Machine Learning and Hamilton-Jacobi-Bellman Equation for Optimal Decumulation: a Comparison Study

ArXiv ID: 2306.10582 “View on arXiv”

Authors: Unknown

Abstract

We propose a novel data-driven neural network (NN) optimization framework for solving an optimal stochastic control problem under stochastic constraints. Customized activation functions for the output layers of the NN are applied, which permits training via standard unconstrained optimization. The optimal solution yields a multi-period asset allocation and decumulation strategy for a holder of a defined contribution (DC) pension plan. The objective function of the optimal control problem is based on expected wealth withdrawn (EW) and expected shortfall (ES) that directly targets left-tail risk. The stochastic bound constraints enforce a guaranteed minimum withdrawal each year. We demonstrate that the data-driven approach is capable of learning a near-optimal solution by benchmarking it against the numerical results from a Hamilton-Jacobi-Bellman (HJB) Partial Differential Equation (PDE) computational framework.

Keywords: stochastic control, neural networks, Hamilton-Jacobi-Bellman (HJB), defined contribution (DC) pension, expected shortfall (ES), Pension / Retirement Planning

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: The paper employs advanced mathematics, including stochastic optimal control, HJB-PDEs, and neural network approximations, yielding a high math complexity score. Its empirical rigor is substantial due to a direct benchmark against a numerical PDE ground truth, robustness tests on out-of-sample data, and explicit discussion of implementation, though it lacks live trading or full dataset disclosure, placing it in the Holy Grail quadrant.
  flowchart TD
    A["Research Goal: Solve Optimal Decumulation<br/>for DC Pension Plans"] --> B["Methodology: NN Framework<br/>vs. HJB PDE Benchmark"]
    B --> C["Data & Inputs:<br/>Market Scenarios & Stochastic Constraints"]
    C --> D{"Computational Process"}
    D --> E["NN: Data-Driven<br/>Optimization & Training"]
    D --> F["HJB: Numerical<br/>PDE Solver"]
    E & F --> G["Key Outcomes:<br/>Near-optimal Strategy<br/>Validated vs. Benchmark"]