On the Cauchy problem for integro-differential operators in H\"older classes and the uniqueness of the martingale problem

The existence and uniqueness in H\"older spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^{d+{\alpha}} with a bounded nondegenerate m, H\"older in x and measurable in y. The result is applied to prove the uniqueness of the corresponding martingale problem.