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Large and Moderate deviation principles for the Multivalued McKean-Vlasov SDEs with jumps

Lingyan Cheng, Caihong Gu, Wei Liu, Fengwu Zhu

TL;DR

The paper addresses how to derive large and moderate deviation principles for multivalued McKean–Vlasov SDEs with jumps driven by Lévy noise, under non-Lipschitz coefficients. It employs the weak convergence method together with Bihari's inequality to handle law-dependent drift/diffusion and the multivalued operator $A$. The main results provide an LDP for $X^\varepsilon$ with a variational rate function $I(g)=\inf_{(\phi,\psi)\in S, g=Y^u}(Q_1(\phi)+Q_2(\psi))$ and a corresponding MDP under a vanishing scale, both under precise structural assumptions. These findings extend deviation principles to interacting particle systems with jumps and constraint-like operators, enabling exponential tail and moderate-deviation analyses in this broad MMVSDE setting.

Abstract

By using the weak convergence method, we establish the large and moderate deviation principles for the multivalued McKean-Vlasov SDEs with non-Lipschitz coefficients driven by Lévy noise in this paper. The Bihari's inequality is used to overcome the challenges arising from the non-Lipschitz conditions on the coefficients.

Large and Moderate deviation principles for the Multivalued McKean-Vlasov SDEs with jumps

TL;DR

The paper addresses how to derive large and moderate deviation principles for multivalued McKean–Vlasov SDEs with jumps driven by Lévy noise, under non-Lipschitz coefficients. It employs the weak convergence method together with Bihari's inequality to handle law-dependent drift/diffusion and the multivalued operator . The main results provide an LDP for with a variational rate function and a corresponding MDP under a vanishing scale, both under precise structural assumptions. These findings extend deviation principles to interacting particle systems with jumps and constraint-like operators, enabling exponential tail and moderate-deviation analyses in this broad MMVSDE setting.

Abstract

By using the weak convergence method, we establish the large and moderate deviation principles for the multivalued McKean-Vlasov SDEs with non-Lipschitz coefficients driven by Lévy noise in this paper. The Bihari's inequality is used to overcome the challenges arising from the non-Lipschitz conditions on the coefficients.
Paper Structure (18 sections, 17 theorems, 199 equations)

This paper contains 18 sections, 17 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Notation and Preliminaries
  5. Framework
  6. Energy-Constrained Spaces
  7. Wasserstein distance
  8. Maximal Monotone Operator
  9. Solutions to multivalued McKean-Vlasov SDEs with jumps
  10. Large deviation principle
  11. Bihari's inequality
  12. Bihari's inequality
  13. Large and Moderate deviation principles
  14. Large deviation principle
  15. Moderate deviation principle
  16. ...and 3 more sections

Key Result

Proposition 2.3

Let $(X,K)$ be a pair of functions with $X \in \mathcal{D}([0,T],\overline{D(A)})$ and $K \in V_0$. Then the following statement are equivalent:

Theorems & Definitions (41)

  • Remark 2.1
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Definition 2.2
  • Lemma 2.4
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 31 more