PREDICTION FOR REWARD PROCESSES

We consider D(t) = [J(t), X (t), Z(rho)(t)], t = 0, where J (t) is a semi-Markov process, X (t) is the age process for J (t), and Z(rho) (t) is a reward process on J (t). In this article we produce a prediction formula to predict Z(rho) (t + s) based on the history D(t), the sigma-field generated by {D(u), u <= t}. In addition, we derive the conditional distribution of the excess life gamma(t) given [J (t), X (t)], which involves the quantity h(j,t), the probability that a sojourn in state j exceeds t.