The LDP of McKean-Vlasov stochastic differential equations with H\"{o}lder continuous conditions and integrable conditions

TL;DR

This paper establishes the large deviation principle for both non-degenerate and degenerate McKean-Vlasov stochastic differential equations with H"{o}lder continuous drifts, introducing a new rate function definition and addressing multiple solutions in the skeleton equation.

Contribution

It extends LDP results to MVSDEs with H"{o}lder continuous drifts, including degenerate cases, and proposes a novel rate function applicable under weaker conditions.

Findings

LDP holds for non-degenerate MVSDEs with H"{o}lder drifts

LDP established for degenerate MVSDEs with H"{o}lder drifts

A new rate function reduces to the traditional one under Lipschitz conditions

Abstract

In this paper, we first study the large deviation principle (LDP) for non-degenerate McKean-Vlasov stochastic differential equations (MVSDEs) with H\"{o}lder continuous drifts by using Zvonkin's transformation. When the drift only satisfies H\"{o}lder condition, the skeleton equation may have multiple solutions. Among these solutions, we find one that ensures the MVSDEs satisfy the LDP. Moreover, we introduce a new definition for the rate function that reduces to traditional rate function if the drift satisfies the Lipschitz condition. Secondly, we study the LDP for degenerate MVSDEs with H\"{o}lder continuous drifts.