Paper: arXiv 2509.13091

Authors: Matteo Buttarazzi, Tiziano De Angelis, Gabriele Stabile

Abstract

This paper addresses the problem of determining the optimal time for an individual to convert retirement savings into a lifetime annuity. The individual invests their wealth into a dividend-paying fund that follows the dynamics of a geometric Brownian motion, exposing them to market risk. At the same time, they face an uncertain lifespan influenced by a stochastic mortality force. The latter is modelled as a piecewise deterministic Markov process (PDMP), which captures sudden and unpredictable changes in the individual’s mortality force. The individual aims to maximise expected lifetime linear utility from consumption and bequest, balancing market risk and longevity risk under an irreversible, all-or-nothing annuitization decision. The problem is formulated as a three-dimensional optimal stopping problem and, by exploiting the PDMP structure, it is reduced to a sequence of nested one-dimensional problems. We solve the optimal stopping problem and find a rich structure for the optimal annuitization rule, which cover all parameter specifications. Our theoretical analysis is complemented by a numerical example illustrating the impact of a single health shock on annuitization timing, along with a sensitivity analysis of key model parameters.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical treatment of an optimal annuitization problem using advanced continuous-time mathematics, optimal stopping, and piecewise deterministic Markov processes. However, as a purely mathematical finance paper, it relies on stylized numerical examples and lacks real-world empirical data validation or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal Annuitization Timing<br>(Max Utility under Market & Longevity Risk)"] --> B["Inputs & Dynamics:<br>Market (GBM) + Stochastic Mortality (PDMP)"]
    B --> C["Formulate 3D Optimal Stopping Problem"]
    C --> D["Dimensionality Reduction:<br>Nested Sequence of 1D Problems"]
    D --> E["Numerical Example & Sensitivity Analysis"]
    E --> F["Outcomes: Complete Theoretical Map of<br>Optimal Rules & Impact of Health Shocks"]