Semi-Markov and reward fields

We introduce semi-Markov fields and provide formulations for the basic terms in the semi-Markov theory. In particular we define and consider a class of associated reward fields. Then we present a formula for the expected reward at any multidimensional time epoch. The formula is indeed new even for the classical semi-Markov processes. It gives the expected cumulative reward for fairly large classes of reward functions; in particular, it provides the formulas for the expected cumulative reward given in Masuda and Sumitau (1991), Soltani (1996) and Soltani and Khorshidian (1998).