  The finite state semi-Markov process is a generalization over the Markov
chain in which the sojourn time distribution is any general distribution. In
this article we provide a sufficient stochastic maximum principle for the
optimal control of a semi-Markov modulated jump-diffusion process in which the
drift, diffusion and the jump kernel of the jump-diffusion process is modulated
by a semi-Markov process. We also connect the sufficient stochastic maximum
principle with the dynamic programming equation. We apply our results to finite
horizon risk-sensitive control portfolio optimization problem and to a
quadratic loss minimization problem.
