Prediction for Reward Processes

Abstract

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

Keywords:

• Conditional expectation functions
• Current life
• Excess life
• Prediction
• Reward processes
• Semi-Markov processes

Mathematics Subject Classification:

• Primary 60K15
• Secondary 60K05

ACKNOWLEDGMENTS

The authors express their sincere thanks to referees for their careful readings of the article and providing valuable comments. The authors also thank B. Heydaridad, statistics master graduate from Shiraz University, for his verification of the algebraic work of Examples 3.1 and 4.2.