Stabilities of delay stochastic McKean-Vlasov equations in the G-framework

This article focuses on a type of stochastic McKean-Vlasov equation in the G-framework (G-SMVE) and addresses the stability issue for delay stochastic McKean-Vlasov equations in the G-framework (G-SMVDEs). Distribution dependence and uncertainty prevent us from directly applying the stochastic analysis method for stochastic McKean-Vlasov equations (SMVEs) to G-SMVEs directly. To overcome this difficulty, we introduce definitions, including the derivative of a function with a law under the G-expectation and the Lions derivatives. Then we construct a new G-Itô formula for G-SMVEs according to the G-Itô formula for stochastic differential equations in the G-framework (G-SDEs) and the Ito formula for SMVEs. Using the new G-Itô formula and the Lyapunov functional method, we investigate the moment exponential stability and almost sure asymptotic stability of G-SMVDEs. To overcome the difficulty in obtaining the distribution dependence of the exact solution to the G-SMVDE, we introduce the empirical measure and the corresponding interacting particle system, and then prove the stability equivalence between the underlying G-SMVDE and the corresponding interacting particle system. Two examples are used to confirm our theoretical results.