The Annals of Applied Probability

Exact simulation of the Wright–Fisher diffusion

Paul A. Jenkins and Dario Spanò

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The Wright–Fisher family of diffusion processes is a widely used class of evolutionary models. However, simulation is difficult because there is no known closed-form formula for its transition function. In this article, we demonstrate that it is in fact possible to simulate exactly from a broad class of Wright–Fisher diffusion processes and their bridges. For those diffusions corresponding to reversible, neutral evolution, our key idea is to exploit an eigenfunction expansion of the transition function; this approach even applies to its infinite-dimensional analogue, the Fleming–Viot process. We then develop an exact rejection algorithm for processes with more general drift functions, including those modelling natural selection, using ideas from retrospective simulation. Our approach also yields methods for exact simulation of the moment dual of the Wright–Fisher diffusion, the ancestral process of an infinite-leaf Kingman coalescent tree. We believe our new perspective on diffusion simulation holds promise for other models admitting a transition eigenfunction expansion.

Article information

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1478-1509.

Received: June 2016
First available in Project Euclid: 19 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 60H35: Computational methods for stochastic equations [See also 65C30] 60J60: Diffusion processes [See also 58J65] 92D15: Problems related to evolution

Monte Carlo simulation Wright–Fisher diffusion exact algorithm Fleming–Viot process diffusion bridge retrospective simulation Kingman’s coalescent population genetics


Jenkins, Paul A.; Spanò, Dario. Exact simulation of the Wright–Fisher diffusion. Ann. Appl. Probab. 27 (2017), no. 3, 1478--1509. doi:10.1214/16-AAP1236.

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