Dynamic programming for semi-Markov modulated SDEs

We consider a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation of diffusive type modulated by a semi-Markov process with a finite state space. The time horizon is both deterministic and finite. Within such setup, we provide a detailed proof of the dynamic programming principle and use it to characterize the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We illustrate our results with an application to Mathematical Finance: the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.

Automated summary

In this paper, we study a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation modulated by a semi-Markov process. We provide a detailed proof of the dynamic programming principle and show that the value function can be characterized as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We illustrate our results with an application to the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.