Statistics and Probability Letters 82 (2012) 1669–1676

Contents lists available at SciVerse ScienceDirect

Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

Almost sure exponential stability of the θ-method for stochasticdifferential equationsLin Chen, Fuke Wu ∗

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China

a r t i c l e i n f o

Article history:Received 18 September 2011Received in revised form 5 May 2012Accepted 6 May 2012Available online 14 May 2012

Keywords:Stochastic differential equationsAlmost sure exponential stabilityθ-methodSemimartingale convergence theoremOne-sided linear growth

a b s t r a c t

Our previous work shows that the backward Euler–Maruyama (BEM) method mayreproduce the almost sure stability of stochastic differential equations (SDEs) withoutthe linear growth condition for the drift coefficient (see Wu et al. (2010)) but theEuler–Maruyama (EM)method cannot. It is well known that the θ-method is more generaland may be specialized as the BEM and EM by choosing θ = 1 and θ = 0. Then it isvery interesting to examine the interval in which the θ-method holds the same stabilityproperty as the BEMmethod. This paper shows that when θ ∈ (1/2, 1], the θ-methodmayreproduce the almost sure stability of the exact solution of SDEs. Finally, a two-dimensionalexample is presented to illustrate this result.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Stability theory of numerical solutions is one of central problems in numerical analysis. Stability analysis of numericalmethods for stochastic differential equations (SDEs) has recently received a great deal of attention. Due to the stochasticnature, the stability concepts of numerical schemes for SDEs mainly include, for example, moment stability (M-stability)and almost sure stability (or the trajectory stability (T -stability)). There is an extensive literature concerned with momentstability (for example, Baker and Buckwar (2005), Burrage et al. (2000), Burrage and Tian (2000), Higham (2000), Highamet al. (2007), Kloeden and Platen (1992), Mao (2007), Pang et al. (2008) and Saito and Mitsui (1996)). Regarding the almostsure stability of numerical methods for SDEs, it was shown, by the Chebyshev inequality and the Borel–Cantelli lemma,that the moment exponential stability implies almost sure exponential stability under certain conditions (for example, seeHigham et al. (2007) and Pang et al. (2008)). Higham and his coauthors (Higham, 2000; Higham et al., 2007) directly studiedthe numerical sequence and obtained almost sure stability by the strong law of large numbers.

Noting that there are similar expressions for the continuous and discrete semimartingale convergence theorems, Rodkinaand Schurz (2005) obtained the sufficient conditions for almost sure asymptotic stability of both exact and numericalsolutions of linear SDEs. Mao et al. (2011) and Wu et al. (2010) obtained almost sure exponential stability of numericalsolutions produced by the BEMmethod for nonlinear SDEs. It is well-known that the θ-method is more general and may bespecialized as the BEMmethod by choosing θ = 1. Then it is very interesting to examine the interval in which the θ-methodholds the same stability property as the BEMmethod.

Consider the following n-dimensional nonlinear SDE:

dx(t) = f (x(t), t)dt + g(x(t), t)dw(t), t ≥ 0 (1.1)

∗ Corresponding author. Tel.: +86 27 87557195; fax: +86 27 87543231.E-mail addresses: cl18971072943@163.com (L. Chen), wufuke@mail.hust.edu.cn (F. Wu).

0167-7152/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2012.05.004

1670 L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676

with initial data ξ ∈ LbF0(Ω; Rn), f , g : C(Rn

× R+; Rn) are Borel measurable, and w(t) is a scalar Brownian motion. Asa standing hypothesis, we shall impose the following local Lipschitz condition (cf. Mao (1994, 1997)) on the coefficients fand g .

Assumption 1. Both f and g satisfy the local Lipschitz condition. That is, for each j = 1, 2, . . . , there exists a constant cj > 0such that

|f (x, t)− f (x, t)| ∨ |g(x, t)− g(x, t)| ≤ cj|x − x|

for all t ≥ 0 and those x, x ∈ Rn with |x| ∨ |x| ≤ j.

In this paper, we will address the following question:– If Eq. (1.1) is almost surely exponentially stable, whether there exists an interval such that when θ is in this interval, theθ-method may reproduce this stability?

This paper will show that for θ ∈ (1/2, 1], the θ-method can reproduce the almost sure stability of the exact solution ofEq. (1.1).

In the next section, we shall give some necessary notations and lemmas for the use of this paper. Then Section 3 willdiscuss the almost sure exponential stability of the exact solution to Eq. (1.1) and its θ-method approximation under theone-sided Lipschitz condition. The last section gives one numerical example and simulations to illustrate our result.

2. Notations and lemmas

Throughout this paper, unless otherwise specified, we use the following notations. Let | · | be the Euclidean norm in Rn.If A is a vector or matrix, its transpose is denoted by AT . If A is a matrix, its trace norm is denoted by |A| =

trace(ATA). Let

R+ = [0,∞). Denoted by L(Ω; Rn) the space of variables fromΩ to Rn. Let LbF0(Ω; Rn) be the family of all F0-measurable

bounded Rn-valued random variables. The inner product of X, Y ∈ Rn is denoted by ⟨X, Y ⟩ or XTY .Let (Ω,F , P) be a complete probability space with a filtration Ftt≥0 satisfying the usual conditions, that is, it is right

continuous and increasingwhileF0 contains all P-null sets. Letw(t) be a scalar Brownianmotion defined on this probabilityspace. Let C2(Rn

; R+) denote the family of all function V (x) from Rn to R+ which are continuous twice differentiable, anddefine an operator LV : Rn

× R+ → R by

LV (x, t) = Vx(x)f (x, t)+12trace[gT (x, t)Vxx(x)g(x, t)] (2.1)

for all t ≥ 0, x ∈ Rn, where

Vx(x) =

∂V (x)∂x1

,∂V (x)∂x2

, . . . ,∂V (x)∂xn

, Vxx(x) =

∂2V (x)∂xi∂xj

n×n

.

If x(t) is a solution of Eq. (1.1), by the Itô formula,dV (x(t)) = LV (x(t), t)dt + Vx(x(t))g(x(t), t)dw(t). (2.2)

The following lemmas will play important roles in this paper.

Lemma 1 (Continuous Semimartingale Convergence Theorem, See Liptser and Shiryaev (1989) and Mao (1997)). Let A(t),U(t)be two Ft-adapted increasing processes on t ≥ 0 with A(0) = U(0) = 0 a.s. Let M(t) be a real-valued local martingale withM(0) = 0 a.s. Let ζ be a nonnegative F0-measurable random variable. Assume that X(t) is nonnegative and

X(t) = ζ + A(t)− U(t)+ M(t) for t ≥ 0.

If limt→∞ A(t) < ∞ a.s. then for almost all ω ∈ Ω ,

limt→∞

X(t) < ∞ and limt→∞

U(t) < ∞,

that is, both X(t) and U(t) converge to finite random variables.

Lemma 2 (Discrete Semimartingale Convergence Theorem, See Rodkina and Schurz (2005) and Shiryaev (1996)). Let Ai, Ui betwo sequences of nonnegative random variables such that both Ai and Ui areFi−1-measurable for i = 1, 2, . . . , and A0 = U0 = 0a.s. Let Mi be a real-value local martingale with M0 = 0 a.s. Let ζ be a nonnegative F0-measurable random variable. Assume thatXi is a nonnegative semimartingale with the Doob–Mayer decomposition

Xi = ζ + Ai − Ui + Mi.

If limi→∞ Ai < ∞ a.s. then for almost all ω ∈ Ω ,

limi→∞

Xi < ∞ and limi→∞

Ui < ∞,

that is, both Xi and Ui converge to finite random variables.

L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676 1671

Lemma 3 (See Wang et al. (2008)). Let h(x) is a function from Rn to Rn. There exists a nonnegative constant σ such that

xTh(x) ≤ −σ |x|2

for any x ∈ Rn. Then for any 0 ≤ k1 ≤ k2 ∈ R, the inequality

|x − k1h(x)| ≤1 + σk11 + σk2

|x − k2h(x)|

holds.

In the following sections we will employ these lemmas to establish the almost sure asymptotic stability theorem forexact solutions to Eq. (1.1) and its θ-method approximation.

3. Stability of the exact solution and the θ-method approximation

Applying the θ-method (cf. Higham (2000)) to Eq. (1.1) yields the approximate solutionx0 = ξ,xn+1 = xk + θ f (xn+1, (n + 1)∆)∆+ (1 − θ)f (xn, n∆)∆+ g(xn, n∆)∆wn, n ≥ 0 (3.1)

where∆ is the stepsize and∆wn := w((n + 1)∆)− w(n∆) is the Brownian increment.To be precise, let us give the definition on the almost sure exponential stability of SDEs and their numerical

approximations.

Definition 1. The solution x(t, ξ) to Eq. (1.1) is said to be almost surely exponentially stable if

lim supt→∞

1tlog |x(t, ξ)| < 0 a.s. (3.2)

for any initial data ξ ∈ LbF0(Ω; Rn).

Definition 2. The approximate solution xn to Eq. (3.1) is said to be almost surely exponentially stable if

lim supn→∞

1n∆

log |xn| < 0 a.s. (3.3)

for any bounded variables ξ .

In this section, our aim is to examine the interval in which the θ-method can reproduce the almost sure exponentialstability of the exact solution of Eq. (1.1). Let us state a theorem which does not only give the existence-and-uniquenessresult of the solution but also provides us with a criterion on the almost sure exponential stability of the exact solution(please see Mao and Rassias (2005) for the existence-and-uniqueness result andMao (1999, 2002) for almost sure stability).

Theorem 1. Let Assumption 1 hold. Assume that there are two nonnegative constants σ , σ0 such that

xT f (x, t) ≤ −σ |x|2, (3.4)

|g(x, t)|2 ≤ σ0|x|2, (3.5)

for all x ∈ Rn and t ≥ 0. If

2σ > σ0, (3.6)

then for any given initial data ξ ∈ LbF0(Ω; Rn), there exists a unique global solution to Eq. (1.1) and this solution, denoted by

x(t; ξ), has property that

lim supt→∞

1tlog |x(t, ξ)| ≤ −

γ

2< 0 a.s. (3.7)

where γ = 2σ − σ0.

Let us now examine the stability of the θ-method approximate solution (3.1). Note that if 0 < θ ≤ 1, the θ-method issemi-implicit. To ensure that this scheme is well defined, let us impose the following one-sided Lipschitz condition on f :there exists a constant λ such that for any x1, x2 ∈ Rn and t ≥ 0,

⟨x1 − x2, f (x1, t)− f (x2, t)⟩ ≤ λ|x1 − x2|2. (3.8)

Under this condition, if θλ∆ < 1, then the θ-method (3.1) is well defined (see Hairer andWanner (1996) andMao and Yuan(2006)). In the following, let us keep this condition.

1672 L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676

Theorem 2. Let conditions (3.4)–(3.6) and (3.8) hold. Let θ ∈ (1/2, 1], a∗= ((2σ − σ0)/2σ) ∧ (2θ − 1), γ = 2a∗σ , and

ε ∈ (0, γ /2) be arbitrary. Then there exists∆∗ > 0 such that if ∆ < ∆∗, then for any given finite-valuedF0-measurable randomvariables ξ , the approximate solution xn defined by (3.1) has property that

lim supn→∞

1n∆

log |xn| ≤ −γ

2+ ε < 0 a.s. (3.9)

Proof. By (3.1) and (3.4), for any a ∈ (0, a∗), we have

|xn+1 − (1 − θ + a)∆f (xn+1, (n + 1)∆)|2

≤ |xn+1 − (1 − θ + a)∆f (xn+1, (n + 1)∆)|2 + (2θ − 1 − a)2∆2|f (xn+1, (n + 1)∆)|2

− 2⟨xn+1 − (1 − θ + a)∆f (xn+1, (n + 1)∆)− xn+1, (2θ − 1 − a)∆f (xn+1, (n + 1)∆)⟩= |xn + (1 − θ)∆f (xn, n∆)+ g(xn, n∆)∆wn|

2+ 2(2θ − 1 − a)∆xTn+1f (xn+1, (n + 1)∆)

≤ |xn − (1 − θ)∆f (xn, n∆)|2 + σ0|xn|2|∆wn|2+ 4(1 − θ)∆xTn f (xn, n∆)

+ 2(2θ − 1 − a)∆xTn+1f (xn+1, (n + 1)∆)+ 2⟨xn + (1 − θ)∆f (xn, n∆), g(xn, n∆)∆wn⟩.

By Lemma 3, we have

|xn − (1 − θ)∆f (xn, n∆)|2 ≤

1 + (1 − θ)σ∆

1 + (1 − θ + a)σ∆

2

|xn − (1 − θ + a)∆f (xn, n∆)|2.

Letting An = |xn − (1 − θ)∆f (xn, n∆)|2, we therefore have that

An+1 ≤

1 + (1 − θ)σ∆

1 + (1 − θ + a)σ∆

2

An − 4(1 − θ)∆σ |xn|2 − 2(2θ − 1 − a)∆σ |xn+1|2+ σ0∆|xn|2 + m∆

n ,

wherem∆n = σ0|xn|2(|∆wn|

2−∆)+ 2⟨xn + (1 − θ)∆f (xn, n∆), g(xn, n∆)∆wn⟩.

For any positive constant C > 1, we have

C (n+1)∆An+1 − Cn∆An ≤ C (n+1)∆

1 + (1 − θ)σ∆

1 + (1 − θ + a)σ∆

2

− C−∆

An −∆[4(1 − θ)σ |xn|2

+ 2(2θ − 1 − a)σ |xn+1|2− σ0|xn|2] + m∆

n

,

which implies that

Cn∆An ≤ A0 + 2(2θ − 1 − a)σ∆|x0|2 −

n−1i=0

C−∆

−

1 + (1 − θ)σ∆

1 + (1 − θ + a)σ∆

2C (i+1)∆Ai

−∆

n−1i=0

C (i+1)∆[4(1 − θ)σ + 2(2θ − 1 − a)σC−∆

− σ0]|xi|2 + Mn,

where

Mn =

n−1i=0

C (i+1)∆m∆i

= σ0

n−1i=0

C (i+1)∆|xi|2(|∆wi|

2−∆)+ 2

n−1i=0

C (i+1)∆⟨xi + (1 − θ)∆f (xi, i∆), g(xi, i∆)∆wi⟩

=: σ0mn + 2mn.

Noting that E[(|∆wn |2−∆)|Fn∆] = 0 and xn is Fn∆-measurable, we have

E[mn|F(n−1)∆] = mn−1 + E[Cn∆|xn−1 |

2(|∆wn−1 |2−∆)|F(n−1)∆]

= mn−1 + Cn∆|xn−1|

2E[(|∆wn−1 |2−∆)|F(n−1)∆]

= mn−1,

which implies that mn is a martingale. Clearly, mn is also a martingale. These imply that Mn = σ0mn + 2mn is a martingalewith M0 = 0.

L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676 1673

Let us now introduce two functions

ϕ(a, C) = 4(1 − θ)σ + 2(2θ − 1 − a)σC−∆− σ0, (3.10)

ψ(a, C) = C−∆−

1 + (1 − θ)σ∆

1 + (1 − θ + a)σ∆

2

. (3.11)

Then we have

Cn∆An ≤ const − ψ(a, C)n−1i=0

C (i+1)∆Ai − ϕ(a, C)∆n−1i=0

C (i+1)∆|xi|2 + Mn,

where const represents some constant whose precise value is not important.By the definition of a∗ and condition (3.6), we have ϕ(a, 1) > 0, ψ(a, 1) > 0 and ϕ2(a, C) < 0, ψ2(a, C) < 0 for any

fixed a ∈ (0, a∗) and C > 1, where

ϕ2(a, C) =dϕ(x, y)

dy

x=a,y=C

, ψ2(a, C) =dψ(x, y)

dy

x=a,y=C

,

which implies that there exists a unique C∗∆ > 1 such that

ϕ(a, C∗

∆) ∧ ψ(a, C∗

∆) = 0.

Choosing C = C∗∆, it is obvious that C

n∆|xn|2 ≤ Cn∆An ≤ const + Mn. By Lemma 2, we have

lim supn→∞

Cn∆|xn|2 ≤ ∞ a.s. (3.12)

On the other hand, for any fixed C > 1, ϕ(a, C) is decreasing in a and ψ(a, C) is increasing in a. These imply that thereexist a unique a∆ ∈ (0, a∗) and a unique C∆ > 1, such that if a∆ = 2θ − 1,

ϕ(a∆, C∆) ≥ 0, ψ(a∆, C∆) = 0 (3.13)

and if a∆ < 2θ − 1,

ϕ(a∆, C∆) = ψ(a∆, C∆) = 0. (3.14)

Introduce a positive constant η such that C = eη , and define

ϕ∆(a, η) = ϕ(a∆, eη), ψ∆(a, η) =ψ(a∆, eη)

∆.

Letting η = η∆ = log C∆, a = a∆, by (3.13) and (3.14), if a∆ = 2θ − 1,

ϕ∆(a, η) ≥ 0, ψ∆(a, η) = 0,

and if a∆ < 2θ − 1,

ϕ∆(a, η) = ψ∆(a, η) = 0.

Noting that lim∆→0(1 − e−η∆)/∆ = η, we have lim∆→0 ϕ∆(a, η) = lim∆→0(2σ − σ0 − 2σa∆) and lim∆→0 ψ∆(a, η) =

lim∆→0(2σa∆ − η∆), which implies that

lim∆→0

a∆ =2σ − σ0

2σ∧ (2θ − 1) = a∗ and lim

∆→0η∆ = lim

∆→02σa∆ = γ .

We therefore have that, for any positive ε ∈ (0, γ /2), there exists ∆∗ > 0 such that λθ∆∗ < 1 and for any ∆ < ∆∗,η∆ > γ − 2ε. By the definition of η∆, (3.12) is equivalent to

lim supn→∞

eη∆n∆|xn|2 ≤ ∞ a.s.

We therefore obtain that for any∆ < ∆∗,

lim supn→∞

1n∆

log |xn| ≤ −γ

2+ ε < 0 a.s.

as required.

Remark 3. In Mao et al. (2011) and Wu et al. (2010), the BEM method (θ = 1) for nonlinear SDEs may reproduce thealmost sure exponential stability of exact solutions of SDEs. This theorem extends the previous results and shows that whenθ ∈ (1/2, 1], θ-method has the similar almost surely exponential stability to the BEMmethod (θ = 1). This implies that theresult of this paper is more general than the existing results in Mao et al. (2011) and Wu et al. (2010).

1674 L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676

X1

solution x1

time t0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 1. Numerical solution x1 of Eq. (4.1) (theta = 0.6).

time t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

solution x2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

X2

Fig. 2. Numerical solution x2 of Eq. (4.1) (theta = 0.6).

4. Numerical example

In this section, we give a numerical example and its simulations with two different θ to illustrate the almost sureexponential stability of the theta scheme for the SDEs whose coefficients are nonlinear.

Example 1. Consider the following two-dimensional SDE:dx1(t) = (−5x1(t)− 2x31(t))dt + (1.5x1(t)+ 0.5x2(t))dω(t),dx2(t) = (2x1(t)− 5x2(t)− x32(t))dt + (−0.5x1(t)− 1.5x2(t))dω(t),

(4.1)

with initial data x1(0) = 3, x2(0) = 4.

Let xT = (x1, x2), f T (x, t) = (−5x1 − 2x31, 2x1 − 5x2 − x32), gT (x, t) = (1.5x1 + 0.5x2,−0.5x1 − 1.5x2). Clearly,

⟨x − x, f (x, t)− f (x, t)⟩ ≤ 4|x − x|2,xT f (x, t) ≤ −4|x|2,|g(x, t)|2 ≤ 4|x|2.

The coefficients of Eq. (4.1) satisfy conditions (3.4)–(3.6) and (3.8). It is also easy to compute γ = 4.By Theorem 1, the exact solutions to Eq. (4.1) are almost surely exponentially stable. Applying the theta scheme and

choosing∆ = 10−4 and θ = 0.6, then we have a∗= 0.5 ∧ 0.2 = 0.2 and γ = 8 × 0.2 = 1.6. The simulations of Eq. (4.1)

are as following.By Figs. 1 and 2, we know that the numerical solution tends to zero, when θ = 0.6.Applying the theta scheme and choosing∆ = 10−4 and θ = 0.9, thenwe have a∗

= 0.5∧0.8 = 0.5 and γ = 8×0.5 = 4.The simulations of Eq. (4.1) are as following.

By Figs. 3 and 4, we know that the numerical solution tends to zero. These examples show that the theta scheme canreproduce the stability of the exact solutions to Eq. (4.1) with θ ∈ (0.5, 1].

L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676 1675

time t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

solution x1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

X1

Fig. 3. Numerical solution x1 of Eq. (4.1) (theta = 0.9).

time t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

solution x2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

X2

Fig. 4. Numerical solution x2 of Eq. (4.1) (theta = 0.9).

Acknowledgments

The authors would like to thank the referees for their detailed comments and helpful suggestions. This research wassupported in part by the National Science Foundations of China (Grant Nos. 11001091 and 61134012) and in part by Programfor New Century Excellent Talents in University.

References

Baker, C.T.H., Buckwar, E., 2005. Exponential stability in p-th mean of solutions, and of convergent Eulertype solutions, of stochastic delay differentialequations. J. Comput. Appl. Math. 184, 404–427.

Burrage, K., Burrage, P., Mitsui, T., 2000. Numerical solutions of stochastic differential equations-implematation and stability issues. J. Comput. Appl. Math.125, 171–182.

Burrage, K., Tian, T., 2000. A note on the stability propertis of the Euler methods for solving stochastic differential equations. NZ J. Math. 29, 115–127.Hairer, E., Wanner, G., 1996. Solving Ordinary Differnential II: Stiff and Differential-Algebraic Problems, second ed. Springer, Berlin.Higham, D.J., 2000. Mean-square and asymptotic stability of the stochastic theta methods. SIAM J. Numer. Anal. 38, 753–769.Higham, D.J., Mao, X., Yuan, C., 2007. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM

J. Numer. Anal. 45, 592–607.Kloeden, P.E., Platen, E., 1992. The Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Liptser, R.Sh., Shiryaev, A.N., 1989. Theory of Martingale. Kluwer Academic Publishers, Dordrecht.Mao, X., 2007. Exponential stability of equidistant Euler–Maruyama approximations of stochastic differential delaly equations. J. Comput. Appl. Math. 200,

297–316.Mao, X., 1994. Exponential Stability of Stochastic Differential Equation. Marcel Dekker, New York.Mao, X., 1997. Stochastic Differential Equations and their Applications. Horwood, Chichester.Mao, X., 1999. LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 236, 350–369.Mao, X., 2002. A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125–142.Mao, X., Rassias, M., 2005. Kasminskii-type theorem for stochastic differential delay equations. Stoch. Anal. Appl. 23, 1045–1069.Mao, X., Shen, Y., Alison, G., 2011. Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential

equations. J. Comput. Appl. Math. 235, 1213–1226.Mao, X., Yuan, C., 2006. Stochastic Differential Equations with Markovian Switching. Imperial College Press, London.Pang, S., Deng, F., Mao, X., 2008. Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential

equations. J. Comput. Appl. Math. 213, 127–141.

1676 L. Chen, F. Wu / Statistics and Probability Letters 82 (2012) 1669–1676

Rodkina, A., Schurz, H., 2005. Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R1 .J. Comput. Appl. Math. 180, 13–31.

Saito, Y., Mitsui, T., 1996. Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 2254–2267.Shiryaev, A.N., 1996. Probability. Springer, Berlin.Wang, W., Wen, L., Li, S., 2008. Nonlinear stability of θ-methods for neutral differential equations in Banach space. J. Appl. Math. Comput. 198, 742–753.Wu, F., Mao, X., Szpruch, L., 2010. Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numer. Math. 115,

681–697.