Drift reduction method for SDEs driven by inhomogeneous singular L{é}vy noise
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 onedimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is onedimensional 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{ö}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.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.06595
 arXiv:
 arXiv:2208.06595
 Bibcode:
 2022arXiv220806595K
 Keywords:

 Mathematics  Probability