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([2]) defined the operator of projection <TEX>$S_t$</TEX> on limiting diffusion and new measure <TEX>$dQ=S_tdP$</TEX>. and showed the martingale property on this operator and measure. Let <TEX>$P_{\rho}$</TEX> be the unique solution of the martingale problem for <TEX>$\mathcal{L}_0$</TEX> starting at <TEX>${\rho}$</TEX> and <TEX>${\pi}_1,{\pi}_2,{\cdots},{\pi}_n$</TEX> the projection of <TEX>$E^n$</TEX> on <TEX>$x_1,x_2,{\cdots},x_n$</TEX>. In this note we define <TEX>$$dQ_{\rho}=S_tdP_{\rho}$$</TEX> and show that <TEX>$Q_{\rho}$</TEX> solves the martingale problem for <TEX>$\mathcal{L}_{\pi}$</TEX> starting at <TEX>${\rho}$</TEX>.