Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes

TL;DR

This paper derives sharp two-sided estimates and regularity properties for the transition density of a diagonal system of SDEs driven by cylindrical lpha-stable processes, extending understanding of such stochastic systems.

Contribution

The paper constructs the transition density for diagonal SDE systems driven by cylindrical lpha-stable processes and provides sharp estimates and regularity results, using the Chen and Zhang method.

Findings

Established existence of the transition density for the system.

Derived sharp two-sided estimates for the density.

Proved Hf6lder and gradient regularity of the density.

Abstract

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 $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and H\"older 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 H\"older and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.