Paper: arXiv 2311.12169

Abstract

This paper examines the retirement decision, optimal investment, and consumption strategies under an age-dependent force of mortality. We formulate the optimization problem as a combined stochastic control and optimal stopping problem with a random time horizon, featuring three state variables: wealth, labor income, and force of mortality. To address this problem, we transform it into its dual form, which is a finite time horizon, three-dimensional degenerate optimal stopping problem with interconnected dynamics. We establish the existence of an optimal retirement boundary that splits the state space into continuation and stopping regions. Regularity of the optimal stopping value function is derived and the boundary is proved to be Lipschitz continuous, and it is characterized as the unique solution to a nonlinear integral equation, which we compute numerically. In the original coordinates, the agent thus retires whenever her wealth exceeds an age-, labor income- and mortality-dependent transformed version of the optimal stopping boundary. We also provide numerical illustrations of the optimal strategies, including the sensitivities of the optimal retirement boundary concerning the relevant model’s parameters.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced stochastic control, duality, and free-boundary analysis with a three-dimensional degenerate problem and nonlinear integral equations, making it highly mathematically complex. However, it focuses solely on theoretical analysis and numerical illustrations without any backtesting, datasets, or empirical implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Model Retirement Choice<br>under Age-dependent Mortality"] --> B["Methodology:<br>Combined Stochastic Control<br>& Optimal Stopping"]
  B --> C["Key Step:<br>Dual Formulation<br>(3D Degenerate Problem)"]
  C --> D["Computation:<br>Nonlinear Integral Equation<br>for Optimal Boundary"]
  D --> E["Numerical Solution<br>& Boundary Computation"]
  E --> F["Findings:<br>Retirement Trigger: Wealth ><br>Age/Income/Mortality Boundary"]
  F --> G["Outcomes:<br>Sensitivity Analysis<br>of Retirement Boundary"]