Estimates for the difference between approximate and exact solutions to stochastic differential equations in the G-framework

This article investigates the Euler-Maruyama approximation procedure for stochastic differential equations in the framework of G-Browinian motion with non-linear growth and non-Lipschitz conditions. Subject to non-linear growth condition, it is revealed that the Euler-Maruyama approximate solutions are bounded in M^2_G.In view ofnon-linear growth and non-uniform Lipschitz conditions,we give estimates for the difference between the exact solution Z(t) and approximate solutions Zq(t) of SDEs in the framework of G-Brownia nmotion.


Introduction
The stochastic differential equations (SDEs) theory has been used in several disciplines of sciences and engineering. In biological sciences, they are utilized to model the achievement of stochastic changes in reproduction on population processes [1,2]. In space, SDEs describe the transport of cosmic rays. They can be used to model the climate and weather. The percolation of fluid through absorbent structures and water catchment can be modelled by SDEs [3]. They are now very common in mechanical, computer, chemical and electrical engineering etc. By virtue of the growth and Lipschitz conditions, SDEs in the framework of G-Brownian motion were studied by Peng [4,5]. He derived the existence and uniqueness results in view of the contraction principle technique. With Picard's iteration scheme, the stated theory was developed by Gao [6]. By virtue of the Caratheodory approximation procedure, the existencuniqueness results were achieved by Faizullah [7]. The mentioned theory was extended to integral Lipschitz conditions by Bai and Lin [8]. Subject to the discontinuous coefficients, Faizullah derived that SDEs in the G-framework possess more than one solutions [9]. In the present article, we investigate the Euler-Maruyama approximation procedure for SDEs in the framework of G-Browinian motion with non-linear growth and non-Lipschitz conditions. Let 0 ≤ t 0 ≤ t ≤ T < ∞ and consider the following stochastic differential equation in the G-framework on t ∈ [t 0 , T] with given initial condition Z(t 0 ) = Z 0 . The quadratic variation process of G-Browniam motion For all x ∈ R n , the given coefficients g(., x), h(., x) and w(., x) belongs to space M 2 G ([t 0 , T]; R n ). The integral form of Equation (1) is given as the following where Z 0 ∈ R d is a given initial condition. All through the present article, we assume the following assump- where the function ϒ(.) : R + → R + is non-decreasing and concave with ϒ(0) = 0, ϒ(r) > 0 for r > 0 and Since ϒ is concave and ϒ(0) = 0, for all r ≥ 0, where C and D are positive constants. For every t ∈ [t 0 , T] and g(t, 0), h(t, 0), w(t, 0) ∈ L 2 , where M is a positive constant. Assumptions (3) and (6) are known as non-uniform Lipschitz and weakened linear growth conditions respectively. The current paper is organized in three more sections. Section 2 contains several basic results and concepts such as the definitions of G-expectation, G-Brownian motion, Ito's integral, Hölder's inequality, Doobs martingale's inequality and Gronwall's inequality etc. Section 3 presents the idea of Euler-Maruyama approximate solutions for SDEs in the G-framework. This section consists of an important result, which reveals that the Euler-Maruyama approximate solutions are bounded. In Section 4, we prove an important lemma, which is utilized in the main theorem. This section gives estimates for the difference between an exact solution Z(t) and approximate solutions Z q (t) of SDEs in the framework of G-Brownian motion.

Preliminaries
Building on the previous ideas of G-Brownian motion theory, this section is devoted to the preliminary notions and results required for the subsequent sections of this article. For more details on the concepts of G-Brownian theory, readers are suggested to consult the papers [10][11][12][13][14][15][16][17]. Let be a given fundamental non-empty set. Suppose H be a space of linear real functions defined on satisfying that (i) 1 ∈ H (ii) for every d ≥ 1, Z 1 , Z 2 , . . . , Z n ∈ H and ϕ ∈ C b.Lip (R n ) it holds ϕ(Z 1 , Z 2 , . . . , Z n ) ∈ H i.e., with respect to Lipschitz bounded functions, H is stable. Then ( , H, E) is a subexpectation space, where E is a sub-expectation defined as the following.
Moreover, let be the space of all R n -valued continuous paths (w t ) t≥0 starting from zero. Also, suppose that associated with the below distance, is a metric Fix T ≥ 0 and set where

Definition 2.2:
A d-dimensional stochastic process {W(t)} t≥0 satisfying the below properties is called a G-Brownian motion (7). Then the G-quadratic variation process { W t } t≥0 and G-Itô's integral I(μ) are respectively defined by The following two lemmas are borrowed from the book [18]. They are called as Hölder's and Gronwall's inequalities respectively. Lemma 2.4: Let p, q > 1, 1/p + 1/q = 1 and g, h ∈ L 2 .
For more details of the following (Burkholder-Davis-Gundy (BDG) inequalities) two lemmas, see [6]. Just for simplicity, all through this article we take k 1 = k 2 = 1.

Lemma 3.1: Let conditions
and M,C,D are already defined positive constants.

Proof: In view of the inequality
Apply subexpectation on both sides. Then in view of the Holder inequality (2.4), Lemmas 2.6 and 2.7, we proceed as the following E|g(s,Z q (s))| 2 ds E|h(s,Z q (s))| 2 ds Utilizing conditions (3) and (6), the last inequality yields In view of the notionZ q (s), we obtain where G 1 = 4E|Z 0 | 2 + 16T(T + 2)(M + C) and G 2 = 16D(T + 2). Consequently, an application of Gronwall's inequality gives The proof is complete.

Remark 3.2: Lemma 3.1 shows that for every
By an identical way as used in lemma 3.1, one can prove that for any T > 0, where K is a positive constant.

Estimates for the difference between approximate and exact solutions to SDEs in the G-framework
We now derive an important lemma, which will be utilized in the next theorem. Here we present estimates for the difference between an exact and approximate solutions for SDEs in the G-framework.

Lemma 4.1:
Assume that the hypothesis of Lemma 3.1 hold. Let t 0 ≤ r < t ≤ T. For all q ≥ 1, where H 1 = 12(T + 2)(M + C + KD) and M,C,D,K are already defined positive constants.
Proof: Let t 0 ≤ r < t ≤ T. For any q ≥ 1, Equation (9) becomes Use the inequality | 3 i=1 a i | 2 ≤ 4 3 i=1 |a i | 2 and apply subexpectation on both sides. Then in view of the Holder inequality (2.4), Lemmas 2.6 and 2.7, we proceed as the following In view of the notion ofZ q (s), we have By virtue of Lemma 3.1, we get Consequently, where H 1 = 12(T + 2)(M + C + KD). The proof stands completed.

Remark 4.2:
Using identical arguments as used in Lemma 4.1, one can prove that where H 1 is a positive constant. (3) and (6) hold. Then for all q ≥ 1 and any T > 0,

Theorem 4.3: Let
where C, D and H 1 are positive constants. (2) and (9) we derive

Proof: Using the fundamental inequality
Apply subexpectation on both sides.
By an application of the Grownwall's inequality we derive Using Lemma 4.1, we estimate N as follows substituting the value of N in (14) provides, The proof stands completed.

Conclusion
In recent years, the importance of SDEs has become more apparent due to their applications in modelling real life phenomena. Subject to the Lipschitz conditions, the existence theory for stochastic functional differential equations (SFDEs) in the G-framework was developed by Ren, Bi and Sakthivel [19]. The stated theory was extended to non-uniform Lipschitz conditions by Faizullah [20][21][22] and to discontinuous coefficients by Faizullah, Rahman, Afzal and Chohan [23]. Further, Faizullah established the pth moment estimates for the stated equations [24,25]. It is expected that the techniques used in the present paper can be used in several different directions such as to find estimates for the difference between the exact and approximate solutions for the above stated SFDEs in the G-framework, neural stochastic differential equations driven by G-Brownian motion [26] and stochastic differential equations with piecewise arguments in the G-framework etc. We hope that the current study will play a key role to establish a framework for the above mentioned problems.

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

Funding
This work is supported by the Commonwealth Scholarship Commission in the United Kingdom with CSC ID: PKRF-2017-429.