) {\rm d}t<+\infty$。对任意给定实数$p>1$，令$\rho_\cdot\geq \mu_\cdot+\frac\theta{2(p-1)}\nu_\cdot^2$（而非Zhang、Li、Hu与Fan[2026, arXiv:2603.13873v1]中使用的$\rho_\cdot\geq\mu_\cdot+\frac\theta{2[1\wedge(p-1)]}\nu_\cdot^2$）为某常数$\theta>1$对应的实值过程，满足$\int_0^\tau|\rho_t|{\rm d}t<+\infty$。我们在加权因子为$e^{\int_0^t \rho_r{\rm d}r}$的加权$L^p$空间中开展研究。在此框架下，我们建立了关于BSDEs加权$L^p$解的若干创新性结果：一个存在性结果、一个存在唯一性结果、一个最小（最大）解的存在唯一性结果，以及两个比较定理。这些发现统一并改进了部分已有结果。本文采用了一些新颖思路来应对一般无界随机系数与一般加权空间带来的挑战。
英文摘要：
This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs in short) with a general random terminal time $τ$ 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 $μ_\cdot\in\R$ and $ν_\cdot\in\R_+$ satisfying $\int_0^τ(|μ_t|+ν^2_t) {\rm d}t<+\infty$. For any given real $p>1$, let $ρ_\cdot\geq μ_\cdot+\fracθ{2(p-1)}ν_\cdot^2$ (instead of $ρ_\cdot\geqμ_\cdot+\fracθ{2[1\wedge(p-1)]}ν_\cdot^2$ used in Zhang, Li, Hu and Fan [2026, arXiv:2603.13873v1]) be a real-valued process for some constant $θ>1$ such that $\int_0^τ|ρ_t|{\rm d}t<+\infty$. We work within a weighted $L^p$ space with the weighting factor $e^{\int_0^t ρ_r{\rm d}r}$. 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.
