On the martingale extension of limiting diffusion in population genetics

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 $\rho$.