Existence, Uniqueness, and Stability of distribution-dependent Stochastic Differential Equations Driven by G-Brownian Motion

McKean-Vlasov stochastic differential equations (MVSDEs), also known as mean-field
SDEs, are essential for describing systems where particle coefficients depend on both the
individual state and the marginal distribution of the population. While classical theory is
well-established, this study explores these equations within the G-expectation framework
to address model uncertainty and volatility ambiguity.
We investigate the G-MVSDE under the assumption of Lipschitz continuity for the
drift and diffusion coefficients with respect to the state variable and the 2-Wasserstein
distance. To establish the existence and uniqueness of solutions, the Picard successive
approximation scheme is employed. The convergence of this iterative process is rigorously
analyzed using G-stochastic calculus, specifically leveraging the subadditive properties of
G-expectation and Burkholder-Davis-Gundy (BDG) type inequalities.
The research demonstrates that the Picard iteration sequence forms a Cauchy se-
quence in the complete space L2(Ω, C([0, T ], R^d)), ensuring convergence to a unique strong
solution. Furthermore, we establish the stability of these solutions by deriving quanti-
tative bounds. By applying Gronwall’s inequality, we prove that the solution depends
continuously on small perturbations in initial conditions and coefficients.
This study advances the theoretical foundation of nonlinear diffusions under G-Brownian
motion. By confirming that G-MVSDEs are well-posed and stable, this work provides a
robust framework for applications in mean-field games, financial risk management, and
large-scale interacting particle systems where uncertainty is a primary factor.