Well-posedness of some non-linear stable driven SDEs

TL;DR

This paper establishes the well-posedness of certain non-linear McKean-Vlasov SDEs driven by symmetric alpha-stable Lévy processes, even with unbounded drifts, under mild regularity conditions.

Contribution

It introduces a methodology to prove well-posedness for non-linear SDEs with unbounded drifts in super-critical cases, expanding existing theoretical frameworks.

Findings

Proved well-posedness for non-linear SDEs driven by alpha-stable Lévy processes.

Established strong well-posedness results under mild regularity assumptions.

Extended analysis to include unbounded drift terms in super-critical regimes.

Abstract

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $R^d$ under some mild H{\"o}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $\alpha \in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.