Paper: arXiv 2610.06504

Authors: Zhaojie Ren, Sheng Wang, Tak Kwong Wong, Sheung Chi Phillip Yam

Abstract

This article studies a retirement planning problem from a new perspective in which a retiree delegates an initial lump sum to a professional fund manager. The fund is managed dynamically to deliver lifelong benefits while satisfying a guarantee that, at all times, wealth remains above a prescribed solvency level and the benefit rate remains above a minimum level. We formulate this problem as a continuous-time stochastic control model, in which the controls are the benefit rate and the portfolio allocation, and the objective incorporates general utility functions over benefits, management fees, and terminal bequest. Conventional approaches such as standard viscosity solution and martingale methods are not immediately applicable here due to the simultaneous influence of (i) the control variable nature of the benefit rate; (ii) the guarantee constraints; and (iii) the running wealth-dependent utility. We develop a new approach that deals directly with the associated fully nonlinear Hamilton-Jacobi-Bellman equation and establish the unique existence of a classical solution. One particular trick for resolving this equation is to apply an intricate transform that diverts the former to a semilinear auxiliary equation. A major contribution is to establish the existence of the solution to the auxiliary equation and characterize its precise growth, derivative, and boundary behavior. These results are crucial for establishing the verification theorems and deriving the optimal strategy. Besides, numerical experiments indicate that guarantee constraints induce more conservative policies, particularly at low wealth levels, while providing effective downside protection for both wealth and benefits. The associated well-being loss relative to the unconstrained framework remains small, suggesting that incorporating guarantees into retirement products is practically appealing.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly complex mathematical framework for a novel retirement planning problem. While it includes numerical experiments, the empirical rigor is limited to these simulations rather than real-world backtesting or extensive data validation. The core contribution lies in its theoretical development.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimize Retirement Planning with Guarantees] --> B{Methodology: Continuous-Time Stochastic Control Model};
    B --> C[Inputs: Initial Lump Sum, Utility Functions, Guarantee Levels];
    C --> D[Computational Process: Solve HJB Equation via Intricate Transform];
    D --> E[Key Findings: Unique Classical Solution, Optimal Strategies, Impact of Guarantees];