Supercritical McKean-Vlasov SDE driven by cylindrical $\alpha$-stable process

TL;DR

This paper investigates the well-posedness and approximation of supercritical McKean-Vlasov SDEs driven by cylindrical $ ext{alpha}$-stable processes, establishing strong solutions, propagation of chaos, and convergence of Euler schemes.

Contribution

It provides the first well-posedness results for supercritical McKean-Vlasov SDEs driven by cylindrical $ ext{alpha}$-stable noise with $eta$-H"older drifts, and analyzes particle and Euler approximations.

Findings

Established strong and weak well-posedness under specified conditions.

Proved strong propagation of chaos for the particle system.

Demonstrated convergence of Euler approximations and commutation properties.

Abstract

In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $\alpha$-stable process in $\mathbb{R}^d$ with $\alpha \in (0,1)$: $$ \mathord{{\rm d}} X_t = (K * \mu_{t})(X_t)\mathord{{\rm d}}t + \mathord{{\rm d}} L_t^{(\alpha)}, \quad X_0 = x \in \mathbb{R}^d, $$ where $K: \mathbb{R}^d \to \mathbb{R}^d$ is a $\beta$-order H\"older continuous function, and $\mu_t$ represents the time marginal distribution of the solution $X$. We establish both strong and weak well-posedness under the conditions $\beta \in (1 - \alpha/2, 1)$ and $\beta \in (1 - \alpha, 1)$, respectively. Additionally, we demonstrate strong propagation of chaos for the associated interacting particle system, as well as the convergence of the corresponding Euler approximations. In particular, we prove a commutation property between the particle approximation and the Euler approximation.