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Stability for multivalued McKean-Vlasov stochastic differential equations

Huijie Qiao, Jun Gong

Abstract

The work concerns multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the existence and uniqueness of strong solutions for multivalued McKean-Vlasov stochastic differential equations with non-Lipschitz coefficients. Then, the classical Itô's formula is extended to that for multivalued McKean-Vlasov stochastic differential equations. Finally, the asymptotic stability of second moments and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function are shown.

Stability for multivalued McKean-Vlasov stochastic differential equations

Abstract

The work concerns multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the existence and uniqueness of strong solutions for multivalued McKean-Vlasov stochastic differential equations with non-Lipschitz coefficients. Then, the classical Itô's formula is extended to that for multivalued McKean-Vlasov stochastic differential equations. Finally, the asymptotic stability of second moments and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function are shown.

Paper Structure

This paper contains 6 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Maximal monotone operators
  5. Derivatives for functions on ${\mathcal{P}}_{2}({\mathbb R}^d)$
  6. The existence and uniqueness of strong solutions

Key Result

Lemma 2.2

For $X\in C([0,T],\overline{{\mathcal{D}}(A)})$ and $K\in {\mathscr V}_{0}$, the following statements are equivalent:

Theorems & Definitions (17)

  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 7 more