Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability

Abstract

This work considers McKean–Vlasov equations driven by Lévy noise in which the coefficients rely not only on the state of the unknown process but also on its probability distribution, and researches the well-posed, attractiveness and stability of solutions with resolvent operator theory. Under global Lipschitz condition, the well-posed of mild solutions is first studied for McKean–Vlasov integro-differential equations by using contraction mapping principle. Then according to Cauchy–Schwartz inequality and Itô isometry formula, one gains global attracting set and quasi-invariant set of solutions of McKean–Vlasov integro-differential equations and also obtained sufficient conditions of stability with continuous dependence on the coefficient of solutions. Moreover, one attains mean-square stability of solutions by Gronwall inequality and almost sure stability by applying Borel–Cantelli lemma and Chebyshev inequality for the aforementioned equation. Finally two examples are presented to verify our results.