The martingale problem for a class of nonlocal operators of diagonal type

We consider systems of stochastic differential equations of the form $$\d X_t^i = \sum_{j=1}^d A_{ij}(X_{t-}) \d Z_t^j$$ for $i=1,\dots,d$ with continuous, bounded and non-degenerate coefficients. Here $Z_t^1,\dots,Z_t^d$ are independent one-dimensional stable processes with $\alpha_1,\dots,\alpha_d\in(0,2)$. In this article we research on uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the uniqueness of weak solutions in the case of diagonal coefficient matrices.