Abstract
In this paper, we aim to develop the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs for short) with non-Lipschitz coefficients. By the properties of G-Brownian motion and stochastic inequality, we prove that the solution of the averaged G-SDEs converges to that of the standard one in the mean-square sense and also in capacity. Finally, two examples are presented to illustrate our theory.
Highlights
1 Introduction The averaging principle for a dynamical system is important in problems of mechanics, control, and many other areas
The first rigorous results were obtained by Bogoliubov and Mitropolsky [3], and further developments were studied by Hale [9]
Denis et al [6] obtained some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by G-expectation
Summary
The averaging principle for a dynamical system is important in problems of mechanics, control, and many other areas. As is known to all, a lot of problems in theory of differential systems can be solved effectively by the averaging principle. With the developing of stochastic analysis theory, many authors began to study the averaging principle for differential systems with perturbations and extended the averaging theory to the case of stochastic differential equations (SDEs). For the potential applications in uncertainty problems, risk measures, and the superhedging in finance, the theory of nonlinear expectation has been developed. Peng [20] established a framework of G-expectation theory and G-Brownian motion. Denis et al [6] obtained some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by G-expectation. Luo and Wang [17] studied the sample solution of
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