Drift reduction method for SDEs driven by inhomogeneous singular L{\'e}vy noise

TL;DR

This paper investigates stochastic differential equations driven by inhomogeneous symmetric Lévy noise, establishing uniqueness, regularity, and transition density representations, with a novel reduction method for handling drift terms.

Contribution

It introduces a new reduction technique to analyze SDEs with drift driven by inhomogeneous Lévy noise, providing regularity and density results.

Findings

Unique weak solutions under certain conditions

Hölder regularity of transition semigroup

Representation of transition probability density

Abstract

We study SDE $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \geq 0 $$ where $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump L\'evy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric $\alpha_i$-stable process ($\alpha_i \in (0,2)$ and they are not necessarily equal). Under certain assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probability density and we establish H{\"o}lder regularity of the corresponding transition semigroup. The method we propose is based on a reduction of an SDE with a drift term to another SDE without such a term but with coefficients depending on time variable. Such a method have the same spirit with the classic characteristic method and seems to be of independent interest.