AI 中文总结
本文提出配对模糊框架，结合Girsanov模型不确定性与现金次可加风险评价，用上反射BSDE刻画最优停止值，并开发深度学习方案求解反射二次BSDE，数值实验展示贴现率与熵模糊性对美式期权停止决策的影响。
AI 中文摘要
我们研究了在概率模型和贴现率同时存在模糊性的动态风险度量下的最优停止问题。我们引入了一个配对模糊框架，将Girsanov模型不确定性与现金次可加风险评价相结合，并通过一个上反射倒向随机微分方程（BSDE）来刻画停止值。我们建立了所得停止算子的结构性质，并研究了与熵风险度量相关的二次驱动项，在几个基准情形中获得了显式停止规则。随后，我们开发了一种针对反射二次BSDE的深度学习方案。收敛性分析利用离散反射和截断将二次问题简化为全局Lipschitz系统，并将反射BSDE离散化估计与神经网络逼近误差相结合。针对美式期权的数值实验展示了贴现率和熵模糊性对停止值和执行决策的影响。
英文摘要
We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.
