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
$$d{Q}_{\rho }=S_{t}d{P}_{\rho }$$
and show that $Q_{\rho }$ solves the martingale problem for ${\mathcal{L}}_{\pi }$ starting at $\rho$.