Stochastic flow for SDEs with jumps and irregular drift term

TL;DR

This paper establishes pathwise uniqueness and the existence of stochastic flows for non-degenerate SDEs driven by Levy noise with jumps, even with irregular, Holder continuous drift terms, extending previous results to broader classes of noise.

Contribution

It improves assumptions on Levy noise in SDEs with irregular drift, including relativistic and truncated stable processes, broadening the scope of stochastic flow existence results.

Findings

Proves pathwise uniqueness for SDEs with Holder continuous drift and Levy noise.

Establishes existence of stochastic flows under relaxed noise assumptions.

Extends results to relativistic and truncated stable processes.

Abstract

We consider non-degenerate SDEs with a $\beta$-Holder continuous and bounded drift term and driven by a Levy noise $L$ which is of $\alpha$-stable type. If $\alpha \in [1,2)$ and $\beta \in (1 - \frac{\alpha}{2},1) $ we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise $L$. In our previous paper $L$ was assumed to be non-degenerate, $\alpha$-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes of temperated stable processes.