Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large deviation principle for a two-time-scale McKean-Vlasov model with jumps

Xiaoyu Yang, Yong Xu

Abstract

This work focus on the large deviation principle for a two-time scale McKean-Vlasov system with jumps. Based on the variational framework of the McKean-Vlasov system with jumps, it is turned into weak convergence for the controlled system. Unlike general two-time scale system, the controlled McKean-Vlasov system is related to the law of the original system, which causes difficulties in qualitative analysis. In solving this problem, employing asymptotics of the original system and a Khasminskii-type averaging principle together is efficient. Finally, it is shown that the limit is related to the Dirac measure of the solution to the ordinary differential equation.

Large deviation principle for a two-time-scale McKean-Vlasov model with jumps

Abstract

This work focus on the large deviation principle for a two-time scale McKean-Vlasov system with jumps. Based on the variational framework of the McKean-Vlasov system with jumps, it is turned into weak convergence for the controlled system. Unlike general two-time scale system, the controlled McKean-Vlasov system is related to the law of the original system, which causes difficulties in qualitative analysis. In solving this problem, employing asymptotics of the original system and a Khasminskii-type averaging principle together is efficient. Finally, it is shown that the limit is related to the Dirac measure of the solution to the ordinary differential equation.
Paper Structure (6 sections, 4 theorems, 123 equations)

This paper contains 6 sections, 4 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Notations, Assumptions and Main Results
  3. Preliminaries and Notations
  4. Assumptions and Main Results
  5. Preliminary Lemmas
  6. Proof of the Main Result Theorem \ref{['thm']}

Key Result

Theorem 2.1

Assume (A1)--(A4), $\delta =o (\varepsilon)$, we let $\varepsilon \to 0$. The slow variable $X^{\varepsilon ,\delta}$ of two-time scale McKean-Vlasov model (1) satisfies the large deviation principle on $\mathbf{D}$ with the good rate function $I: \mathbf{D}\rightarrow [0, \infty)$ where $S_{\xi }:=\{(\psi,\phi)\in \mathbf{S}: \xi=\mathcal{G}^{0}(\psi,\phi)\}$ for $\xi\in\mathbf{D}$.

Theorems & Definitions (4)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3