Transition density estimates for diagonal systems of SDEs driven by cylindrical α-stable processes

We consider the system of stochastic differential equation \$dX_t = A(X_{t-}) dZ_t\$, \$X_0 = x\$, driven by cylindrical \$\alpha\$-stable process \$Z_t\$ in \$R^d\$. We assume that \$A(x) = (a_{ij}(x))\$ is diagonal and \$a_{ii}(x)\$ are bounded away from zero, from infinity and Holder continuous. We construct transition density \$p^A(t,x,y)\$ of the process \$X_t\$ and show sharp two-sided estimates of this density. We also prove Holder and gradient estimates of \$x \to p^A(t,x,y)\$. Our approach is based on the method developed by Chen and Zhang.