Optimal portfolio in a regime-switching model

In this paper we derive the solution of the classical Merton problem, i.e. maximizing the utility of the terminal wealth, in the case when the risky assets follow a diffusion model with switching coefficients. We show that the optimal portfolio is a generalisation of the corresponding one in the classical Merton case, with portfolio proportions which depend on the market regime. We perform our analysis via the classical approach with the Hamilton-Jacobi-Bellman equation. First we extend the mutual fund theorem as presented in Bjork (1998) to our framework. Then we show explicit solutions for the optimal strategies in the particular cases of exponential, logarithm and power utility functions.