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

Paper: arXiv 2410.05524

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.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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)"]

Cited References (21)

1. Deep Learning for Constrained Utility Maximisation (2020) — link
2. Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan (2020) — link
3. Speculative trading, prospect theory and transaction costs (2019) — link
4. Non-Concave Utility Maximization without the Concavification Principle (2019) — link
5. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019) — link
6. Utility Maximization Under Trading Constraints with Discontinuous Utility (2019) — link
7. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations (2018) — link
8. Solving high-dimensional partial differential equations using deep learning (2017) — link

0 of 21 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.