Regularity of Generalized Mean-Field G-SDEs

We study the regularity properties of the unique solution of a generalized mean-field G-SDE. More precisely, we consider a generalized mean-field G-SDE with a square-integrable random initial condition, establish its first- and second-order Fréchet differentiability in the stochastic initial condition, and specify the G-SDEs of the respective Fréchet derivatives. The first- and second-order Fréchet derivatives are obtained for locally Lipschitz coefficients admitting locally Lipschitz first- and second-order Fréchet derivatives respectively. Our approach heavily relies on the Grönwall inequality, which leverages the Lipschitz continuity of the coefficients.