Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 143079, 14 pages
doi:10.1155/2011/143079
Research Article
Almost Surely Asymptotic Stability of Exact and
Numerical Solutions for Neutral Stochastic
Pantograph Equations
Zhanhua Yu
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
Correspondence should be addressed to Zhanhua Yu, yuzhanhua@yeah.net
Received 26 March 2011; Revised 29 June 2011; Accepted 29 June 2011
Academic Editor: Nobuyuki Kenmochi
Copyright q 2011 Zhanhua Yu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph
equations NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions,
we show that the backward Euler method BEM with variable stepsize can preserve the almost
surely asymptotic stability. Numerical examples are demonstrated for illustration.
1. 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
f t, xt, x qt .
xt − N x qt
1.1
Taking the environmental disturbances into account, we are led to the following neutral
stochastic pantograph equation NSPE
d xt − N x qt
f t, xt, x qt dt g t, xt, x qt dBt,
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
2
Abstract and Applied Analysis
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.
2. Neutral Stochastic Pantograph Equation
Throughout this paper, unless otherwise specified, we use the following notations. Let
Ω, F, {Ft }t≥0 , P be a complete probability space with filtration {Ft }t≥0 satisfying the usual
conditions i.e., it is right continuous, and F0 contains all P -null sets. Bt is a scalar
Brownian motion defined on the probability space. | · | denotes the Euclidean norm in Rn .
or matrix, its
The inner product of x, y in Rn is denoted by x, y or xT y. If A is a vector
transpose is denoted by AT . If A is a matrix, its trace norm is denoted by |A| traceAT A.
Let L1 0, T ; Rn denote the family of all Rn -value measurable Ft -adapted processes f T
{ft}0≤t≤T such that 0 |ft|dt < ∞ w.p.1. Let L2 0, T ; Rn denote the family of all Rn -value
T
measurable Ft -adapted processes f {ft}0≤t≤T such that 0 |ft|2 dt < ∞ w.p.1.
Consider an n-dimensional neutral stochastic pantograph equation
d xt − N x qt
f t, xt, x qt dt g t, xt, x qt dBt,
2.1
on t ≥ 0 with F0 -measurable bounded initial data x0 x0 . Here 0 < q < 1, f : R ×Rn ×Rn →
Rn , g : R × Rn × Rn → Rn , and N : Rn → Rn .
Let CRn ; R denote the family of continuous functions from Rn to R . Let C1,2 R ×
Rn ; R denote the family of all nonnegative functions V t, x on R × Rn which are
continuously once differentiable in t and twice differentiable in x. For each V ∈ C1,2 R ×
Rn ; R , define an operator LV from R × Rn × Rn to R by
LV t, x, y Vt t, x − N y Vx t, x − N y f t, x, y
1
trace g T t, x, y Vxx t, x − N y g t, x, y ,
2
2.2
Abstract and Applied Analysis
3
where
∂V t, x
∂V t, x
Vx t, x ,...,
,
∂x1
∂xn
∂2 V t, x
Vxx t, x .
∂xi ∂xj
∂V t, x
,
Vt t, x ∂t
2.3
n×n
To be precise, we first give the definition of the solution to 2.1 on 0 ≤ t ≤ T .
Definition 2.1. A Rn -value stochastic process xt on 0 ≤ t ≤ T is called a solution of 2.1 if it
has the following properties:
1 {xt}0≤t≤T is continuous and Ft -adapted;
2 ft, xt, xqt ∈ L1 0, T ; Rn , gt, xt, xqt ∈ L2 0, T ; Rn :
3 x0 x0 , and 2.1 holds for every t ∈ 0, T with probability 1.
A solution xt is said to be unique if any other solution xt is indistinguishable from it, that
is,
P {xt xt, 0 ≤ t ≤ T } 1.
2.4
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 ∈ Rn , and t ∈ 0, T ,
f t, x, y − f t, x, y 2 ∨ g t, x, y − g t, x, y 2 ≤ L |x − x|2 y − y2 ,
2.5
and for all x, y ∈ Rn , and t ∈ 0, T ,
f t, x, y 2 ∨ g t, x, y 2 ≤ K 1 |x|2 y2 .
2.6
Assumption 2.3. Assume that there is a constant κ ∈ 0, 1 such that
Nx − N y ≤ κx − y,
∀x, y ∈ Rn .
Under Assumptions 2.2 and 2.3, the following results can be derived.
2.7
4
Abstract and Applied Analysis
Lemma 2.4. Let Assumptions 2.2 and 2.3 hold. Let xt be a solution to 2.1 with F0 -measurable
bounded initial data x0 x0 . Then
E
sup |xt|
0≤t≤T
2
√ 1 − κκ 3 1 − κ
6KT 4T
2
1 E|x0 | exp
√ .
√ 2
1 − κ 1 − κ
1 − κ 1 − κ
≤
2.8
The proof of Lemma 2.4 is similar to Lemma 6.2.4 in 21, so we omit the details.
Theorem 2.5. Let Assumptions 2.2 and 2.3 hold, then for any F0 -measurable bounded initial data
x0 x0 , 2.1 has a unique solution xt 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.
Lemma 2.6. Let {Ai } and {Ui } be two sequences of nonnegative random variables such that both
Ai and Ui are Fi -measurable for i 1, 2, . . ., and A0 U0 0 a.s. Let Mi be a real-valued local
martingale with M0 0 a.s. Let ζ be a nonnegative F0 -measurable random variable. Assume that
{Xi } is a nonnegative semimartingale with the Doob-Mayer decomposition
Xi ζ Ai − Ui Mi .
2.9
If limi → ∞ Ai < ∞ a.s., then for almost all ω ∈ Ω:
lim Xi < ∞,
i→∞
lim Ui < ∞,
i→∞
2.10
that is, both Xi and Ui converge to finite random variables.
3. 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 F0 -measurable bounded initial data x0 .
Moreover, we always assume that ft, 0, 0 0, gt, 0, 0 0, N0 0 in the following
sections. Therefore, 2.1 admits a trivial solution xt 0.
To be precise, let us give the definition on the almost surely asymptotic stability of
2.1.
Definition 3.1. The solution xt to 2.1 is said to be almost surely asymptotically stable if
lim xt 0 a.s.
t→∞
for any bounded F0 -measurable bounded initial data x0.
3.1
Abstract and Applied Analysis
5
Lemma 3.2. Let ρ : R → 0, ∞ and z : 0, ∞ → Rn be a continuous functions. Assume that
ρt
1
σ1 : lim sup < ,
κ
t → ∞ ρ qt
σ2 : lim sup ρtzt − N z qt < ∞.
3.2
t→∞
Then,
lim sup ρt|zt| ≤
t→∞
σ2
.
1 − κσ1
3.3
Proof. Using the idea of Lemma 3.1 in 9, we can obtain the desired result.
Lemma 3.3. Suppose that 2.1 has a continuous unique global solution xt for given F0 -measurable
bounded initial data x0 . Let Assumption 2.3 hold. Assume that there are functions U ∈ C1,2 R ×
Rn ; R , w ∈ CRn ; R , and four positive constants λ1 > λ2 , λ3 , λ4 such that
LU t, x, y ≤ −λ1 wx qλ2 w y ,
t, x, y ∈ R × Rn × Rn ,
U t, x − N y ≤ λ3 wx λ4 w y , t, x ∈ R × Rn .
3.4
Then, for any ε ∈ 0, γ ∗ lim sup tγ
t→∞
∗
−ε
U t, xt − N x qt
<∞
a.s.,
3.5
where γ ∗ is positive and satisfies
∗
λ1 λ2 q−γ .
3.6
That is,
lim U t, xt − N x qt
0
t→∞
a.s.
3.7
Proof. Choose V t, xt tγ Ut, xt − Nxqt for t, x ∈ R × Rn 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.
Theorem 3.4. Suppose that 2.1 has a continuous unique global solution xt for given F0 measurable bounded initial data x0 . Let Assumption 2.3 hold. Assume that there are four positive
constants λ1 − λ4 such that
2
T f t, x, y ≤ −λ1 |x|2 λ2 y ,
2 x−N y
g t, x, y 2 ≤ λ3 |x|2 λ4 y2
3.8
6
Abstract and Applied Analysis
for t ≥ 0 and x, y ∈ Rn . If
λ1 − λ3 >
λ2 λ4
,
q
3.9
then, the global solution xt to 2.1 is almost surely asymptotically stable.
Proof. Let Ut, x wx |x|2 . Applying Lemma 3.3 and Lemma 3.2 with ρ 1, we can
obtain the desired conclusion.
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.
4. Almost Surely Asymptotic Stability of the Backward Euler Method
To define the BEM for 2.1, we introduce a mesh H {m; t−m , t−m1 , . . . , t0 , t1 , . . . , tn , . . .} as
follows. Let hn tn1 − tn , h−m−1 t−m . Set t0 γ0 > 0 and tm q−1 γ0 . We define m − 1 grid
points t1 < t2 < · · · < tm−1 in t0 , tm by
ti t0 iΔ0 ,
for i 1, 2, . . . , m − 1,
4.1
where Δ0 tm − t0 /m and define the other grid points by
tkmi q−k ti ,
for k −1, 0, 1, . . . , i 0, 1, 2, . . . , m − 1.
4.2
It is easy to see that the grid point tn satisfies qtn tn−m for n ≥ 0, and the step size hn satisfies
qhn hn−m ,
for n ≥ 0,
lim hn ∞.
n→∞
4.3
For the given mesh H, we define the BEM for 2.1 as follows:
Yn1 − NYn1−m Yn − NYn−m hn ftn1 , Yn1 , Yn1−m gtn , Yn , Yn−m ΔBn ,
n ≥ −m,
Y−m − NY−m−m x0 − Nx0 h−m−1 ft−m , Y−m , Y−m−m 4.4
g0, x0 , x0 Bt−m .
Here, Yn n ≥ −m is an approximation value of xtn and Ftn -measurable. ΔBn Btn1 −
Btn is the Brownian increment. The approximations Yn−m n −m, −m 1, . . . , −1 are
calculated by the following formulae:
Yn−m 1 − θn x0 θn Y−m ,
n −m, −m 1, . . . , −1,
4.5
where θn qtn /t−m . As a standard hypothesis, we assume that the BEM 4.4 is well defined.
Abstract and Applied Analysis
7
To be precise, let us introduce the definition on the almost surely asymptotic stability
of the BEM 4.4.
Definition 4.1. The approximate solution Yn to the BEM 4.4 is said to be almost surely
asymptotically stable if
lim Yn 0 a.s.
n→∞
4.6
for any bounded F0 -measurable bounded initial data x0 .
Theorem 4.2. 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
lim Yn 0 a.s.
n→∞
4.7
That is, the approximate solution Yn to the BEM 4.4 is almost surely asymptotically stable.
Proof. Set Y n Yn − NYn−m . For n ≥ 0, from 4.4, we have
2 2
Y n1 − hn ftn1 , Yn1 , Yn1−m Y n gtn , Yn , Yn−m ΔBn .
4.8
Then, we can obtain that
2 2
2
Y n1 ≤ Y n 2hn Y n1 , ftn1 , Yn1 , Yn1−m gtn , Yn , Yn−m ΔBn 2 Y n , gtn , Yn , Yn−m ΔBn ,
4.9
which subsequently leads to
2 2
Y n1 ≤ Y n 2hn Y n1 , ftn1 , Yn1 , Yn1−m 2
gtn , Yn , Yn−m hn mn ,
4.10
where
2 mn 2 Y n , gtn , Yn , Yn−m ΔBn gtn , Yn , Yn−m ΔBn2 − hn .
4.11
By conditions 3.8 and 3.9, we have
2 2
Y n1 ≤ Y n − λ1 hn |Yn1 |2 λ2 hn |Yn1−m |2
λ3 |Yn |2 λ4 |Yn−m |2 hn mn .
4.12
8
Abstract and Applied Analysis
Using the equality |a b|2 ≤ 2|a|2 2|b|2 , we obtain that
2 1
Y n1 ≥ |Yn1 |2 − |NYn1−m |2 ,
2
2
Y n ≤ 2|Yn |2 2|NYn−m |2 .
4.13
Inserting these inequalities to 4.12 and using Assumption 2.3 yield
1
λ1 hn |Yn1 |2 ≤ 2 λ3 hn |Yn |2 κ2 λ2 hn |Yn1−m |2
2
2κ2 λ4 hn |Yn−m |2 mn .
4.14
Let An 1 2λ1 hn , Bn 3 − 2λ1 hn 2λ3 hn , Cn 2κ2 2λ2 hn , and Dn 4κ2 2λ4 hn . Using
these notations, 4.14 implies that
|Yn1 |2 − |Yn |2 ≤
Bn
Cn
Dn
2
mn .
|Yn |2 |Yn1−m |2 |Yn−m |2 An
An
An
An
4.15
Then, we can conclude that
|Yn |2 ≤ |Y0 |2 n−1
Bi
i0
Ai
|Yi |2 n−1
Ci
i0
Ai
|Yi1−m |2 n−1
Di
i0
Ai
|Yi−m |2 n−1
2
mi .
A
i
i0
4.16
Note that
n−1
Ci
Ai
i0
|Yi1−m |2 n−1
Di
i0
Ai
2
|Yi−m | n−m
Cim−1
|Yi |2
A
i−m1 im−1
−1
n−1
n−1
Cim−1
Cim−1
Cim−1
|Yi |2 |Yi |2 −
|Yi |2 ,
A
A
A
im−1
im−1
im−1
i0
i−m1
in−m1
n−m−1
i−m
Dim
|Yi |2
Aim
−1
n−1
n−1
Dim
Dim
Dim
|Yi |2 |Yi |2 −
|Yi |2 .
A
A
A
im
im
im
i−m
i0
in−m
4.17
Abstract and Applied Analysis
9
We, therefore, have
|Yn |2 n−1
n−1
Cim−1
Dim
|Yi |2 |Yi |2
A
A
in−m im
in−m1 im−1
≤ |Y0 |2 −1
−1
Cim−1
Dim
|Yi |2 |Yi |2
A
A
im−1
im
i−m
i−m1
4.18
n−1 Bi
i0
n−1
2
Cim−1 Dim
mi .
|Yi |2 Ai Aim−1 Aim
A
i
i0
Similar to 4.15, from 4.4, we can obtain that
B−1
C−1
|Y−1 |2 |Y−m |2
A−1
A−1
2
D−1 2
21 − θ−1 2 |x0 |2 2θ−1
m−1 ,
|Y−m |2 A−1
A−1
Bn−1
Cn−1 21 − θn 2 |x0 |2 2θn2 |Y−m |2
|Yn |2 − |Yn−1 |2 ≤
|Yn−1 |2 An−1
An−1
Dn−1
2
21 − θn−1 2 |x0 |2 2θn−1
|Y−m |2
An−1
|Y0 |2 − |Y−1 |2 ≤
|Y−m |2 − |x0 |2 ≤
2
mn−1 ,
An−1
4.19
−m 1 ≤ n ≤ −1,
B−m−1 D−m−1
C−m−1 2
21 − θ−m 2 |x0 |2 2θ−m
|x0 |2 |Y−m |2
A−m−1
A−m−1
2
m−m−1 ,
A−m−1
where An , Bn , Cn , Dn n −m − 1, . . . , −1 are defined as before,
mn 2 Yn − NYn−m , gtn , Yn , Yn−m ΔBn
2 gtn , Yn , Yn−m ΔBn2 − hn , −m ≤ n ≤ −1,
m−m−1
4.20
2 2 x0 − Nx0 , g0, x0 , x0 Bt−m g0, x0 , x0 B2 t−m − t−m .
From 4.19, we have
|Y0 |2 ≤ A|x0 |2 B|Y−m |2 −1
−1
Bi
2
mi ,
|Yi |2 A
A
i−m1 i
i−m−1 i
4.21
10
Abstract and Applied Analysis
where
A1
−1
−2
2Di 2Ci B−m−1 D−m−1 1 − θi 2 1 − θi1 2 ,
A−m−1
Ai
Ai
i−m
i−m−1
−1
−2
2Di 2
2Ci 2
B−m B−1
B
θi θ .
A−m A−1 i−m Ai
Ai i1
i−m−1
4.22
Obviously A > 0. By 4.18 and 4.21, we can obtain that
|Yn |2 n−1
n−1
Cim−1
Dim
|Yi |2 |Yi |2
A
A
im−1
im
in−m
in−m1
n−1 D0
Bi Cim−1 Dim
2
2
≤ A|x0 | B |Y−m | |Yi |2 Mn ,
A0
Ai Aim−1 Aim
i−m1
4.23
n−1
where Mn i−m−1 2/Ai mi . Similar to the proof in 18, we can obtain that Mn is a
martingale with M−m−1 0. Note that him−1 ≤ him and him hi /q for i ≥ −m. Then,
we have
3 6κ2 − 2λ1 hi − λ3 hi − λ2 him−1 − λ4 him Bi Cim−1 Dim
≤
Ai Aim−1 Aim
1 2λ1 hi
1
λ2 λ 4
−
≤
3 6κ2 − 2 λ1 − λ3 −
hi .
1 2λ1 hi
q
q
4.24
Using the condition 3.9 and limi → ∞ hi ∞, we obtain that there exists an integer i∗ such
that
Bi Cim−1 Dim
≥ 0,
Ai Aim−1 Aim
Bi Cim−1 Dim
< 0,
Ai Aim−1 Aim
i ≤ i∗ ,
4.25
∗
i>i .
Set U−m−1 0,
U−m
⎧
D0
⎪
⎪
|Y−m |2
⎨ B
A
0
⎪
⎪
⎩0
D0
> 0,
A0 D0
if B ≤ 0,
A0
if
B
Abstract and Applied Analysis
⎧
n−1 ⎪
Bi Cim−1 Dim
⎪
⎪
|Yi |2
⎪
⎨U−m Ai Aim−1 Aim
i−m1
Un i∗ ⎪
Bi Cim−1 Dim
⎪
⎪
U
⎪
|Yi |2
−m
⎩
A
A
A
i
im−1
im
i−m1
Vn ⎧
⎪
0
⎪
⎨
⎪
⎪
⎩−
11
if − m 1 ≤ n ≤ i∗ 1,
if n > i∗ 1,
if − m − 1 ≤ n ≤ i∗ 1,
n−1 Bi Cim−1 Dim
|Yi |2
A
A
A
i
im−1
im
ii∗ 1
if n > i∗ 1.
4.26
Obviously,
lim Un U−m n→∞
i∗ Bi Cim−1 Dim
|Yi |2 < ∞,
A
A
A
i
im−1
im
i−m1
a.s.
4.27
Moreover, 4.23 implies that
|Yn |2 ≤ C|x0 |2 Un − Vn Mn .
4.28
Here C max{A, 1}. According to 4.27, using Lemma 2.6 yields
lim sup |Yn |2 < ∞
n→∞
lim Vn < ∞ a.s.
a.s.,
n→∞
4.29
Then, we have
lim −
i→∞
Bi Cim−1 Dim
|Yi |2 0 a.s.
Ai Aim−1 Aim
4.30
Note that
lim −
i→∞
Bi Cim−1 Dim
Ai Aim−1 Aim
λ1 − λ2 − λ3 − λ4
> 0.
λ1
4.31
We therefore obtain that
lim |Yn |2 0 a.s.
n→∞
4.32
Then, the desired conclusion is obtained. This completes the proof.
5. Numerical Experiments
In this section, we present numerical experiments to illustrate theoretical results of stability
presented in the previous sections.
12
Abstract and Applied Analysis
2.5
2
Yn
1.5
1
0.5
0
−0.5
0
2
4
6
8
10
12
tn
Yn (ω1 )
Yn (ω2 )
Yn (ω3 )
Figure 1: Almost surely asymptotic stability with x0 2, t0 0.01, m 2.
10
8
Yn
6
4
2
0
−2
0
1
2
3
4
5
6
7
8
tn
Yn (ω1 )
Yn (ω2 )
Yn (ω3 )
Figure 2: Almost surely asymptotic stability with x0 10, t0 1, m 1.
Consider the following scalar problem:
1
d xt − x0.5t −8xt x0.5tdt sinx0.5tdBt,
2
x0 x0 .
t ≥ 0,
5.1
Abstract and Applied Analysis
13
For the test 5.1, we have λ1 11, λ2 4, λ3 0, and λ4 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 Yn ω1 , Yn ω2 , Yn ω3 using the BEM 4.4 with x0 2, t0 0.01, m 2. In Figure 2, three different paths
Yn ω1 , Yn ω2 , Yn ω3 of BEM approximations are computed with x0 10, t0 1, m 1.
The results demonstrate that these paths are asymptotically stable.
Acknowledgments
The author would like to thank the referees for their helpful comments and suggestions.
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