A laurent series for the resolvent of a strongly continuous stochastic semi-group

Abstract

Let (P t )t≥0 be a standard stochastic semi-group of Markov transition operators, continuous in the strong operator topology at t=0 and ∞. Let (R λ)λ>0 be the corresponding resolvent. We show R λ=λ−1 P *+σ k=0 ∞ (−λ)k H k+1, assuming P * is a uniform limit of P t , at infinity and H=∫ 0 ∞ (P t−P *)dt.This Laurent expansion is of interest in the theory of controlled Markov processes. Suppose (X t )i≥0 is a Markov process having transitions (P t ) and describing the evolution of some controlled system. Costs are accrued at a rate u(x) whenever the system is in state X t =x. Then R λ u is an expected total discounted cost, where a dollar at time t is discounted to a present value of e−λt . Our result expands this total discounted cost as a Laurent series in the interest rate λ.More details are given for finite state Markov chains and diffusion processes on compact intervals.