On the averaging principle for SDEs driven by G-Brownian motion with non-Lipschitz coefficients

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.


Introduction
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. The first rigorous results were obtained by Bogoliubov and Mitropolsky [3], and further developments were studied by Hale [9]. 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). We refer the reader to Bao et al. [2], Chen et al. [5], Golec and Ladde [8], Khasminskii [11,12], Liptser and Stoyanov [14], Liu et al. [15], Stoyanov and Bainov [23], Wu and Yin [25], Xu et al. [28,29], Xu and Miao [26,27], Luo et al. [16], and the references therein.
On the other hand, 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. After that, the theory of G-SDEs has drawn increasing attention and has been studied subsequently by many authors. For example, Gao et al. [7] investigated the existence of the solution and large deviations for G-SDEs. Hu et al. [10] studied the regularity of the solution for the backward SDEs driven by G-Brownian motion. Luo and Wang [17] studied the sample solution of G-SDEs and obtained a new kind of comparison theorem. In the G-framework, Zhang and Chen [31] considered the quasi-sure exponential stability of semi-linear G-SDEs. By means of G-Lyapunov function method, Li et al. [13], Ren et al. [22], and Yin et al. [30] established the moment stability and the quasi-sure stability for nonlinear G-SDEs.
Compared with classical Brownian motion, the structure of G-Brownian motion is very complex. G-Brownian motion is not defined on a probability space but on the Gexpectation space. A natural question is as follows: Is there an averaging principle for SDEs driven by G-Brownian motion? In this paper, we shall investigate the averaging principle of nonlinear G-SDEs where B t is one-dimensional G-Brownian motion, B t is the quadratic variation process of the G-Brownian motion B t . Our main objective is to show that the solution of the averaged equation converges to that of the standard equation in the sense of mean square and capacity.
It is worth noting that most existing works of research on the averaging principle of SDEs require that the coefficients of SDEs are global Lipschitz continuous. In fact, the global Lipschitz condition imposed on [5,8,11,14,16,23,25,28] is quite strong when one discusses practical applications in the real world. For the past few years, many scholars have devoted themselves to finding some weaker conditions to study the averaging principle of stochastic system. Recently, some works on the averaging principle of stochastic system (see [18,24,29]) have been obtained under the Yamada-Watanabe condition: For any x, y ∈ R n and t ≥ 0, where k(·) is a continuous increasing concave function from R + to R + such that k(0) = 0 and 0 + 1 k(x) dx = ∞. However, this condition is somewhat restrictive because it does require that the control function k(x) of the modules of the continuity of the coefficients is concave, while this restriction excludes Eq. (4.4) of Example 4.3. In fact, f (t, x) of (4.4) does not satisfy condition (1.2) because log x -1 < (log x -1 ) 2 for x ≤ η. In this paper, we use the non-Lipschitz condition which arose in the study of the Brownian motion on the group of diffeomorphisms of the circle [1] to study the averaging principle for Eq. (1.1). Compared with (1.2), one will find that in our paper the coefficients f , h, and g of Eq. (1.1) are not assumed to be controlled by the concave functions. Thus, the conditions here are weaker than those of [18,24,29], and some results in [18,24,29] are generalized and improved.
The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminaries about G-Brownian motion. In Sect. 3, we establish the stochastic averaging principle of Eq. (1.1). By the Burkholder-Davis-Gundy inequality and some useful lemmas to be established, we prove that the solution of the averaged equation will converge to that of the standard equation in the sense of mean square and capacity. Finally, two illustrative examples are given in Sect. 4.

Preliminaries
Let us first recall some basic definitions and lemmas about G-Brownian motions. For more details, please see, e.g., [6,7,20].
Let be a given nonempty set, and let H be a linear space of real-valued functions defined on . We assume that H satisfies that c ∈ H for any constant c and |X| ∈ H for all X ∈ H.
whereX is an independent copy of X, y d = z means y and z are identically distributed.
is called a G-Brownian motion if the following conditions are satisfied: Now, let = C 0 (R + ) be the space of all real-valued continuous paths (w t ) t≥0 with w 0 = 0 equipped with the distance Consider the canonical process B t (w) = (w t ) t≥0 . For any T ≥ 0, we define is the space of all bounded, real-valued, and Lipschitz continuous functions on R n . Peng [20] defined the sublinear expectationÊ on ( , L ip ( )) so that the canonical process B t is a G-Brownian motion. This sublinear expectation is known as a G-expectation. For each p ≥ 1, L p G ( ) denotes the completion of L ip ( ) under the norm · p = (Ê| · | p ) 1 p . For a given pair of T > 0 and p ≥ 1, define , the Bochner integral and Itô integral are defined by respectively.
For the G-Brownian motion, we define the quadratic variation process of B t by Then Q 0,T can be uniquely extended to M 1 G (0, T). We still denote this mapping by .
Let B( ) be a Borel σ -algebra of . It was proved in [6] that there exists a weakly compact family P of probability measures defined on ( , B( )) such that

Stochastic averaging principle
In this section, we study the averaging principle of G-SDEs. Let us consider the standard form of Eq. (1.1): with the initial condition x ε (0) = x 0 ∈ R n . Here, f , h, g : R + × R n → R n are given functions and ε ∈ [0, ε 0 ] is a positive small parameter with ε 0 being a fixed number. In this paper, the following hypotheses are imposed on the coefficients f , h, and g.

Assumption 3.1
For any x, y ∈ R n and t ≥ 0, and where L is a positive constant and k i (·) are two positive continuous functions bounded on Letf ,h,ḡ : R n → R n be three functions, satisfying (3.2), (3.3), and (3.5) with respect to x. We also assume that the following condition is satisfied.
Remark 3.4 Under Assumptions 3.1-3.2, it is easy to conclude that the standard SDEs driven by G-Brownian motion (3.1) and the averaged one (3.6) have a unique solution, respectively (see Qiao [21]). Now, we present our main results which are used for revealing the relationship between the processes x ε (t) and y ε (t). x ε (t)y ε (t) 2 ≤ δ 1 . (3.7) In order to prove our main result, we need to introduce the following lemmas. In what follows, C > 0 is a constant which can change its value from line to line.