Sufficient stochastic maximum principle for the optimal control of semi-Markov modulated jump-diffusion with application to Financial optimization

TL;DR

This paper develops a sufficient stochastic maximum principle for controlling semi-Markov modulated jump-diffusion processes, extending stochastic control theory and applying it to financial portfolio optimization and risk management.

Contribution

It introduces a new maximum principle for semi-Markov modulated jump-diffusions and links it with dynamic programming, enhancing control strategies in complex stochastic systems.

Findings

Derived a sufficient maximum principle for semi-Markov modulated jump-diffusions.

Connected the maximum principle with dynamic programming equations.

Applied the results to financial portfolio optimization and risk-sensitive control.

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.