Optimal Investment and Consumption in a Stochastic Factor Model

ArXiv ID: 2509.09452 “View on arXiv”

Authors: Florian Gutekunst, Martin Herdegen, David Hobson

Abstract

In this article, we study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem, and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Itô diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including – for the first time – the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.

Keywords: Stochastic Factor Models, Hamilton-Jacobi-Bellman (HJB), Incomplete Markets, Power Utility, Numerical Methods

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, focusing on advanced stochastic calculus, HJB equations, and asymptotic analysis to prove existence and uniqueness of solutions, with minimal emphasis on backtesting or implementation details.

  flowchart TD
    A["Research Goal:<br>Optimal Investment & Consumption<br>in Stochastic Factor Model"] --> B{"State Space Type?"}
    B -->|Finite| C["Methodology: Dynamic Programming<br>Finite State Space"]
    B -->|Infinite Interval| D["Methodology: HJB Equation<br>Sub- & Supersolution Theory"]
    C --> E["Data/Inputs:<br>Transition Probabilities &<br>Utility Parameters"]
    D --> F["Data/Inputs:<br>Itô Diffusion Dynamics &<br>Market Parameters"]
    E --> G["Computational Process:<br>Efficient Numerical Algorithm<br>(Value Function Iteration)"]
    F --> H["Computational Process:<br>Asymptotic Analysis &<br>Verification Arguments"]
    G --> I["Key Outcomes:<br>Well-posed Problem<br>& Fast Discretization Scheme"]
    H --> I
    F --> J["Specific Application:<br>First Rigorous Verification<br>of Heston Model"]
    J --> I