Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps

Abstract: We study existence of densities for solutions to stochastic differential equations with H\"older continuous coefficients and driven by a $d$-dimensional L\'evy process $Z=(Z_{t}){t\geq 0}$, where, for $t>0$, the density function $f{t}$ of $Z_{t}$ exists and satisfies, for some $(\alpha_{i}){i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/\alpha{i}}\int\limits {\mathbb{R}^{d}}|f{t}(z+e_{i}h)-f_{t}(z)|dz\leq C|h|,\ \ h\in \mathbb{R},\ \ i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basis vectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic $(\alpha_{1},\dots,\alpha_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.