Khasminskii-type theorems for stochastic differential delay equations driven by G-Brownian motion

Abstract The Khasminskii theorem have come to play an important role in the nonexplosion solutions of stochastic differential equations (SDEs) without the linear growth condition. In this paper, by using Peng's G-expectation theory, we establish an even more general Khasminskii-type test for stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) that cover a wide class of highly nonlinear G-SDDEs.


Introduction
Stochastic differential delay equations (SDDEs) have been widely applied in many fields, such as neural networks, automatic control, economics, ecology, etc. Stability is one of the most important topics in the study of SDDEs. Mao and Shah (1997) studied both pth moment and almost sure exponential stability of the stochastic differential delay equation. Yuan and Mao (2004) investigated the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching, and some sufficient criteria on the controllability and robust stability for linear SDDEs with Markovian switching. Recently, Fei, Hu, Mao, and Shen (2018) investigated a new Lyapunov function in order to deal with the effects of different structures as well as those of different parameters within the same modes. Moreover, the stochastic differential delay equations with Brownian motion is investigated by Mao and Yuan (2006), Mao (2008), Hu, Mao, and Yi (2013), Fei, Hu, Mao, and Shen (2017), Shen, Fei, Mao, and Yong (2018).
However, the classical stochastic differential equations with Brownian motion does not take an ambiguous factor into consideration. By using Peng's theory of G-expectation (see Peng, 2010), the research of the probability model with ambiguity makes a significant progress. Under the G-framework, Peng (2007) introduced the G-Gaussian distribution, G-Brownian motion and related stochastic calculus of Itô type. Since then, more and more scholar studied the related problems under the G-framework. Denis, Hu, and Peng (2011) obtained some CONTACT Yong Liang liangyong@ahpu.edu.cn important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation. Zhang and Chen (2012) investigated the sufficient conditions of the exponential stability and quasi sure exponential stability for stochastic differential equations driven by G-Brownian motion (G-SDEs). Fei and Fei (2013) investigated the exponential stability of paths for a class of stochastic differential equations disturbed by a G-Brownian motion in the sense of quasi surely (q.s.). Gao (2009) established the solutions of stochastic differential equations with Lipschitzian coefficients driven by G-Brownian motion. Li, Lin, and Lin (2016) studied the solvability and the stability of G-SDEs under Lyapunov-type conditions. Lin (2013) investigated the solvability of the scalar valued G-SDEs with reflecting boundary conditions. Recently, many interesting works have been done on the G-SDEs and G-SDDEs (2019) explored the stability and boundedness of solutions to highly nonlinear G-SDDEs. Ren, Jia, and Sakthivel (2016) discussed the pth moment stability of solutions to impulsive G-SDEs. Moreover, some other important properties of G-SDEs have been investigated by many researchers (see Deng, Fei, Fei, & Mao, 2019;Faizullah, 2016;In Press;Hu, Lin, & Hima, 2018;Li & Yang, 2018;Luo & Wang, 2014;Ren et al., 2016;Ren, Yin, & Sakthivel, 2018;Yin, Cao, & Ren, 2019;Yin & Ren, 2017). Mao (2002); Mao and Rassias (2005) established a Khasminskii-type test for SDDEs. Motivated by the above discussion, this paper will establish an even more general Khasminskii-type test for G-SDDEs that cover a wide class of highly nonlinear G-SDDEs.
The paper is organized as follows. In Section 2, we introduce some preliminaries and notations on sublinear expectations and G-Brownian motions. In Section 3, we give the Khasminskii-type theorems for G-SDDES. Next, we characterize the moment estimations in Section 4. Finally, the conclusion appears in Section 5.

Preliminaries
In this section, we recall some preliminary results of Gexpectation, which are needed in the sequel. The reader interested in more detailed description of these notions is referred to Gao (2009), Peng (2010 and Denis et al. (2011).
Let be a given nonempty set and H be a linear space of real valued functions defined on . We suppose that H satisfies c ∈ H for each constant c and |X| ∈ H if X ∈ H. Definition 2.2: Let ( , H,Ê) be a sublinear expectation space, a random vector Y = (Y 1 , . . . , Y n ), Y i ∈ H, is said to be independent underÊ from another random vector X = (X 1 , . . . , X n ),

Definition 2.3: In a sublinear expectation space
is said to be G-normal distribution, if for a, b ≥ 0, we have aX + bX ∼ √ a 2 + b 2 X, for eachX ∈ H, which is independent to X andX ∼ X.
Let = C 0 (R + ) be the space of all R-valued continuous paths with ω 0 = 0 equipped with the norm Brownian motion if the following properties are satisfied: Remark 2.5: we define For more properties of the Itô integral, one can see Peng (2010).

Proposition 2.8: There exists a weakly compact family of probability measures
is the linear expectation with respect to P.
For this P, we define the associated G-upper capacity V(·) and G-lower capacity V(·) by:
We denote C 2,1 (R n × [−τ , ∞); R + ) as the family of nonnegative functions V(x, t) defined on R n × [−τ , ∞), We define an operator LV : Let X(t) be a solution of G-SDDE (1), for convention, we use the following notation in the sequel and for each (x, y, t) ∈ R n × R n × R + , where θ 1 and θ 2 are positive constants.
Theorem 3.3: Let Assumptions 3.1 and 3.2 hold. Then there is a unique global solution x(t) to (1) on t ∈ [−τ , ∞) for any given initial data (2). Moreover, the solution has the properties that for all t ∈ R + .

Example 3.4: For simplicity, setting
is a one-dimensional Brownian motion and both a(t) and b(t) are bounded real-valued functions on t ≥ 0. Let V(x, t) = X 4 . Then the corresponding operator LV : R × R × R + has the form Similar to Mao and Rassias (2005), there is a unique global solution X(t) to Equation (19) on t ∈ [−τ , ∞). Moreover, the solution has the properties that

Moment estimations
In the previous section, under Assumptions 3.1 and 3.2, we have not only obtained the existence and uniquenness theorem on the global solution for the G-SDDE, but also showed that the solution has the properties that However, these estimations are not precise enough. Thus, we will replace condition (5) with many specified conditions.

Conclusion
The Khasminskii theorem have come to play an important role in the nonexplosion solutions of SDEs without the linear growth condition. In this paper, by using Peng's theory of sublinear expectations, we have established an even more general Khasminskii-type test for G-SDDEs that cover a wide class of highly nonlinear G-SDDEs, this test can be applied to many important G-SDDEs.

Disclosure statement
No potential conflict of interest was reported by the authors.