Optimal Investment and Consumption in a Stochastic Factor Model

2025-09-11

AI Summary

This paper addresses the optimal investment and consumption problem for power-utility investors in an infinite-horizon, incomplete stochastic factor model, covering both discrete (finite) and continuous (infinite) state spaces. Methodologically, it develops a sub/supersolution theory for boundary-free second-order ODEs under a Heston-type framework, integrating Hamilton–Jacobi–Bellman (HJB) analysis, Itô diffusion modeling, and asymptotic characterization. The contribution is threefold: (i) it establishes, for the first time, rigorous well-posedness of this class of nonlinear stochastic control problems; (ii) it derives explicit upper and lower bounds for the value function; and (iii) it designs a high-accuracy, fast discretization algorithm achieving close alignment between theoretical and numerical solutions. Crucially, the work breaks from conventional boundary assumptions, constructing a general analytical framework applicable to diffusion-based environments—thereby substantially enhancing both computational tractability and theoretical rigor of long-horizon optimization under incomplete information.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Optimal investment and consumption for power utility investors
• Characterizing well-posedness in stochastic factor models
• Solving HJB equations without boundary values in finance

Innovation

• Finite state space numerical algorithm
• Open interval HJB equation theory
• Efficient discretisation approximation scheme