On Mean Field Stochastic Differential Equations Driven by $$G$$ -Brownian Motion with Averaging Principle

Abstract

In a sublinear space \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\), we consider Mean Field stochastic differential equations (\(G\)-MFSDEs in short), called also \(G\)-McKean–Vlasov stochastic differential equations, which are SDEs where coefficients depend not only on the state of the unknown process but also on its law. We mean by law of a random variable \(X\) on \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\), the set \(\left\{P_{X}:P\in\mathcal{P}\right\}\), where \(P_{X}\) is the law of \(X\) with respect to \(P\) and \(\mathcal{P}\) is the family of probabilities associated to the sublinear expectation \(\widehat{\mathbb{E}}\). In this paper, we study the existence and uniqueness of the solution of \(G\)-MFSDE by using the fixed point theorem. To this end, we introduce a new type Kantorovich metric between subsets of laws and adapted Lipchitz and linear growth conditions. Furthermore, we prove the validity of the averaging principle and obtain convergence theorem where the solution of the averaged \(G\)-MFSDE converges to that of the standard one in the mean square sense.