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Second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps: existence, uniqueness and averaging principle

Chungang Shi

Abstract

In the paper, a class of second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps with non-Lipschitz conditions is considered. The existence and uniqueness of the mild solution is established by means of the Carath${\rm \acute{e}}$odory approximation technique. Furthermore, an averaging principle is obtained between the solution of the second-order McKean-Vlasov stochastic evolution equation and that of the simplified equation in mean square sense.

Second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps: existence, uniqueness and averaging principle

Abstract

In the paper, a class of second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps with non-Lipschitz conditions is considered. The existence and uniqueness of the mild solution is established by means of the Carathodory approximation technique. Furthermore, an averaging principle is obtained between the solution of the second-order McKean-Vlasov stochastic evolution equation and that of the simplified equation in mean square sense.
Paper Structure (4 sections, 8 theorems, 82 equations)

This paper contains 4 sections, 8 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Existence and Uniqueness
  4. Averaging principle

Key Result

Proposition 3

Assume that $A$ is the infinitesimal generator of a cosine family $\{C(t):t\in\mathbb{R}\}$. Then the following properties hold, (1) There exists $M_{A}\geq 1$ and $\omega\geq0$ such that $\|C(t)\|\leq M_{A}e^{\omega|t|}$ and hence $\|S(t)\|\leq M_{A}e^{\omega|t|}$. (2) $A\int_{s}^{r}S(u)x=[C(r)-C(s

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Remark 4
  • Lemma 5: Ma
  • Lemma 6: A
  • Lemma 7: RS1
  • Remark 8
  • Definition 9
  • Lemma 10
  • ...and 7 more