The martingale problem for a class of stable-like processes

Abstract

Let $\alpha \in (0,2)$ and consider the operator $$Lf(x)=\int [f(x+h)-f(x)-1_{(|h|\le 1)}\nabla f(x)\cdot h]\frac{A(x,h)}{|h{|}^{d+\alpha }}dh,$$ 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.