The martingale problem for a class of stable-like processes

Let α ∈ ( 0 , 2 ) and consider the operator L f ( x ) = ∫ [ f ( x + h ) − f ( x ) − 1 ( | h | ≤ 1 ) ∇ f ( x ) ⋅ h ] A ( x , h ) | h | d + α d h for f ∈ C 2 ( R d ) , where the ∇ f ( x ) ⋅ h term is omitted if α < 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.