The Well-Posedness and Stabilities of Mean-Field Stochastic Differential Equations Driven by G-Brownian Motion

Abstract

Abstract. This article focuses on a type of mean-field stochastic differential equation driven by G-Brownian motion (G-MFSDEs for short). We obtain the well-posedness of G-MFSDEs by truncating their space and distribution variables to obtain an Euler-like sequence, and then prove the Euler-like sequence to be a Cauchy sequence. G-Brownian motion and distribution dependence which is defined via the G-expectation prevent us from directly applying the stability analysis techniques developed for G-SDEs and MFSDEs to G-MFSDEs. To overcome these difficulties, we establish the G-Itô formula for G-MFSDEs, then, using this as our basis, we investigate the moment exponential stability and the global stability in probability of G-MFSDEs. We also stabilize an unstable G-MFSDE by adding a discrete-time state observation feedback control into the drift coefficient. Two examples are presented to check our theory results.

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