CRRA utility maximization over a finite horizon in an exponential Lévy model with finite activity

We study a dynamic portfolio optimization problem over a finite horizon with n risky securities and a risk-free asset. The prices of the risky securities are modelled by ordinary exponentials of jump-diffusions. The goal is to maximize the expected discounted utility from both consumption up to the final horizon and terminal wealth. We prove a verification theorem that characterizes the value function and the optimal policy by means of a regular solution of a HJB partial integro-differential equation. The verification theorem is used to obtain closed-form expressions for the value function and the optimal policy considering power and exponential utility functions.