The martingale problem for a class of stable-like processes

Abstract

Let $\alpha\in (0,2)$ and consider the operator $$L f(x) =\int [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}} dh, $$ where the $\nabla f(x)\cdot h$ term is omitted if $\alpha<1$. We consider the martingale problem corresponding to the operator $L$ and under mild conditions on the function $A$ prove that there exists a unique solution.