Optimal portfolio under ratio-type periodic evaluation in incomplete markets with stochastic factors

ArXiv ID: 2401.14672 “View on arXiv”

Authors: Unknown

Abstract

This paper studies a type of periodic utility maximization for portfolio management in an incomplete market model, where the underlying price diffusion process depends on some external stochastic factors. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon. For both power and logarithmic utilities, we formulate the auxiliary one-period optimization problems with modified utility functions, for which we develop the martingale duality approach to establish the existence of the optimal portfolio processes and the dual minimizers can be identified as the “least favorable” completion of the market. With the help of the duality results in the auxiliary problems and some fixed point arguments, we further derive and verify the optimal portfolio processes in a periodic manner for the original periodic evaluation problems over an infinite horizon.

Keywords: portfolio optimization, periodic utility maximization, martingale duality, incomplete markets, stochastic factors, General Financial Instruments

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats
  • Why: The paper is highly theoretical, employing advanced stochastic calculus, duality methods, and fixed point theory to solve a portfolio optimization problem in an incomplete market, which drives up the math complexity. Conversely, the summary and excerpt contain no empirical data, backtests, or implementation details, focusing purely on theoretical proofs and existence results.
  flowchart TD
    A["Research Goal"] --> B["Methodology"]
    B --> C["Computational Process"]
    C --> D["Outcomes"]
    
    A --> A1["Optimal Portfolio for Periodic<br>Utility Maximization"]
    
    B --> B1["Martingale Duality Approach<br>for Auxiliary Problems"]
    B --> B2["Fixed Point Arguments<br>for Infinite Horizon"]
    
    C --> C1["Identify 'Least Favorable'<br>Market Completion"]
    C --> C2["Formulate & Solve<br>Periodic Optimization"]
    
    D --> D1["Existence of<br>Optimal Portfolio Processes"]
    D --> D2["Explicit Construction<br>for Power/Logarithmic Utilities"]