Well-posedness of some non-linear stable driven SDEs

Noufel Frikha; Valentin Konakov; Stéphane Menozzi

doi:10.3934/dcds.2020302

Article Contents

2021, Volume 41, Issue 2: 849-898. Doi: 10.3934/dcds.2020302

This issue Previous Article Asymptotic dynamics of a system of conservation laws from chemotaxis Next Article A generalization of the Babbage functional equation

Well-posedness of some non-linear stable driven SDEs

1.
Université de Paris, Laboratoire de Probabilités, Statistiques et Modélisation (LPSM), UMR 8001, F-75013 Paris, France
2.
Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation
3.
Laboratoire de Modélisation Mathématique d'Evry (LaMME), UMR CNRS 8070, Université d'Evry Val d'Essonne, Université Paris-Saclay, 23 Boulevard de France 91037 Evry, France
4.
Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation

^* Corresponding author: Stéphane Menozzi
^* Corresponding author: Stéphane Menozzi

Received: October 2019

Revised: June 2020

Early access: August 2020

Published: February 2021

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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 Lévy processes with values in $ {{{\mathbb R}}}^d $ under some mild Hölder 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.

Keywords:

McKean-Vlasov stable SDEs,
non-linear martingale problem,
weak uniqueness,
strong uniqueness,
perturbative techniques of parametrix type.

Mathematics Subject Classification: Primary 60H10, 60G46; Secondary 60H30, 35R09.

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References

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