On Control of Drawdown: Robust Invariance and Optimality

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.