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Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises

Chang-Song Deng, Xing Huang

Abstract

We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric $α$-stable Lévy process ($1/2<α\leq1$), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case $α\in(0,1]$. The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach's fixed point theorem.

Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises

Abstract

We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric -stable Lévy process (), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case . The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach's fixed point theorem.
Paper Structure (4 sections, 5 theorems, 53 equations)

This paper contains 4 sections, 5 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Some preparations
  3. Proof of Theorem \ref{['EUS']}
  4. Appendix

Key Result

Theorem 1.1

Assume $(A)$ or $(A')$. Then E1 is strongly/weakly well-posed in $\mathscr P_{\eta}$, and the solution satisfies $\mathscr L_{X_\cdot}\in C([0,T];\mathscr P_{\eta})$ and

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['EUS']}
  • ...and 3 more