AI 中文总结
本文针对无限时域下带非线性评估的最优停止问题，基于百慕大策略证明动态规划原理、ε-最优停时与最优停时的存在性，给出Doob型收敛结果并提供BSDEs相关示例。
AI 中文摘要
本文研究无限时域、非负收益及由两个指标S和τ索引的非线性评估ρ_{S,τ}下的最优停止问题，其中S为评估时间，τ为收益披露时间，智能体的停止策略被限定在所谓百慕大停时集Θ中。在对非线性评估ρ和收益的适当假设下，证明该框架满足动态规划原理；研究了ε-最优停时与最优停时的存在性，得出ε-最优停时存在，且价值族首次触及收益的时刻为最优当且仅当该时刻有限的结论；还针对仅依赖第一指标的ρ_{S,τ}=ρ_S的非负(Θ,ρ)-上鞅，给出Doob型收敛结果；并提供了一个来自无限时域倒向随机微分方程（BSDEs）的示例。
英文摘要
In this paper, we consider an optimal stopping problem with infinite horizon, non-negative pay-offs and non-linear evaluations $ρ_{S,τ}$ indexed by two indices: $S$ and $τ$, where $S$ is the time of evaluation and $τ$ is the time when the pay-off is revealed. The agent's stopping strategies are constrained to be in the set of so-called Bermudan stopping times $Θ$. Under suitable assumptions on the non-linear evaluations $ρ$ and on the pay-off, we show that a dynamic programming principle holds in this framework. We investigate the existence of $\varepsilon$-optimal stopping times, as well as the existence of optimal stopping times. We show that an $\varepsilon$-optimal stopping time exists. We also prove that the first time when the value family hits the pay-off is optimal if and only if it is finite. We also provide Doob's type convergence for non-negative \emph{$(Θ, ρ)$}-supermartingales in the case where $ρ_{S,τ}=ρ_S$ depends on the first index only. We provide an example from BSDEs with infinite horizon.
