A Finite-Horizon Optimal Investment And Consumption Problem Using Regime-Switching Models

This paper is concerned with a finite-horizon optimal investment and consumption problem in continuous-time regime-switching models. The market consists of one bond and n ≥ 1 correlated stocks. An investor distributes his/her wealth among these assets and consumes at a non-negative rate. The market parameters (the interest rate, the appreciation rates and the volatilities of the stocks) and the utility functions are assumed to depend on a continuous-time Markov chain with a finite number of states. The objective is to maximize the expected discounted total utility of consumption and the expected discounted utility from terminal wealth. We solve the optimization problem by applying the stochastic control methods to regime-switching models. Under suitable conditions, we prove a verification theorem. We then apply the verification theorem to a power utility function and obtain, up to the solution of a system of coupled ordinary differential equations, an explicit solution of the value function and the optimal investment and consumption policies. We illustrate the impact of regime-switching on the optimal investment and consumption policies with numerical results and compare the results with the classical Merton problem that has only a single regime.