Bulletin of the
Korean Mathematical Society BKMS

In allelic model $X=(x_1 ,x_2 ,\cdots ,x_d )$, \[ M_f(t)=f(p(t))-\int_0^t Lf(p(t))ds \] is a $P$-martingale for diffusion operator $L$ under the certain conditions. In this note, we can show existence and uniqueness of solution for stochastic differential equation and martingale problem associated with mean vector. Also, we examine that if the operator related to this martingale problem is connected with Markov processes under certain circumstance, then this operator must satisfy the maximum principle.

Keywords: countable allelic model, martingale problem, stochastic differential equation, mean vector, second order differential operator

MSC numbers: 92D25, 60H30, 47A05