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From Semi-Infinite Constraints to Structured Robust Policies: Optimal Gain Selection for Financial Systems

Chung-Han Hsieh

TL;DR

The paper tackles robust gain selection for financial trading under semi-infinite uncertainty by introducing a double linear policy that splits capital between long and short components. It proves that robust positivity constraints reduce to two structured policies—the balanced and complementary—enabling an explicit, tractable solution and a graphical method to compute optimal gains. The approach generalizes mean-variance optimization by incorporating worst-case robustness across uncertain return parameters and provides proof of existence and (often) uniqueness of the optimal solution. Empirical results, including Monte Carlo tests and rolling-horizon analyses on historical data, demonstrate improved risk-adjusted performance and downside protection compared with non-robust strategies. The framework offers a scalable, interpretable toolkit for robust trading under uncertainty with potential extensions to dependent returns and transaction costs.

Abstract

This paper studies the robust optimal gain selection problem for financial trading systems, formulated within a \emph{double linear policy} framework, which allocates capital across long and short positions. The key objective is to guarantee \emph{robust positive expected} (RPE) profits uniformly across a range of uncertain market conditions while ensuring risk control. This problem leads to a robust optimization formulation with \emph{semi-infinite} constraints, where the uncertainty is modeled by a bounded set of possible return parameters. We address this by transforming semi-infinite constraints into structured policies -- the \emph{balanced} policy and the \emph{complementary} policy -- which enable explicit characterization of the optimal solution. Additionally, we propose a novel graphical approach to efficiently solve the robust gain selection problem, drastically reducing computational complexity. Empirical validation on historical stock price data demonstrates superior performance in terms of risk-adjusted returns and downside risk compared to conventional strategies. This framework generalizes classical mean-variance optimization by incorporating robustness considerations, offering a systematic and efficient solution for robust trading under uncertainty.

From Semi-Infinite Constraints to Structured Robust Policies: Optimal Gain Selection for Financial Systems

TL;DR

The paper tackles robust gain selection for financial trading under semi-infinite uncertainty by introducing a double linear policy that splits capital between long and short components. It proves that robust positivity constraints reduce to two structured policies—the balanced and complementary—enabling an explicit, tractable solution and a graphical method to compute optimal gains. The approach generalizes mean-variance optimization by incorporating worst-case robustness across uncertain return parameters and provides proof of existence and (often) uniqueness of the optimal solution. Empirical results, including Monte Carlo tests and rolling-horizon analyses on historical data, demonstrate improved risk-adjusted performance and downside protection compared with non-robust strategies. The framework offers a scalable, interpretable toolkit for robust trading under uncertainty with potential extensions to dependent returns and transaction costs.

Abstract

This paper studies the robust optimal gain selection problem for financial trading systems, formulated within a \emph{double linear policy} framework, which allocates capital across long and short positions. The key objective is to guarantee \emph{robust positive expected} (RPE) profits uniformly across a range of uncertain market conditions while ensuring risk control. This problem leads to a robust optimization formulation with \emph{semi-infinite} constraints, where the uncertainty is modeled by a bounded set of possible return parameters. We address this by transforming semi-infinite constraints into structured policies -- the \emph{balanced} policy and the \emph{complementary} policy -- which enable explicit characterization of the optimal solution. Additionally, we propose a novel graphical approach to efficiently solve the robust gain selection problem, drastically reducing computational complexity. Empirical validation on historical stock price data demonstrates superior performance in terms of risk-adjusted returns and downside risk compared to conventional strategies. This framework generalizes classical mean-variance optimization by incorporating robustness considerations, offering a systematic and efficient solution for robust trading under uncertainty.
Paper Structure (23 sections, 11 theorems, 61 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 11 theorems, 61 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. The Double Linear Policy and System Dynamics
  4. Cumulative Gain and Robust Positive Expectation
  5. The Robust Optimal Gain Selection Problem
  6. Robust Positivity and Structural Results
  7. Expected Gain and Variance
  8. Characterization of Robust Positive Expectation
  9. Monotonicity Lemmas
  10. Solving Robust Optimal Gain Selection
  11. Existence and Uniqueness
  12. Graphical Approach to Finding the Optimum
  13. Parameterization for the Balanced Policy
  14. Parameterization for the Complementary Policy
  15. Merging Balanced and Complementary Policies
  16. ...and 8 more sections

Key Result

Lemma 2.3

For stage $k=0,1,\dots$, the double linear policy $\pi(k)$ leads to all-time state positivity with probability one; i.e., $\mathbb{P}(V(k) >0) = 1.$ Moreover, the double linear policy is cash-financed, meaning that $| \pi(k) | \leq V(k)$ for all $K_i \in \mathcal{K}$ with $i \in \{L, S\}$ and $k \ge

Figures (6)

  • Figure 1: Block Diagram of the Double Linear Policy Scheme.
  • Figure 2: Uneven $\alpha$ May Lead to Failing RPE. A Visualization of $\mathbb{E}[ { \mathcal{G}}_k ( \tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{2}; X )]$.
  • Figure 3: Efficient Frontiers on Mean-Standard Deviation Plane: Double Linear Policy Versus Standard Linear Feedback.
  • Figure 4: Mean-Standard Deviation Curves: Balanced Policy and Complementary Policy.
  • Figure 5: Out-of-Sample Trading Performance with TSLA from 2015 to 2025 Using Rolling Optimization with a 60-Day Window Size: Upper Panel Shows Portfolio Value Over Time; Lower Panel Depicts Time-Varying Optimal Triples for the Double Linear Policy.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: State Positivity and Cash-Finance
  • proof
  • Definition 2.4: RPE with Risk Control
  • Remark 2.5
  • Remark 2.7
  • Lemma 3.1: Nominal Expected Gain and Variance
  • proof
  • Remark 3.2
  • ...and 30 more