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.

This research was supported in part by the National Science Foundation under Grant GK-21460.