On Control of Drawdown: Robust Invariance and Optimality
Abstract
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.
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BibTeX
@misc{hsieh2026control,
title = {{On Control of Drawdown: Robust Invariance and Optimality}},
author = {Chung-Han Hsieh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23272},
eprint = {2609.23272},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23272},
}
RIS
TY - UNPB
TI - On Control of Drawdown: Robust Invariance and Optimality
AU - Chung-Han Hsieh
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23272
UR - https://arxiv.org/abs/2609.23272
AB - 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.
ER -