Existence of optimal delay-dependent control for finite-horizon continuous-time Markov decision process

This paper intends to study the optimal control problem for the continuous-time Markov decision process with denumerable states and compact action space. The admissible controls depend not only on the current state of the jumping process but also on its history. By the compactification method, we show the existence of an optimal delay-dependent control under some explicit conditions, and further establish the dynamic programming principle. Moreover, we show that the value function is the unique viscosity solution of certain Hamilton-Jacobi-Bellman equation which does not depend on the delay-dependent control policies. Consequently, under our explicit conditions, there is no impact on the value function to make decision depending on or not on the history of the jumping process.Comment: 22 page