Paper: arXiv 2212.05317

Authors: Giorgio Ferrari, Shihao Zhu

Abstract

We propose a tractable dynamic framework for the joint determination of optimal consumption, portfolio choice, and healthcare irreversible investment. Our model is based on a Merton’s portfolio and consumption problem, where, in addition, the agent can choose the time at which undertaking a costly lump sum health investment decision. Health depreciates with age and directly affects the agent’s mortality force, so that investment into healthcare reduces the agent’s mortality risk. The resulting optimization problem is formulated as a stochastic control-stopping problem with a random time-horizon and state-variables given by the agent’s wealth and health capital. We transform this problem into its dual version, which is now a two-dimensional optimal stopping problem with interconnected dynamics and finite time-horizon. Regularity of the optimal stopping value function is derived and the related free boundary surface 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 invests into healthcare whenever her wealth exceeds an age- and health-dependent transformed version of the optimal stopping boundary. We also provide the numerical illustrations of the optimal strategies and some financial implications are discussed.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical extension of the classic Merton problem by incorporating health capital and optimal stopping via free boundary analysis. While mathematically strong with explicit continuous-time derivations and numerical solutions, it lacks empirical validation against real-world market or health demographic data.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Merton Problem with Irreversible Healthcare Investment] --> B[Model Setup: Wealth, Health Capital & Health-Dependent Mortality]
    B --> C[Formulate Stochastic Control-Stopping Problem]
    C --> D[Apply Dual Control Transformation to 2D Optimal Stopping Problem]
    D --> E[Derive Free Boundary & Solve Nonlinear Integral Equation Numerically]
    E --> F[Outcomes: Wealth Thresholds for Health Investment & Optimal Financial Strategies]