Korean J. Math. Vol. 22 No. 1 (2014) pp.29-36
DOI: https://doi.org/10.11568/kjm.2014.22.1.29

On the martingale extension of limiting diffusion in population genetics

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Won Choi


The limiting diffusion of special diploid model can be defined as a discrete generator for the rescaled Markov chain. Choi( \cite{Choi2} ) defined the operator of projection $S_t$ on limiting diffusion and new measure $dQ=S_t dP$. and showed the martingale property on this operator and measure. Let $P_\rho$ be the unique solution of the martingale problem for $\mathcal L_0$ starting at $\rho$ and
$\pi_1 ,\pi_2 , \cdots, \pi_n$ the projection of $E^n$ on $x_1 , x_2 ,\cdots, x_n$. In this note we define
$$ dQ_\rho =S_t dP_\rho $$
and show that $Q_\rho$ solves the martingale problem for $\mathcal L_\pi$ starting at $\rho$.

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Supporting Agencies

This research was supported by Incheon National University Research Grant 2013-2014.


[1] W.Choi, On the limiting diffusion of special diploid model in population genetics, Bull. Korean Math. Soc. 42 (2) (2005), 397–404. Google Scholar

[2] W.Choi, On the martingale property of limiting diffusion in special diploid model, J. Appl. Math. info. 31 (1) (2013), 241–246. Google Scholar

[3] A.Shimizu, Stationary distribution of a diffusion process taking values in proba- bility distributions on the partitions, Proceedings of a Workshop held in Nagoya, (1985), 100-114. Google Scholar