Communications in Mathematical Sciences

Volume 14 (2016)

Number 7

Bounds for the expected value of one-step processes

Pages: 1911 – 1923

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n7.a6

Authors

Benjamin Armbruster (Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois, U.S.A.)

Ádám Besenyei (Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary; and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Budapest, Hungary)

Péter L. Simon (Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary; and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Budapest, Hungary)

Abstract

Mean-field models are often used to approximate Markov processes with large statespaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{ 0,1, \dotsc , N \}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. While the mean-field model is a well known approximation, this lower bound is new, and unlike an asymptotic result, these bounds can be used for finite $N$. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.

Keywords

mean-field model, exact bounds, one-step processes, ODE

2010 Mathematics Subject Classification

34C11, 60J75, 92D30

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Published 14 September 2016