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 Hölder continuous coefficients and driven by a $d$-dimensional Lévy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(α_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/α_{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 $(α_{1},\dots,α_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.