Existence of densities for stochastic differential equations driven by L\'evy processes with anisotropic jumps

TL;DR

This paper investigates the existence of probability density functions for solutions to stochastic differential equations driven by anisotropic Lévy processes, extending known results to more general jump structures with Hölder continuous coefficients.

Contribution

It establishes the existence of densities for SDE solutions driven by anisotropic Lévy processes under new conditions, including Hölder continuity and anisotropic jump behavior.

Findings

Density functions exist for solutions under specified conditions.

Results cover anisotropic stable laws and subordinate Brownian motions.

Provides a framework for analyzing densities in anisotropic jump processes.

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}.