Portfolio Optimization with Allocation Constraints and Stochastic Factor Market Dynamics

Paper: arXiv 2303.09835

Authors: Marcos Escobar-Anel, Michel Kschonnek, Rudi Zagst

Abstract

We study the expected utility portfolio optimization problem in an incomplete financial market where the risky asset dynamics depend on stochastic factors and the portfolio allocation is constrained to lie within a given convex set. We employ fundamental duality results from real constrained optimization to formally derive a dual representation of the associated HJB PDE. Using this representation, we provide a condition on the market dynamics and the allocation constraints, which ensures that the solution to the HJB PDE is exponentially affine and separable. This condition is used to derive an explicit expression for the optimal allocation-constrained portfolio up to a deterministic minimizer and the solution to a system of Riccati ODEs in a market with CIR volatility and in a market with multi-factor OU short rate.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using HJB PDEs, duality theory, and Riccati ODEs to solve constrained portfolio optimization under stochastic factor dynamics. However, it lacks empirical backtesting, market data application, and numerical validation, making it purely analytical.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Utility Maximization with Convex Constraints & Stochastic Factors] --> B[Inputs: Stochastic Dynamics CIR Volatility / OU Short Rate & Convex Allocation Set]
    B --> C[Methodology: Apply Convex Duality to HJB PDE]
    C --> D[Identify Separation Condition for Exponentially Affine HJB Solution]
    D --> E[Derive System of Matrix Riccati ODEs]
    E --> F[Outcomes: Explicit Optimal Portfolio Strategy via Deterministic Minimization & Riccati Solutions]