Explicit solutions to utility maximization problems in a regime-switching market model via Laplace transforms

TL;DR

This paper derives explicit solutions for utility maximization in a regime-switching market using Laplace transforms, providing a novel analytical approach for hybrid stochastic systems with practical financial applications.

Contribution

It introduces a Laplace transform method to explicitly solve the Hamilton-Jacobi-Bellman equation in regime-switching models, expanding analytical tools in financial mathematics.

Findings

Explicit solutions for two- and three-state regimes

The Laplace transform method simplifies complex HJB equations

Method adaptable to other hybrid systems

Abstract

We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled by a continuous-time finite-state Markov chain. We consider an investor with a Constant Relative Risk Aversion (CRRA) utility function. We deduce the associated Hamilton-Jacobi-Bellman equation to construct the solution and the optimal trading strategy and verify optimality by showing that the value function is the unique constrained viscosity solution of the HJB equation. By means of a Laplace transform method, we show how to explicitly compute the value function and illustrate the method with the two- and three-states cases. This method is interesting in its own right and can be adapted in other applications involving hybrid systems and using other types of transforms with basic properties similar to the Laplace transform.