On the pth moment estimates of solutions to stochastic functional differential equations in the G-framework

The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder’s inequality, Bihari’s inequality, Gronwall’s inequality and Burkholder–Davis–Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.

which were established by him included G-Itô's integral, G-Itô's formula and G-quadratic variation process B . A new and interesting phenomenon that is related to the G-Brownian motion is the fact that its quadratic variation process, which is also a continuous process, has got stationary and independent increments. Therefore, it continues to qualify for being termed as a Brownian motion. Thus, the idea of G-framework-related stochastic differential equations was initiated (Peng 2006(Peng , 2008. Due to the applicability of the theory, many authors published their work on this emerging phenomenon in a short span of time (Bai and Lin 2014;Denis et al. 2010;Xua and Zhang 2009). As important as the existence theory, moment estimate is one of the most useful and basic schemes of analyzing dynamic behavior of SFDEs. It is also worth noting that the pth moment of the solution for such SDEs driven by G-Brownian motion with non-linear growth condition has not been fully explored, which remains an interesting research topic. This article will fill the mentioned gap. We present the analysis for the solution to the following SFDE in the G-framework with initial data Y t 0 = ζ satisfying It is understood that Y(t) is the value of stochastic process at time t and Y t = {Y (t + θ): − ρ ≤ θ ≤ 0, ρ > 0}, indicates BC([−ρ, 0]; R)-valued stochastic process, which is a collection of continuous and bounded real valued functions ϕ defined on [−ρ, 0] having norm �ϕ� = sup −ρ≤θ ≤0 | ϕ(θ) |. The coefficients κ, and µ are Borel measurable real valued functions on [0, T ] × BC([−ρ, 0] (Faizullah et al. 2016). The rest of the paper is organized as follows: "Preliminaries" section is devoted to some basic definitions and results. "pth Moment estimates for SFDEs in the G-framework" section presents the pth moment estimates for SFDEs in the G-framework, under non-linear growth condition. "Continuity of pth moment for SFDE in the G-framework" section shows that the pth moment of solution to SFDE is continuous. The path-wise asymptotic estimates are given in "Path-wise asymptotic estimate" section.

Preliminaries
In this section some fundamental notions and results are given, which are used in the forthcoming sections of this paper. For more detailed literature of G-expectation, see the papers Denis et al. (2010), Faizullah (2012, Li andPeng (2011), Song (2013) and book Peng (2010).
Definition 1 Let H be a linear space of real valued functions defined on a nonempty basic space . Then a sub-linear expectation E is a real valued functional on H with the following features: For any real constant γ , (1) Let C b.Lip (R l×d ) denotes the set of bounded Lipschitz functions on R l×d and denotes the collection of processes of the following type: where the above process is defined on a partition π T = {t 0 , t 1 , . .
Then (B t ) t≥0 is known as G-Brownian motion.
Lemma 3 If 1 q + 1 r = 1 for any q, r > 1, g ∈ L 2 and h ∈ L 2 then gh ∈ L 1 and The following two lemmas are borrowed from the book Mao (1997).
Theorem 7 Let Y ∈ L p . Then for each ǫ > 0, In the above Theorem 7, Ĉ is known as capacity defined by Ĉ (H ) = sup P∈P P(H ), where P is a collection of all probability measures on (�, B(�) and H ∈ B(�), which is Borel σ-algebra of . Also, we remind Ĉ (H) = 0 means that set H is polar and a property holds quasi-surely (q.s. in short) means that it holds outside a polar set. The rest of the paper is organized as follows. In "Preliminaries" section, the pth moment estimates are studied. In "pth Moment estimates for SFDEs in the G-framework" section, continuity of pth moment is shown. In "Continuity of pth moment for SFDE in the G-framework" section, path-wise asymptotic estimates for SFDEs driven by G-Brownian motion are given.

pth Moment estimates for SFDEs in the G-framework
Let Eq. (1) admit a unique solution Y(t). Assume that a non-linear growth condition holds, which is given as follows.

Theorem 8 Assume that the non-linear growth condition
, c 2 and c 3 are positive constants.

Continuity of pth moment for SFDE in the G-framework
In the next theorem, under non-linear growth condition, it is shown that the pth moment of the solution to SFDE in the G-framework (1) is continuous.
Proof By using the inequality (a + b + c) p ≤ 3 p−1 (a p + b p + c p ), Eq. (1) follows Applying G-expectation on both sides, using the BDG inequalities (Gao 2009), Holder's inequality and non-linear growth condition, we proceed as follows

Theorem 10 Assume that the non-linear growth condition (3) holds. Then
Proof For each k = 1, 2, . . . , using the non-linear growth condition in a similar fashion as in Theorem 8, Eq. (10) we obtain, Recall that E is a sub-linear expectation. Unlike a classical expectation, it is not based on a particular probability space. So, instead of probability, we use a different concept known as capacity. Thanks to Theorem 7 for any arbitrary ǫ > 0, we have The Borel-Cantelli lemma follows for almost all w ∈ , there exists a random integer k 0 = k 0 (w) such that consequently, we get But ǫ is arbitrary, so The proof is complete.

Conclusion
Generally, we cannot find explicit solutions to nonlinear SDEs. Thus one needs to present the analysis for solutions to these equations. Existence and moment estimates are the most important characteristics for solutions to SDEs. Here, we have used some important inequalities such as Bihari's inequality, Hölder's inequality, Gronwall's inequality and Burkholder-Davis-Gundy (BDG) inequalities to investigate the pth moment estimates for SFDEs driven by G-Brownian motion. Then the asymptotic estimates for these equations have been developed. Furthermore, continuity of pth moment for the solutions to SFDEs in the G-framework has been proved. The G-Brownian motion theory is the generalization of the classical Brownian motion theory. The methodology used to estimate pth moment for SDE is interesting and applicable in various practical applications. For example, pth moment estimates are useful in biological population models (Shang 2013a) and distributed system control (Shang 2012(Shang , 2013b(Shang , 2015. The methods of the pth moment estimation, developed in our paper, can be used to extend the related theory in above mentioned papers.