On a continuously discounted vector valued Markov decision process

The paper deals with continuous time vector valued Markov decision process on a general state space. The vector rewards are continuously discounted at rate. The optimization criterion of the decision process is made from the domination structure determined by a given convex cone L'. By using an affine transformation of the reward in R^m to R, the authors show the existence of an L'-optimal solution under some conditions and then the relations between an L'-optimal solution and an optimal solution of real valued Markov decision process are characterized. Further, the supremum is considered in the case of which an optimal policy does not exist.