Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point

ArXiv ID: 2410.05524 “View on arXiv”

Authors: Unknown

Abstract

We consider the portfolio optimisation problem where the terminal function is an S-shaped utility applied at the difference between the wealth and a random benchmark process. We develop several numerical methods for solving the problem using deep learning and duality methods. We use deep learning methods to solve the associated Hamilton-Jacobi-Bellman equation for both the primal and dual problems, and the adjoint equation arising from the stochastic maximum principle. We compare the solution of this non-concave problem to that of concavified utility, a random function depending on the benchmark, in both complete and incomplete markets. We give some numerical results for power and log utilities to show the accuracy of the suggested algorithms.

Keywords: Portfolio Optimization, Deep Learning, Hamilton-Jacobi-Bellman (HJB) Equation, Stochastic Maximum Principle, S-shaped Utility, Equities

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, focusing on advanced mathematical techniques like stochastic maximum principle, Malliavin calculus, concavification, and deep learning PDE solvers, which is dense and advanced. Empirical rigor is low as it only presents numerical results for algorithm accuracy with power/log utilities, lacking backtesting on real market data or trading strategy implementation.

  flowchart TD
    A["Research Goal:<br>Solve S-shaped Utility Maximization<br>with Random Reference Point"] --> B
    B["Key Inputs:<br>Portfolio Constraints, <br>Random Benchmark Process, <br>S-shaped Utility Function"] --> C
    subgraph C ["Methodology & Computation"]
        direction TB
        C1["Deep Learning: <br>HJB Equation (Primal/Dual)"]
        C2["Stochastic Maximum Principle: <br>Adjoint Equation"]
        C3["Concavified Utility: <br>Comparison Baseline"]
    end
    C --> D["Computational Execution:<br>ML & Numerical Algorithms<br>in Complete/Incomplete Markets"]
    D --> E["Key Outcomes:<br>1. Accurate Solutions via DL<br>2. Comparison to Concave Methods<br>3. Numerical Results (Power/Log Utilities)"]