AI 中文总结
研究连续时间投资组合优化问题，投资者相对不可复制基准评估，引入基准相对回撤持续时间标准，通过推导方程、建立定理等，得到基于投影的最优控制特征及相关结果，为基准跟踪提供新方案和风险管理新结构。
AI 中文摘要
我们研究了一个连续时间的投资组合优化问题，其中投资者相对于一个不可复制的基准进行评估，并试图控制基准相对表现不佳的持续性。我们引入了一个基准相对回撤持续时间标准，该标准对在不利的基准相对表现状态下花费的预期贴现时间进行惩罚。尽管基准相对回撤会导致路径依赖，但我们表明该问题允许一维马尔可夫表示，并推导了相关的汉密尔顿-雅可比-贝尔曼方程。我们获得了基于显式投影的最优反馈控制特征，建立了验证定理，并确定了相关闭环反射扩散允许唯一强解的几何设置。我们的结果为经典基准跟踪公式提供了一个易于处理的面向下行风险的替代方案，并揭示了一种基于投影的基准相对风险管理新控制结构。
英文摘要
We study a continuous-time portfolio optimization problem in which an investor is evaluated relative to a non-replicable benchmark and seeks to control the persistence of benchmark-relative underperformance. We introduce a benchmark-relative drawdown-duration criterion that penalizes the expected discounted time spent in unfavorable benchmark-relative performance states. Despite the path dependence induced by benchmark-relative drawdowns, we show that the problem admits a one-dimensional Markovian representation and derive the associated Hamilton-Jacobi-Bellman equation. We obtain an explicit projection-based characterization of the optimal feedback control, establish a verification theorem, and identify geometric settings under which the associated closed-loop reflected diffusion admits a unique strong solution. Our results provide a tractable downside-risk-oriented alternative to classical benchmark-tracking formulations and reveal a novel projection-based control structure for benchmark-relative risk management.
