On the construction of the solution of mean field stochastic differential equations driven by G-Brownian motion via small delays

This work explores the strong convergence of the Carathéodory approximation scheme, for nonlinear mean-field stochastic differential equations called also McKean-Vlasov stochastic differential equations (MVSDEs), under the framework of G-Brownian motion. Note that the coefficients dependent on the state variable and its marginal distribution . This numerical scheme is defined by a series of stochastic processes described through the sequence of delayed McKean-Vlasov stochastic differential equations driven by G-Brownian motion. The benefit of the Carathéodory iteration scheme is its capability to explicitly resolve each stochastic delay equation by performing successive stochastic integrations over small intervals. We prove that under Lipschitz continuity, the Carathéodory approximations converge to the unique solution of these equations. In this context, we present an alternative proof for the existence and uniqueness of a strong solution of McKean-Vlasov stochastic differential equation driven by G-Brownian motion. This method gives an effective way to construct numerically the unique strong solution of the equation in question.