A boundary preserving numerical algorithm for the Wright-Fisher model with mutation
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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].
KeywordsWright-Fisher model Stochastic differential equations Strong convergence Hölder condition Ion channels Split step Boundary preserving numerical algorithm
Mathematics Subject Classification (2000)65C30 65L20 92D99 60H35 65C20
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- 2.Berkaoui, A., Bossy, M., Diop, A.: Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM Probab. Stat. 12 (2008) Google Scholar
- 4.Dangerfield, C.E., Kay, D., Burrage, K.: Stochastic models and simulation of ion channel dynamics. Proc. Comput. Sci. 1(1), 1581–1590 (2010) Google Scholar
- 11.Halley, W., Malham, S.J.A., Wiese, A.: Positive stochastic volatility simulation (2008). ArXiv e-prints 0802.4411v1
- 12.Halley, W., Malham, S.J.A., Wiese, A.: Positive and implicit stochastic volatility simulation (2009). ArXiv e-prints 0802.4411v2
- 14.Higham, D., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Finance 8, 35–61 (2005) Google Scholar
- 16.Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(1), 500–544 (1952) Google Scholar
- 18.Kahl, C., Schurz, H.: Balanced Milstein methods for ordinary SDEs. Technical report, Department of Mathematics, Southern Illinois University (2005) Google Scholar
- 27.Noble, D.: A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J. Physiol. 160, 317–352 (1962) Google Scholar
- 28.Pueyo, E., Corrias, A., Burrage, K., Rodriguez, B.: From ion channel fluctuations to the electrocardiogram. Implications for cardiac arrhythmogenesis. Biophys. J. (in press) Google Scholar
- 30.Wright, S.: Evolution in Mendelian populations. Genetics 16(2), 97–159 (1931) Google Scholar