Stochastic averaging for multiscale Markov processes with an application to a Wright-Fisher model with fluctuating selection

2015/04/07 by Martin Hutzenthaler, Hutzenthaler, Martin, Peter Pfaffelhuber +3
Economics, Econometrics and Finance · Mathematics · #60F05 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1504.01508

openalex publication_date 2015/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Z = (Z_t)_t∈[0,∞) be an ergodic Markov process and, for every n∈ℕ, let Zⁿ = (Z_{n² t})_t∈[0,∞) drive a process Xⁿ. Classical results show under suitable conditions that the sequence of non-Markovian processes (Xⁿ)_n∈ℕ converges to a Markov process and give its infinitesimal characteristics. Here, we consider a general sequence (Zⁿ)_n∈ℕ. Using a general result on stochastic averaging from [Kur92], we derive conditions which ensure that the sequence (Xⁿ)_n∈ℕ converges as in the classical case. As an application, we consider the diffusion limit of a Wright-Fisher model with fluctuating selection.

Citations

• On the scaling limits of Galton Watson processes in varying environment

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