Mean square exponential stability of stochastic function di ﬀ erential equations in the G-framework

: This research focuses on the stochastic functional di ﬀ erential equations driven by G-Brownian motion (G-SFDEs) with in ﬁ nite delay. It is proved that the trivial solution of a G-SFDE with in ﬁ nite delay is exponentially stable in mean square. An example is also presented to illustrate the e ﬀ ectiveness of the obtained theory.


Introduction
Stochastic differential equations (SDEs), which are often used to describe some stochastic dynamical systems, have been encountered in many science and engineering problems (see, e.g., [1,2]). In reality, these stochastic dynamical systems depend not only on the present but also on the past states. For such systems, stochastic functional differential equations (SFDEs) are used to described them (see, e.g., [3][4][5]).
In the study of stochastic dynamical systems, stability analysis, as a hot topic of stochastic dynamical systems, has aroused great concern (please see monographs [6] and [7]). So far, there are numerous literature on the stability of SFDEs (e.g., [8][9][10][11][12][13][14]). Among them, Zhou et al. [11] investigated the exponential stability of criteria for SFDEs with infinite delay. Pavlovic and Jankovic [12] studied both the pth moment and the almost sure stability on a general decay for SFDEs with infinite delay by Razumikhin approach. Ngoc [13] presented the criteria for the mean square exponential stability of general SFDEs with infinite delay. Li and Xu [14] discussed the exponential stability in mean square of SFDEs and neutral SFDEs with infinite delay by a novel approach.
Motivated by describing measuring finance risk and volatility uncertainty, Peng [15] has developed a theoretical framework of G-expectation. Based on the framework of G-expectation, Peng [15,16] introduced the G-Brownian motion and set up its Itô integral. Hu and Peng [17] found that a weakly compact family of probability measures can be used to represent the G-expectation. Under the G-framework, many efforts have been made to study the stability of SDEs driven by G-Brownian motion (G-SDEs), (see [18][19][20][21][22] and references therein). Faizullah et al. [23] studied the mean square exponential stability of nonlinear neutral G-SFDEs. Pan et al. [24] derived the pth moment exponential stability and quasi-sure (q.s.) exponential stability of impulsive G-SFDEs. Li [25] investigated the mean square stability with general decay rate of nonlinear neutral G-SFDEs. However, there are few results on the exponential stability of G-SFDEs with infinite delay, which motivates the present research. This article proves that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square.
This article is organized as follows. Section 2 presents some preliminaries. Section 3 proves that the trivial solution of a G-SFDE is exponentially stable in mean square under some conditions. Finally, an example is presented to illustrate the obtained results.

Preliminaries
This section briefly recalls some preliminaries in G-framework. More relevant details can be seen in [15,16,26,27].
On a non-empty basic space Ω, we can define a linear space of real-valued functions. We suppose that satisfies ∈ C for each constant C, and if ∈ X , is called a sublinear expectations; if for all ∈ X Y , , ∈ C and ≥ λ 0, it satisfies the following properties: (ii) Constant preserving: (iii) Sub-additivity: (iv) Positive homogeneity: The triple E Ω, ,ˆ ( ) is called a sublinear expectation space. If i ( ) and ii ( ) are satisfied, ⋅ Ê[ ] is called a nonlinear expectation and the triple E Ω, ,ˆ ( ) is relevantly called a nonlinear expectation space. For the details of G-normal distribution, G-expectation, G-conditional expectation, and G-Brownian motion, please see Chapter 3 and Chapter 4 of Peng [16].
Denote by and its countably many union equipped with the norm: where Ê stands for the G-expectation.
We now show the representation theorem of G-expectation as follows.
From the aforementioned lemma, the weakly compact family of probability characterizes the degree of Knightian uncertainty. If is singleton, that is P { }, then the model has no ambiguity, and the G-expectation Ê is the classical expectation. Then, define G-upper capacity ⋅ V ( ) and G-lower capacity ⋅ v( ) by: . A property is said to hold q.s. if it is true outside a polar set.
-q.s. means that it holds P-a.s. for each ∈ P . If an event U satisfies And t represents a filtration generated by G-Brownian motion ≥ ω t t 0 ( ( )) . In the following, we next show the stochastic integral with respect to the quadratic variation of G-Brownian motion.

Main results
In this section, we first give the following notations. Denote by be a generalized sublinear expectation space. Let ≥ ω t t 0 ( ( )) be a one-dimensional G-Brownian motion defined on the sublinear expectation space.
Consider the following G-SFDE The equation is also assumed to satisfy the basic conditions for the existence and uniqueness of the solution, which are the local Lipschitz condition and the linear growth condition. As for the proof of the existence and uniqueness of the solution, we can adopt the technique of [3] analogously.
In the following, the definition of the exponential stability in mean square for the G-SFDE (3.1) with the initial value (3.2) is given.
then the trivial solution of equation (3.1) with the given initial value (3.2) is exponentially stable in mean square.

x t f x t t e x t g x t d ω t e x t h x t ω t e h x t d ω t γe x t t e x t f x t t G e x t g x t e h x t t e x t g x t d ω t e x t h x t ω t e h x t d ω t G e x t g x t e h x t t
Then, we can further gain

x t γ e x s s e x s f x s s e G x s g x s h x s s M
being a G-martingale [16]