Averaging Principles For Stochastic Differential Equations Driven By G-Brownian Motion

A new type of Brownian motion,G-Brownian motion,is constructed under a sublinear expectation(or a nonlinear expectation).The corresponding robust version of It?'s calculus with respect to G-Brownian turns out to be a basic tool for problems of risk measures in finance and,more general,for decision theory under uncertainty.As a result,the G-expectation has become more and more widely used in the financial world.The introduction of the G-Brownian motion makes up for the lack of previous Brownian motion studies.The stochastic averaging method is one of the more common and important method of approximate analysis in the formation of the analytical methods.The stochastic averaging principle is the theoretical basis of this kind of approximate analytic method(stochastic averaging method),which provides a convenient and easy way to study the properties of complex systems.Unfortunately,so far no work has been reported about the averaging principle for SDEs driven by G-Brownian motion.At the same time,considering the limitation of the global or local Lipschitz condition,it is extremely urgent to set up the stochastic averaging principle of SDE driven by G-Brownian motion under non-Lipschitz condition(Non-Lipschitz condition is much weaker than the Lipschitz condition).This point motivates us to carry out the present study.1.We investigate the stochastic averaging principles for stochastic differential equations driven by G-Brownian motion.Under suitable condition,we show that the solutions of original SDEs can be approximated by the solutions of averaged SDEs in the sense of mean square.This averaging principle paves a way for reduction of computational complexity.The implication is: one can ignore the complex original systems and concentrates on the average systems instead.2.We concern stochastic differential equations driven by G-Brownian motion under non-Lipschitz condition which is a much weaker condition with a wider range of applications.Stochastic averaging is established for such non-Lipschitz SDEs where an averaged system is presented to replace the original one in the sense of mean square.