The martingale problem for anisotropic nonlocal operators
Jamil Chaker · arXiv (Cornell University) · 2018
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 existence and uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the existence of weak solutions in the general case and establish uniqueness in the case of diagonal matrices.