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Dynamic Weight Optimization for Double Linear Policy: A Stochastic Model Predictive Control Approach

Tan Chin Hong, Chung-Han Hsieh

Abstract

The Double Linear Policy (DLP) framework guarantees a Robust Positive Expectation (RPE) under optimized constant-weight designs or admissible prespecified time-varying policies. However, the sequential optimization of these time-varying weights remains an open challenge. To address this gap, we propose a Stochastic Model Predictive Control (SMPC) framework. We formulate weight selection as a receding-horizon optimal control problem that explicitly maximizes risk-adjusted returns while enforcing survivability and predicted positive expectation constraints. Notably, an analytical gradient is derived for the non-convex objective function, enabling efficient optimization via the L-BFGS-B algorithm. Empirical results demonstrate that this dynamic, closed-loop approach improves risk-adjusted performance and drawdown control relative to constant-weight and prescribed time-varying DLP baselines.

Dynamic Weight Optimization for Double Linear Policy: A Stochastic Model Predictive Control Approach

Abstract

The Double Linear Policy (DLP) framework guarantees a Robust Positive Expectation (RPE) under optimized constant-weight designs or admissible prespecified time-varying policies. However, the sequential optimization of these time-varying weights remains an open challenge. To address this gap, we propose a Stochastic Model Predictive Control (SMPC) framework. We formulate weight selection as a receding-horizon optimal control problem that explicitly maximizes risk-adjusted returns while enforcing survivability and predicted positive expectation constraints. Notably, an analytical gradient is derived for the non-convex objective function, enabling efficient optimization via the L-BFGS-B algorithm. Empirical results demonstrate that this dynamic, closed-loop approach improves risk-adjusted performance and drawdown control relative to constant-weight and prescribed time-varying DLP baselines.

Paper Structure

This paper contains 17 sections, 3 theorems, 32 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Market and Account Dynamics
  4. The Double Linear Policy and Account Value Dynamics
  5. Survivability and Robust Positive Expectation (RPE)
  6. Problem Formulation
  7. State-Space Model
  8. Stochastic MPC with Mean-Variance Objective
  9. Survivability Considerations
  10. Solving the Stochastic MPC Problem
  11. Analytical Gradient Derivation
  12. Handling the Positive Expectation Constraint
  13. Empirical Illustrations
  14. Impact of Transaction Frictions
  15. Cross-Asset Robustness Check
  16. ...and 2 more sections

Key Result

Lemma C.1

Fix a prediction horizon $H>0$. Suppose the current account values satisfy $V_L(k) >0$ and $V_S(k) \geq 0$. If the weight sequence satisfy $0 \leq w_i \leq w_{\max}$ for all $i \in \{k, k+1, \dots, k+H-1\}$, then for all $h=1,\dots,H$, the system output satisfies $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 3: DLP--SMPC applied to the BTC-USD with cross-validated parameters, $\gamma = 0.1$, $H = 29$, and $L = 18$. (Top): accounts value dynamics; (Bottom) control-weight trajectory over time.
  • Figure 4: Performance comparison of the proposed DLP--SMPC with L-BFGS-B strategy against benchmark strategies. Dashed trajectories represent the integration of transaction cost $\varepsilon = 0.1\%$. Account values are displayed on a log scale.

Theorems & Definitions (10)

  • Remark 1: RPE versus Predicted Positive Expectation
  • Lemma C.1: Trajectory-Wide Predicted Survivability
  • proof
  • Remark 2
  • Lemma C.2: Conditional Moments of Predicted Wealth
  • proof
  • Theorem D.1: Analytical Gradient of the Objective Function
  • proof
  • proof : Proof of Lemma \ref{['lemma: conditional moments']}
  • proof : Proof of Theorem \ref{['theorem: gradient']}