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

Won Choi

Abstract


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 $\rho$.


Subject classification

92D10, 60H30, 60G44

Sponsor(s)

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

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References


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

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

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)


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