Semigroup properties of solutions of SDEs driven by L{\'e}vy processes with independent coordinates

Abstract

We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots, Z_t^{(d)}$ are independent one-dimensional L{\'e}vy processes with characteristic exponents $\psi_1, \ldots, \psi_d$. We assume that each $\psi_i$ satisfies a weak lower scaling condition WLSC($\alpha,0,\underline{C}$), a weak upper scaling condition WUSC($\beta,1,\overline{C}$) (where $0< \alpha \le \beta < 2$) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all $\psi_1, \ldots, \psi_d$ are the same and $\alpha, \beta$ are arbitrary, or (ii) not all $\psi_1, \ldots, \psi_d$ are the same and $\alpha > (2/3)\beta$. We also assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed $\gamma \in (0,\alpha) \cap (0,1]$ the semigroup $P_t$ of the process $X$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-\gamma/\alpha} |x - y|^{\gamma} ||f||_\infty$ for arbitrary bounded Borel function $f$. We also show the existence of a transition density of the process $X$.