Under smoothness conditions on the coefficients, the solution map of a mean-field G-SDE is Fréchet differentiable up to second order, with derivatives characterized as solutions of new G-SDEs.
G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type
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abstract
We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.
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Regularity of Solutions of Mean-Field $G$-SDEs
Under smoothness conditions on the coefficients, the solution map of a mean-field G-SDE is Fréchet differentiable up to second order, with derivatives characterized as solutions of new G-SDEs.