부산대학교 도서관

This paper contributes to the vast literature on stochastic optimal control by extending the dynamic programming principle to the case when the state variable dynamics are determined by a diffusive stochastic differential equation whose coefficients depend on a semi-Markov process with a finite state space. This principle is then used to derive the corresponding Hamilton-Jacobi-Bellman equation and to characterize the value function of the stochastic optimal control under consideration as a viscosity solution of such an equation. A verification theorem is also provided. The results are illustrated on a consumption-investment problem when the asset prices evolve according to a semi-Markov modulated SDE. \par This formalism is well suited for many applications, when both the state variable dynamics and the objective functional depend on a set of known `unknowns' occurring at random instants of time, encapsulated here by the components of the semi-Markov process. In particular in finance, semi-Markov processes allow the asset price dynamics to switch between different states of the financial market, and other modeling flexibilities.