Almost Surely Asymptotic Stability of Exact and Numerical Solutions for Neutral Stochastic Pantograph Equations

and Applied Analysis 3 where Vt t, x ∂V t, x ∂t , Vx t, x ( ∂V t, x ∂x1 , . . . , ∂V t, x ∂xn ) , Vxx t, x ( ∂2V t, x ∂xi∂xj )


Introduction
The neutral pantograph equation NPE plays important roles in mathematical and industrial problems see 1 .It has been studied by many authors numerically and analytically.We refer to 1-7 .One kind of NPEs reads x t − N x qt f t, x t , x qt .

1.1
Taking the environmental disturbances into account, we are led to the following neutral stochastic pantograph equation NSPE d x t − N x qt f t, x t , x qt dt g t, x t , x qt dB t , 1.2 which is a kind of neutral stochastic delay differential equations NSDDEs .
Using the continuous semimartingale convergence theorem cf. 8 , Mao et al. see 9, 10 studied the almost surely asymptotic stability of several kinds of NSDDEs.As most NSDDEs cannot be solved explicitly, numerical methods have become essential.Efficient numerical methods for NSDDEs can be found in 11-13 .The stability theory of numerical solutions is one of fundamental research topics in the numerical analysis.The almost surely asymptotic stability of numerical solutions for stochastic differential equations SDEs and stochastic functional differential equations SFDEs has received much more attention see 14-19 .Corresponding to the continuous semimartingale convergence theorem cf. 8 , the discrete semimartingale convergence theorem cf.17, 20 also plays important roles in the almost surely asymptotic stability analysis of numerical solutions for SDEs and SFDEs see 17-19 .To our best knowledge, no results on the almost surely asymptotic stability of exact and numerical solutions for the NSPE 1.2 can be found.We aim in this paper to study the almost surely asymptotic stability of exact and numerical solutions to NSPEs by using the continuous semimartingale convergence theorem and the discrete semimartingale convergence theorem.We prove that the backward Euler method BEM with variable stepsize can preserve the almost surely asymptotic stability under the conditions which guarantee the almost surely asymptotic stability of the exact solution.
In Section 2, we introduce some necessary notations and elementary theories of NSPEs 1.2 .Moreover, we state the discrete semimartingale convergence theorem as a lemma.In Section 3, we study the almost surely asymptotic stability of exact solutions to NSPEs 1.2 .Section 4 gives the almost surely asymptotic stability of the backward Euler method with variable stepsize.Numerical experiments are presented in the finial section.

Neutral Stochastic Pantograph Equation
Throughout this paper, unless otherwise specified, we use the following notations.Let Ω, F, {F t } t≥0 , P be a complete probability space with filtration {F t } t≥0 satisfying the usual conditions i.e., it is right continuous, and F 0 contains all P -null sets .B t is a scalar Brownian motion defined on the probability space.| • | denotes the Euclidean norm in R n .The inner product of x, y in R n is denoted by x, y or x T y.If A is a vector or matrix, its transpose is denoted by A T .If A is a matrix, its trace norm is denoted by |A| trace A T A .Let L 1 0, T ; R n denote the family of all R n -value measurable F t -adapted processes f {f t } 0≤t≤T such that Let C R n ; R denote the family of continuous functions from R n to R .Let C 1,2 R × R n ; R denote the family of all nonnegative functions V t, x on R × R n which are continuously once differentiable in t and twice differentiable in x.
x 0 , and 2.1 holds for every t ∈ 0, T with probability 1.
A solution x t is said to be unique if any other solution x t is indistinguishable from it, that is, To ensure the existence and uniqueness of the solution to 2.1 on t ∈ 0, T , we impose the following assumptions on the coefficients N, f, and g.Assumption 2.2.Assume that both f and g satisfy the global Lipschitz condition and the linear growth condition.That is, there exist two positive constants L and K such that for all x, y, x, y ∈ R n , and t ∈ 0, T , and for all x, y ∈ R n , and t ∈ 0, T ,

2.6
Assumption 2.3.Assume that there is a constant κ ∈ 0, 1 such that Under Assumptions 2.2 and 2.3, the following results can be derived.
Lemma 2.4.Let Assumptions 2.2 and 2.3 hold.Let x t be a solution to 2.1 with F 0 -measurable bounded initial data x 0 x 0 .Then The proof of Lemma 2.4 is similar to Lemma 6.2.4 in 21 , so we omit the details. 2.9 If lim i → ∞ A i < ∞ a.s., then for almost all ω ∈ Ω: that is, both X i and U i converge to finite random variables.

Almost Surely Asymptotic Stability of Neutral Stochastic Pantograph Equations
In this section, we investigate the almost surely asymptotic stability of 2.1 .We assume 2.1 has a continuous unique global solution for given F 0 -measurable bounded initial data x 0 .Moreover, we always assume that f t, 0, 0 0, g t, 0, 0 0, N 0 0 in the following sections.Therefore, 2.1 admits a trivial solution x t 0. To be precise, let us give the definition on the almost surely asymptotic stability of 2.1 .

3.4
Then, for any ε ∈ 0, γ * lim sup where γ * is positive and satisfies That is, Proof.Choose V t, x t t γ U t, x t − N x qt for t, x ∈ R × R n and γ > 0. Similar to the proof of Lemma 2.2 in 9 , the desired conclusion can be obtained by using the continuous semimartingale convergence theorem cf. 8 .

4.2
It is easy to see that the grid point t n satisfies qt n t n−m for n ≥ 0, and the step size h n satisfies For the given mesh H, we define the BEM for 2.1 as follows:

4.4
Here, Y n n ≥ −m is an approximation value of x t n and F t n -measurable.ΔB n B t n 1 − B t n is the Brownian increment.The approximations Y n−m n −m, −m 1, . . ., −1 are calculated by the following formulae:

4.8
Then, we can obtain that 4.9 which subsequently leads to where

4.11
By conditions 3.8 and 3.9 , we have

4.13
Inserting these inequalities to 4.12 and using Assumption 2.3 yield

4.14
Let and D n 4κ 2 2λ 4 h n .Using these notations, 4.14 implies that Then, we can conclude that

4.17
Abstract and Applied Analysis 9 We, therefore, have

4.18
Similar to 4.15 , from 4.4 , we can obtain that

4.19
where A n , B n , C n , D n n −m − 1, . . ., −1 are defined as before,

4.20
From 4.19 , we have where

4.22
Obviously A > 0. By 4.18 and 4.21 , we can obtain that

4.23
where Similar to the proof in 18 , we can obtain that M n is a martingale with M −m−1 0. Note that h i m−1 ≤ h i m and h i m h i /q for i ≥ −m.Then, we have

4.24
Using the condition 3.9 and lim i → ∞ h i ∞, we obtain that there exists an integer i * such that

4.32
Then, the desired conclusion is obtained.This completes the proof.

Numerical Experiments
In this section, we present numerical experiments to illustrate theoretical results of stability presented in the previous sections.Consider the following scalar problem: x 0.5t −8x t x 0.5t dt sin x 0.5t dB t , t ≥ 0, x 0 x 0 .

5.1
Abstract and Applied Analysis 13 For the test 5.1 , we have λ 1 11, λ 2 4, λ 3 0, and λ 4 1 corresponding to Theorem 3.4.By Theorem 3.4, the solution to 5.1 is almost surely asymptotically stable.Theorem 4.2 shows that the BEM approximation to 5.1 is almost surely asymptotically stable.In Figure 1, We compute three different paths Y n ω 1 , Y n ω 2 , Y n ω 3 using the BEM 4.4 with x 0 2, t 0 0.01, m 2. In Figure 2, three different paths Y n ω 1 , Y n ω 2 , Y n ω 3 of BEM approximations are computed with x 0 10, t 0 1, m 1.The results demonstrate that these paths are asymptotically stable.
Theorem 2.5.Let Assumptions 2.2 and 2.3 hold, then for any F 0 -measurable bounded initial data x 0x 0 , 2.1 has a unique solution x t on t ∈ 0, T .Based on Lemma 6.2.3 in 21 and Lemma 2.4, this theorem can be proved in the same way as Theorem 6.2.2 in 21 , so the details are omitted.The discrete semimartingale convergence theorem cf.17, 20 will play an important role in this paper.Let {A i } and {U i } be two sequences of nonnegative random variables such that both A i and U i are F i -measurable for i 1, 2, . .., and A 0 U 0 0 a.s.Let M i be a real-valued local martingale with M 0 0 a.s.Let ζ be a nonnegative F 0 -measurable random variable.Assume that {X i } is a nonnegative semimartingale with the Doob-Mayer decomposition Proof.Using the idea of Lemma 3.1 in 9 , we can obtain the desired result.
0 -measurable bounded initial data x 0 .Lemma 3.2.Let ρ : R → 0, ∞ and z : 0, ∞ → R n be a continuous functions.Assume that σ 1 : lim sup Lemma 3.3.Suppose that 2.1 has a continuous unique global solution x t for given F 0 -measurable bounded initial data x 0 .Let Assumption 2.3 hold.Assume that there are functions Theorem 3.4.Suppose that 2.1 has a continuous unique global solution x t for given F 0measurable bounded initial data x 0 .Let Assumption 2.3 hold.Assume that there are four positive constants λ 1 − λ 4 such that Theorem 3.4 gives sufficient conditions of the almost surely asymptotic stability of NSPEs 2.1 .Based on this result, we will investigate the almost surely asymptotic stability of the BEM with variable stepsize for 2.1 in the following section.
T f t, x, y ≤ −λ 1 |x| 2 λ 2 y 2 , g t, x, y 2 ≤ λ 3 |x| 2 λ 4 y To be precise, let us introduce the definition on the almost surely asymptotic stability of the BEM 4.4 .Assume that the BEM 4.4 is well defined.Let Assumption 2.3 hold.Let conditions 3.8 and 3.9 hold.Then the BEM approximate solution 4.4 obeys That is, the approximate solution Y n to the BEM 4.4 is almost surely asymptotically stable.
n /t −m .As a standard hypothesis, we assume that the BEM 4.4 is well defined.Proof.Set Y n Y n − N Y n−m .For n ≥ 0, from 4.4 , we have