Sufficient Stochastic Maximum Principle for the Optimal Control of Semi-Markov Modulated Jump-Diffusion with Application to Financial Optimization

Abstract

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.

Keywords:

• Semi-Markov modulated jump diffusions
• Sufficient stochastic maximum principle
• Dynamic programming
• Risk-sensitive control
• Quadratic loss-minimization.

Mathematics Subject Classification:

• 93E20
• 60H30
• 46N1.