A boundary preserving numerical algorithm for the Wright-Fisher model with mutation
Dangerfield, C.E., Kay, D., MacNamara, S., & Burrage, K. (2012) A boundary preserving numerical algorithm for the Wright-Fisher model with mutation. BIT Numerical Mathematics, 52(2), pp. 283-304.
The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].
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|Item Type:||Journal Article|
|Keywords:||Boundary preserving numerical algorithm, Hölder condition, Ion channels, Split step, Stochastic differential equations, Strong convergence, Wright-Fisher model|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000)|
|Divisions:||Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
|Copyright Owner:||Copyright 2012 Springer|
|Copyright Statement:||The original publication is available at SpringerLink http://www.springerlink.com|
|Deposited On:||01 Jul 2012 22:55|
|Last Modified:||04 Jul 2012 05:44|
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