关于回撤控制：鲁棒不变性与最优性
On Control of Drawdown: Robust Invariance and Optimality
浏览论文内容
中文总结 AI 辅助
本文提出多资产系统有限时域控制框架，通过鲁棒不变性定理和回撤调制参数化，实现预设回撤限制下的最优策略，并证明其期望收益优于线性时不变策略。
中文摘要 AI 辅助
缓解回撤（即财富从其运行峰值下降的幅度）是路径依赖风险控制中的一个典型问题。在本文中，我们开发了一个有限时域控制框架，在多资产随机系统中强制执行预设的最大百分比回撤限制。我们的第一个结果是精确的鲁棒不变性定理，该定理刻画了所有在支持收益下保持预设回撤限制的控制动作。我们证明每个鲁棒安全控制都允许一种“回撤调制”形式：当前回撤缓冲与可行归一化方向的乘积。这产生了鲁棒回撤安全策略的完整参数化。此外，在阶段独立收益下，我们证明对所有鲁棒安全因果策略的优化可简化为一个一维贝尔曼递归，并产生最优的鲁棒安全状态反馈策略。最后，我们刻画了满足预设回撤限制的线性时不变（LTI）增益，并证明在相同限制下，最优回撤调制实现的期望收益不低于LTI策略。当LTI策略具有正的期望单阶段净收益时，对于至少两个阶段的时域，严格期望收益改进成立。
英文摘要
Mitigating \emph{drawdown}, the decline in wealth from its running peak, presents a canonical problem in path-dependent risk control. In this paper, we develop a finite-horizon control framework that enforces a prescribed maximum percentage drawdown limit in multi-asset stochastic systems. Our first result is an exact robust-invariance theorem characterizing every control action that preserves a prescribed drawdown limit against all supported returns. We show that every robustly safe control admits a \emph{drawdown-modulated} form: the product of the current drawdown \emph{cushion} and a feasible \emph{normalized direction}. This yields a complete parameterization of robustly drawdown-safe policies. Additionally, under stagewise-independent returns, we show that optimizing over all robustly safe causal policies reduces to a one-dimensional Bellman recursion and yields an optimal robustly safe state-feedback policy. Finally, we characterize the linear time-invariant (LTI) gains satisfying a prescribed drawdown limit and prove that optimal drawdown modulation achieves no lower expected return under the same limit. Strict expected-return improvement holds for horizons of at least two stages whenever the LTI policy has positive expected one-stage net return.
