Convergence of the EM method for NSDEs with time-dependent delay in the G-framework

ABSTRACT Consider a neutral stochastic differential equation (NSDE) with time-dependent delay () in the G-framework where denotes a G-Brownian motion and the quadratic variation process of . We introduce an Euler–Maruyama (EM) method for solving this equation and prove that the EM approximate solution converges to the exact solution with a strong order of the mean square closeness equal to one under the global Lipschitz condition. A numerical example is provided to illustrate the effectiveness of our method.


Introduction
Motivated by the problems of asset pricing, risk measures, financial decisions under model uncertainty, Peng established the framework of G-expectation (G-framework), G-Brownian motion and related stochastic calculus of Itô type (see Peng, 2007Peng, , 2010. Since then, the theory of stochastic differential equations driven by G-Brownian motion (G-SDEs) has been extensively studied (see Denis, Hu, & Peng, 2011;Faizullah, 2016;Luo & Wang, 2014;Ren, Yin, & Sakthivel, 2018;Wei, Zhang, & Luo, 2017;Zhang & Chen, 2012). Luo and Wang (2014) proved that the integration of G-SDE in R can be reduced to the integration of ordinary differential equation parameterized by a variable in ( , F).  investigated the consistency of the least squares estimator (LSE) of the parameter for SDEs under distribution uncertainty and developed an algorithm for estimating the G-expectation. Wei et al. (2017) gave the asymptotic estimates for the solution of G-SDEs. Yang and Zhao (2016) introduced a numerical method for simulating G-Brownian motion. Fei et al. explored the stability and boundedness of solutions to highly nonlinear G-SDDEs in . These results lay the theoretical foundation for further research on the G-SDEs.
CONTACT Yong Liang liangyong@ahpu.edu.cn This article has been republished with minor changes. These changes do not impact the academic content of the article.
We know that the stability of the classical stochastic differential equations is an important topic in the study of stochastic systems (see Fei, Hu, Mao, & Shen, 2019;Shen, Fei, Mao, & Liang, 2018). Recently, many researchers have showed great interests to the stochastic stability in the Gframework (see Hu, Ren, & Xu, 2014;Li, Lin, & Lin, 2016;Ren, Jia, & Sakthivel, 2016;Zhang & Chen, 2012;Zhu, Li, & Zhu, 2017). Zhang and Chen (2012) discussed the exponential stability for G-SDEs. Li et al. (2016) studied the solvability and the stability of G-SDEs under Lyapunov-type conditions. Using the G-Lyapunov function technique to investigate the p-the moment stability of solutions to G-SDEs can be found in the references Hu et al. (2014), Ren et al. (2016), and Yin and Ren (2017). However, in the applications of G-SDEs, most of these equations can not be analytically solved, we have to resort to the numerical methods. Unfortunately, the research and papers on this issue are quite few. Li and Yan (2018) considered the stability of the EM method for solving the delayed Hopfield neural networks under the G-framework. Yang and Li (2019) proved that under global Lipschitz assumption a G-SDE is p-th (0 < p < 1) moment exponentially stable if and only if the stochastic θ-method is also pth moment exponentially stable for sufficiently small step size. In Li and Yang (2018), they also showed that the stochastic θ -numerical solution converges to the exact solution for the neutral stochastic delay differential equation driven by G-Brownian motion (G-NSDDE) if the coefficients of the equation satisfy the global Lipschitz assumption. Moreover,  discussed the stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the EM method.
Based on the above discussions, we are interested in designing a numerical approach for solving the G-NSDE with time-dependent delay such that the numerical solution converges to the true solution in the sense of mean square. In the case where the SDEs are driven by the classical Brownian motion, numerical methods for SDEs or NSDDEs have been discussed by many authors. We refer the reader to Deng, Fei, Liu, and Mao (2019), Feng, Qiu, Meng, and Rong (2019), Liu and Mao (2016), Liu, Li, and Deng (2018), Mo, Deng, and Zhang (2017), Mao, Zhu, and Mao (2015), Tan, Wang, Guo, and Zhu (2014), and Zong and Wu (2016), and the literature cited therein. It is worth mentioning that Milošović (2011) established the convergence in probability of the EM approximate solution for a highly nonlinear NSDEs with timedependent delay under the Khasminskii-type conditions. Naturally, we follow the train of his thought in this paper. However, under the G-framework, distribution uncertainty of G-Brownian motion brings about difficulties for estimating the error between the numerical and the exact solutions, we develop new technique to overcome these by virtue of stochastic analysis technique. Thus, this paper is not simply a trivial extension of the existing results to the more complex models. In reference to the existing results in the literature, we make the following contributions.
(1) A computational method is, for the first time, developed for a class of neutral stochastic differential equations with variable delay under the Gframework.
(2) A comprehensive system model is proposed to account for the phenomena of time-dependent delay and distribution uncertainty of stochastic disturbances.
(3) New mathematical techniques are well applied to solve the difficulties due to G-Brownian motion and variable delay. (4) A numerical example including G-expectation simulation is provided to show the convergence order.
The rest of the paper is organized as follows. In Section 2, we present the essential notations, definitions and propositions which are necessary for the whole work. In Section 3, we introduce the EM method for NSDEs with time-dependent delay in the G-framework. The convergence results are shown in Section 4. In Section 5, we give a numerical example to support our theory. At the end, we conclude the paper and points out some future research.

Preliminaries
Let us begin with the notion of a sublinear expectation space ( , H,Ê), where is a given set and H is a linear space of real valued functions defined on . The space H can be viewed as the space of random variables.
Definition 2.1: A sublinear expectationÊ is a functional E : H → R satisfying: (1) Monotonicity: Denote by H t the filtration generated by G-Brownian motion {B(t)} t≥0 and B (t) the quadratic variation process of B(t).
Let | · | be the Euclidean norm in R d . If A is a vector or matrix, its norm is denoted by |A| = trace(A T A), where A T is the A's transpose. If a is a real number, its integer part is denoted by a . Let τ > 0. Denote by For more details on G-Brownian motion, Itô integral and G-SDEs, one can refer the reference Peng (2010). Before stating the main results, we present two useful propositions.
Proposition 2.2 (Peng, 2010) By Propositions 2.2 and 2.3, we deducê Consider a G-NSDE with time-dependent delay described by the following form: In this paper, we assume that delay function δ : R + :→ [0, τ ] is continuous and there is a positive constant ρ such that Now, we rewrite G-NSDE (5) as the following integral form: For the purpose of the following consideration, we impose the following assumptions:

Assumption 2.4 (Global Lipschitz condition):
There is a positive constant L such that ∀ x 1 , x 2 , y 1 , y 2 ∈ R d ,

Assumption 2.5 (Contractive mapping condition):
There is a constant ν ∈ (0, 1) such that ∀ x, y ∈ R d , Moreover, we suppose that D(0) = 0 which, together with (9), means that Assumption 2.6: There is a positive constant β such that From Assumption 2.4, we can deduce that f, g and h satisfy the linear growth condition, that is, for any

The EM method for NSDEs with time-dependent delay in the G -framework
Let us propose the EM scheme for the G-NSDE (5). For any given time T ≥ 0, there are sufficiently large integers m, n ≥ 0 such that = τ/m = T/n. Set t k = k and δ k = δ(k )/ . Now, we introduce the discrete EM approximate solution Y to (5) as follows: where . The step processes of EM solution are defined by where I A denotes the indicator function of the set A. Then, we define the continuous EM approximate solution Letting t = t k in (14), then we get that which means that the continuous and discrete EM solutions coincide at the grid points. Moreover, on the basis of (14) and (15), we get that, for any t ∈ [t k , t k+1 ),

Main results
In this section, we prove that the order of the mean square closeness of the exact solution x(t) for system (5) and the corresponding EM solution y(t) is equal to one. The following theorem shows the convergence and convergence rate.
In order to establish this theorem, we need to show some useful lemmas. Lemma 4.2: Let p > 1 and Assumption 2.5 hold. Then Proof: Recall the elementary inequality that for any a,b > 0, p > 1 and η > 0, Then, Assumption 2.5 implies that for any η > 0, Rearranging this gives which means the desired assertion. Thus, we complete the proof.
The following lemma shows that y(t) and z 1 (t) are close to each other in the mean square sense.

Proof:
For any t ∈ [t k , t k+1 ), using the linear growth condition, from (16) we get that In view of Lemma 4.4, we have that Then, for any t ∈ [−τ , T] and ∈ (0, 1), we get sup −τ ≤t≤TÊ Thus, we complete the proof.
Lemma 4.6: Suppose that delay function δ(t) satisfies condition (6). Then, for any t ≥ 0, Proof: By the definition of the integer part function, we get that for any a, b ∈ R, a ≤ a, By (6) and (35), we have On the basis of Assumption 2.6 and the triangle inequality, we get that Combining (36) and (37), we have Thus, we complete the proof.
where C 4 is a positive constant independent of .
Remark 4.9: Compared with the results in Li and Yang (2018), time-dependent delay is taken into account to investigate the convergence of the Euler-type method for NSDEs in the G-framework, which generalizes the traditional G-model and adapts to more general delay conditions.

Remark 4.10:
It should be pointed out that when σ = σ , these results reduce to the corresponding classical stochastic ones that were discussed in the reference Milošović (2011). We extend the convergence results of the EM method for NSDE with variable delay to the case of G-framework.

Numerical experiment
In this section, we perform a numerical experiment to confirm our theoretical results. We focus on the error between the true solution x and the corresponding EM solution y at the endpoint T defined by e strong (T) =Ê|x(T) − y(T)|.
We note that Theorem 4.1 implies that Now, we use the algorithm introduced byFei and Fei (2019) to simulate the G-expectation of the absolute error between the exact solution x and EM solution y. Let B(t) ∼ N (0, [σ 2 , σ 2 ]t). The partition of the interval [σ 2 , σ 2 ] is defined as For a fixed Brownian motion B k (t) ∼ N (0, σ 2 k t)(k = 1, 2, . . . , M), we denote the exact solution of (5) driven by B k (t) as x k (t) = x(t; B k (t)), that is, for any t ≥ 0 The corresponding continuous EM solution can be represented as y k (t) = y(t; B k (t)). Discrete EM solution is defined by where B k i = B k ((i + 1) ) − B k (i ) ∼ N (0, σ 2 k ). For k = 1, 2, . . . , M, we perform J sampling to estimate the expectations of absolute error |x k (t) − y k (t)|. In the j-th random sampling (j = 1, 2, . . . , J), we represent x kj (t) and y kj (t) as the solutions defined by x k (t) and y k (t), respectively. To approximate e strong , we define the maximum sample average of absolute error between x and y at time T as follows: In our numerical experiment, we take the J = 500 and M = 6.
From Figure 1, we see that there appears to exist positive constants C and γ such that e strong (T) ≤ C γ for sufficiently small . A least squares fit for log C and γ is computed, producing the value 0.5752 for γ , which is close to the order obtained by (59). Clearly, our simulation results are consistent with the theoretical ones.

Conclusion
In this paper, the convergence of the EM method for NSDEs with variable time lag in the G-framework is studied. The results show that the order of the mean square closeness of the approximate solution y and the exact solution x is equal to one. A numerical experiment including G-expectation simulation is given to demonstrate our theory. Our future research topic will try to relax the global Lipschitz condition to the local one.

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