Optimal Investment and Consumption in Stochastic Factor Models

Summary

This work studies infinite-horizon investment and consumption for an investor with power utility in an incomplete market whose opportunities depend on a stochastic factor. It treats both finite-state factors and factors modeled as Itô diffusions, connecting dynamic portfolio choice to the solution of a Hamilton-Jacobi-Bellman equation.

For finite state spaces, the authors characterize when the optimization problem is well posed and give a numerical method for computing its value function. For diffusion factors on open intervals, they use sub- and supersolution methods for second-order ordinary differential equations without prescribed boundary values. This establishes existence and explicit bounds, while asymptotic analysis supports rigorous verification for several models, including Heston. A fast discretization scheme links the continuous diffusion problem to its finite-state counterpart. The results concern mathematical existence, verification, and computation under the stated model assumptions; they do not provide empirical investment performance or a general solution for arbitrary market dynamics.

Key ideas

• The model considers infinite-horizon investment and consumption with power utility in an incomplete market.
• Finite-state stochastic factors permit a characterization of well-posedness and numerical value-function computation.
• For diffusion factors on open intervals, sub- and supersolutions help establish existence and bounds without boundary values.
• Asymptotic behavior supports verification arguments for multiple models, including Heston.
• A fast discretization scheme connects diffusion-based problems to finite-state approximations.

Tags

# Optimal Investment and Consumption in a Stochastic Factor Model


# Optimal Investment and Consumption in a Stochastic Factor Model









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.

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