Paper: arXiv 2501.12600

Abstract

We propose a scalable, policy-centric framework for continuous-time multi-asset portfolio-consumption optimization under inequality constraints. Our method integrates neural policies with Pontryagin’s Maximum Principle (PMP) and enforces feasibility by maximizing a log-barrier-regularized Hamiltonian at each time-state pair, thereby satisfying KKT conditions without value-function grids. Theoretically, we show that the barrier-regularized Hamiltonian yields O($ε$) policy error and a linear Hamiltonian gap (quadratic when the KKT solution is interior), and we extend the BPTT-PMP correspondence to constrained settings with stable costate convergence. Empirically, PG-DPO and its projected variant (P-PGDPO) recover KKT-optimal policies in canonical short-sale and consumption-cap problems while maintaining strict feasibility across dimensions; unlike PDE/BSDE solvers, runtime scales linearly with the number of assets and remains practical at n=100. These results provide a rigorous and scalable foundation for high-dimensional constrained continuous-time portfolio optimization.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper features highly advanced mathematics including Pontryagin’s Maximum Principle, barrier-regularized Hamiltonians, KKT conditions, and high-dimensional convergence proofs, warranting a near-maximal complexity score. Empirically, it demonstrates scalable algorithms with linear runtime scaling to 100 assets, validates feasibility on canonical constrained problems, and compares against PDE/BSDE baselines, indicating strong implementation readiness despite the absence of explicit backtesting data.

Research Flowchart

  flowchart TD
  subgraph A ["1. Research Goal"]
      direction LR
      G["High-Dim Constrained<br>Portfolio Optimization"]
  end

  subgraph B ["2. Methodology"]
      direction LR
      M1["Neural Policy<br>π(s,t;θ)"] --> M2["Log-Barrier<br>Hamiltonian"]
      M2 --> M3["PMP & KKT<br>Optimization"]
  end

  subgraph C ["3. Inputs & Data"]
      direction LR
      I1["Asset Returns<br>& Dynamics"] --> I2["Constraints<br>Short-sale, Consumption"]
  end

  subgraph D ["4. Computational Process"]
      direction LR
      P1["Gradient Descent<br>on Barrier Hamiltonian"] --> P2["Converge to<br>KKT-Feasible Policy"]
  end

  subgraph E ["5. Key Outcomes"]
      direction LR
      O1["O(n) Scalability<br>vs PDE O(n^2)"] --> O2["Strict Feasibility<br>in High Dimensions"]
      O3["Stable PMP Convergence"]
  end

  A --> B
  C --> B
  B --> D
  D --> E