Approximation of solution for Mckean-Vlasov SDEs under G-Brownian motion

Abstract

This paper investigates the McKean-Vlasov stochastic differential equation (MVSDEs), called also  mean-field stochastic differential equations stochastic differential equations (MFSDEs), driven by G-Brownian motion. Note that the coefficients depend not only on the state variable, but also in its marginal distribution, and the solutions of such equation are known in the literature as nonlinear diffusions. Under the assumption of Lipschitz continuous coefficients, we establish the existence and uniqueness of solutions using the method of successive approximations, also known as Picard iteration scheme. The approach leverages the contractive properties of the iteration scheme to rigorously demonstrate convergence to a unique solution. Furthermore, we analyze the stability of the solution with respect to initial condition and coefficients. By introducing small perturbations, we derive quantitative bounds that highlight the continuous dependence of the solution on these parameters. Our results contribute to the theoretical understanding of McKean-Vlasov SDEs in the context of G-expectation theory. These equations are important due to their significant role in a wide range of fields.