一般滤子空间上由右连左极增过程驱动的带不规则障碍的广义反射倒向随机微分方程
Generalized reflected BSDEs with irregular obstacles driven by RCLL increasing processes on general filtered space
AI总结：
该研究在一般滤子空间上，针对由RCLL增过程驱动的GBSDEs和GRBSDEs，建立了存在唯一性等结果，提出两种方法求解特定GRBSDEs并推广到一般利普希茨情形。
AI中文摘要：
我们在满足通常条件的一般滤子概率空间上研究广义倒向随机微分方程（GBSDEs）和广义反射倒向随机微分方程（GRBSDEs），不假设基础滤子是拟左连续的。这些方程由指定的可预测、有界、非降的右连左极（RCLL）过程\boldsymbol{A}驱动，该过程可作为可能不连续的随机时钟，我们称之为“驱动项”。我们首先建立生成元关于状态变量满足利普希茨连续的GBSDEs的先验估计、稳定性、存在性和唯一性结果。由于\boldsymbol{A}可能存在跳跃，分析通过随机指数\boldsymbol{\beta A}定义的加权空间进行。随后我们研究带可选调节下障碍的GRBSDEs，当生成元与状态变量无关时，我们提出两种互补方法：第一种依赖斯奈尔包络表示和最优停时论证，第二种基于适配障碍右跳不连续性的修正惩罚程序。之后通过适当加权巴拿赫空间中的不动点论证得到一般利普希茨情形的结果。
英文摘要：
We study generalized backward stochastic differential equations (GBSDEs) and generalized reflected backward stochastic differential equations (GRBSDEs) on a general filtered probability space satisfying the usual conditions, without assuming that the underlying filtration is quasi-left-continuous. The equations are driven by a prescribed predictable, bounded, nondecreasing RCLL process \(A\), which acts as a possibly discontinuous stochastic clock, which we call a driver. We first establish a priori estimates, stability, existence, and uniqueness results for GBSDEs whose generator is Lipschitz continuous with respect to the state variable. Since \(A\) may have jumps, the analysis is carried out in weighted spaces defined through the stochastic exponential \(\mathcal{E}(βA)\). We then investigate GRBSDEs with an optional regulated lower obstacle. When the generator is independent of the state variable, we develop two complementary approaches. The first relies on a Snell-envelope representation and optimal stopping arguments, while the second is based on a modified penalization procedure adapted to the discontinuities of the right jumps of the obstacle. The general Lipschitz case is subsequently obtained through a fixed-point argument in an appropriate weighted Banach space.
