For the anisotropic stable JCIR process, the heat kernel exists and obeys a weighted anisotropic Besov bound, the strong Feller property holds, and in the subcritical case convergence to the invariant measure is exponential in total variation.
The martingale problem for a class of nonlocal operators of diagonal type
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abstract
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.
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On the anisotropic stable JCIR process
For the anisotropic stable JCIR process, the heat kernel exists and obeys a weighted anisotropic Besov bound, the strong Feller property holds, and in the subcritical case convergence to the invariant measure is exponential in total variation.