Markov Renewal Programs with Small Interest Rates

Abstract

This paper is concerned primarily with the limiting behavior of v(s) as the interest rate s vanishes. Fortunately, discounting corresponds precisely to taking a LaplaceStieltjes transform, so that a substantial theory can be invoked. In particular, the behavior of v(s) as s O-* 0+ is related through Abelian and Tauberian theorems to the behavior of V(t) as t -+ oo. This paper's basic tool is a partial (Laurent) expansion of v(s) in powers of s, which is obtained in Theorem 1. The hypothesis of this theorem is that the reward structure have finite (n + I)st moment and that the transition structure have finite (n + 2)nd moment. Under these conditions, Theorem 1 verifies that