Exponential Stability of Solutions for Retarded Stochastic Differential Equations Without Dissipativity

Home , Exponential stability

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2017157 DYNAMICAL SYSTEMS SERIES B Volume 22, Number 7, September 2017 pp. 2923–2938

EXPONENTIAL STABILITY OF SOLUTIONS FOR RETARDED STOCHASTIC DIFFERENTIAL EQUATIONS WITHOUT DISSIPATIVITY

Min Zhu∗ 1 College of Traﬃc Engineering Hunan University of Technology Zhuzhou, Hunan 412007, China 2 School of Mathematics and Statistics Central South University Changsha, Hunan 410083, China Panpan Ren Department of Mathematics Swansea University Singleton Park, SA2, 8PP, UK Junping Li School of Mathematics and Statistics Central South University Changsha, Hunan 410083, China

(Communicated by Xiaoying Han)

Abstract. This work focuses on a class of retarded stochastic differential equations that need not satisfy dissipative conditions. The principle technique of our investigation is to use variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. By using variation-of-constants formula and estimating the diffusion coefficients we give sufficient conditions for p-th moment exponential stability, almost sure exponential stability and convergence of solutions from different initial value. Fi- nally, we provide two examples to illustrate the effectiveness of the theoretical results.

1. Introduction. Stochastic dynamical systems have been used to represent the real world behavior and they can reveal the uncertainty of the environment in which the model is operating. Because of their wide application in various sciences such as physics, mechanical engineering, control theory and economics, the theory of stochastic dynamical systems has attracted extensive attention. Moreover, it is because stochastic dynamical systems often run for an extended time that the study of stability properties is considerably important and has been one of the most active areas in stochastic analysis. Especially, there has been much interesting in studying stochastic dynamical systems whose evolution in time is governed by random forces

2010 Mathematics Subject Classiﬁcation. 60H10, 39B82, 60H30, 37H10. Key words and phrases. Retarded stochastic diﬀerential equations, variation-of-constants formula, exponential stability, almost sure exponential stability, dissipativity. ∗ Corresponding author: Min Zhu.

2923 2924 MIN ZHU, PANPAN REN AND JUNPING LI as well as intrinsic dependence on the state over a finite interval of its history. Such systems can be called as retarded stochastic differential equations (SDEs); for example, see the monograph [19] for more details. Because of many systems involving retarded arguments, a large number of interesting results on the existence, uniqueness, stability, other quantitative and qualitative properties of solutions have been reported(see, e.g., [4, 9, 13, 21, 27]). In response to the great needs, there is an extensive literature on stability for retarded SDEs. So far there are numerous approaches to investigate various stability (e.g., moment stability, sample path stability, stability in distribution, stability in probability, etc) for retarded SDEs; see, for instance, [10, 12, 14, 15] by exploiting Razumikhin-type theorems, [5, 3, 22, 24] by making use of the weak convergence method, [16] by employing the semimartingale convergence theorem, [17] by applying the LaSalle-type theorems, and [18] by taking advantage of Borel-Cantelli lemma, to name a few. Nevertheless, most of the existing literature focuses on stability for retarded SDEs under certain dissipativity, which is normally assured by imposing information of the current time with certain decay conditions. In contrast to the rapid progress in stability for SDEs with dissipativity, the study for retarded SDEs without dissipativity is still scarce. Compared with retarded SDEs with dissipativity, as far as SDEs without dissipativity, one of the outstanding issues is the lack of the information at the current time, which makes the goal of investigating stability a very difficult task. Whereas, this work aims to take the challenges and to establish stability for several ranges of retarded SDEs, which do not enjoy dissipative property. The rest of this paper is structured as follows. In section 2 we provide some preliminary results and recall definitions of p-th moment exponential stability and almost sure exponential stability, which lay a good foundation for stability analysis; Section 3 focuses on stability for retarded SDEs driven by Brownian motions; Section 4 is devoted to the stability for retarded SDEs of neutral type. In the last section, to show the effectiveness of our theory, two illustrative examples are provided. 2. Preliminary. For any integer n > 0, let Rn be an n-dimensional Euclidean Pn i i space endowed with the inner product hu, vi := i=1 u v and the Euclidean norm 1 n n m |u| := hu, ui 2 for u, v ∈ R . Denote R ⊗ R by the set of all n × m matrices A endowed with Hilbert-Schmidt norm kAk := ptrace(AT A), in which AT is the transpose of A. Let (Ω, P, F ) be a probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., Ft = Ft+ := ∩s>tFs and F0 contains all P-null sets). Let {W (t)}t≥0 be an m-dimensional Brownian motion defined on (Ω, P, F , {Ft}t≥0). Fix τ ∈ (0, ∞), which will be referred to as the delay or time lag. Let C = C([−τ, 0]; Rn) be the family of all continuous functions ξ :[−τ, 0] → n R equipped with the uniform norm kξk∞ := sup−τ≤θ≤0 |ξ(θ)| for ξ ∈ C . For n X(·) ∈ C([−τ, ∞]; R ), define the segment process Xt ∈ C by Xt(θ) := X(t + θ), θ ∈ [−τ, 0], t ≥ 0. Let µ(·) and ρ(·) be Rn ⊗ Rn-valued finite signed measure on [−τ, 0], ν(·) a measure on [−τ, 0], C the set of all complex numbers and Re(z) the qP 2 real part of z ∈ C. Let k|ρk| = sup1≤k≤n 1≤j≤n kρkjkvar, where kρkjkvar is the total variation of ρkj. To begin with, consider the following deterministic linear retarded equation: Z dY (t) = µ(dθ)Y (t + θ) dt, t > 0,Y0 = ξ ∈ C . (2.1) [−τ,0] STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2925

By the variation-of-constants formula (see, e.g., [8, Theorem 1.2, p.170]), Eq.(2.1) has a unique explicit solution Z Z 0 Y (t; ξ) = r(t)ξ(0) + µ(dθ) r(t + θ − s)ξ(s)ds, [−τ,0] θ where r(t) is the fundamental solution of (2.1) with the initial value r(0) = In×n and r(θ) = 0n×n for θ ∈ [−τ, 0). Set Z n λθ o υ1 := sup Re(λ): λ ∈ C, det λIn×n − e µ(dθ) = 0 , [−τ,0] where det(A) denotes the determinant of an A ∈ Rn ⊗ Rn. Then according to [8,

Theorem 3.2, p.271], for any α1 > υ1, there exists a constant cα1 > 0 such that

α1t kr(t)k ≤ cα1 e , t ≥ −τ, where k · k denotes the operator norm of the matrix. If υ1 < 0, by [1, Lemma 1], for any α ∈ (0, −υ1), there exists a constant Cα > 0 such that −αt kr(t)k ≤ Cαe , t ≥ −τ. (2.2) Next, the deterministic linear retarded equation of neutral type Z Z d Y (t) − ρ(dθ)Y (t + θ) = µ(dθ)Y (t + θ) dt, (2.3) [−τ,0] [−τ,0] with the initial value Y0 = ξ ∈ C , by [8, Theorem 1.1, p.256], has a unique solution 1,2 n {Y (t; ξ)}t≥−τ . Then by virtue of [11, Theorem 2.2], for any ξ ∈ W ([−τ, 0]; R ) (the Sobolev space consisting of functions f :[−τ, 0] 7→ Rn such that f(·) and its distributional derivative f 0(·) belong to L2([−τ, 0]; Rn)), Y (t; ξ) can be expressed explicitly by Z Z Z 0 Y (t; ξ) = G(t)ξ(0)− ρ(dθ)G(t + θ)ξ(0) + µ(dθ) G(t + θ − s)ξ(s)ds [−τ,0] [−τ,0] θ Z Z 0 + ρ(dθ) G(t + θ − s)ξ0(s)ds, [−τ,0] θ (2.4) where G(t) is the fundamental solution of (2.3) with the initial value G(0) = In×n and G(θ) = 0n×n for θ ∈ [−τ, 0). Set Z Z n λθ λθ o υ2 := sup Re(λ): λ ∈ C, det λ In×n − e ρ(dθ) − e µ(dθ) = 0 . [−τ,0] [−τ,0]

According to [8, Theorem 3.2, p.271], for any α2 > υ2 there exists a constant cα2 > 0 such that

α2t kG(t)k ≤ cα2 e , t ≥ −τ.

If υ2 < 0, according to [2, Lemma 1], for any β ∈ (0, −υ2), there exists a constant Cβ > 0 such that −βt kG(t)k ≤ Cβe , t ≥ −τ. (2.5) Before we end this section, we recall some notions on stability and lemmas for later purpose. 2926 MIN ZHU, PANPAN REN AND JUNPING LI

Definition 2.1. The solution process {X(t)} is said to be p-th moment exponentially stable if there is a pair of positive constants κ and K, for any p > 0 and ξ ∈ C such that p p −κpt E|X(t; ξ)| ≤ Kkξk∞e . For p = 2, it is named as exponential stability in mean square. Definition 2.2. The solution process {X(t)} is said to be almost surely exponentially stable if for any ξ ∈ C such that 1 lim sup log |X(t; ξ)| < 0 a.s. t→∞ t n m Lemma 2.3. ([6, Theorem 2.3]) Let {f(t)}t≥0 be an R ⊗ R -valued predictable process. Then, for all p ≥ 1 and t > 0 t 1 t Z 2p p Z 1 2p p E r(t − s)f(s)dW (s) ≤ p(2p − 1) (Ekr(t − s)f(s)k ) ds, 0 0 provided that the integral on the right hand side is finite for each t > 0.

3. Exponential stability for retarded SDEs. In this section, we ﬁrst consider a semi-linear retarded SDE of the form Z dX(t) = µ(dθ)X(t + θ) dt + σ(Xt)dW (t), t > 0,X0 = ξ ∈ C , (3.1) [−τ,0] n m where σ : C 7→ R ⊗ R such that σ(0) = 0n×m is Borel measurable. Throughout this section, we assume that, for any φ, ϕ ∈ C , there exists an L > 0 such that Z kσ(φ) − σ(ϕ)k2 ≤ L |φ(0) − ϕ(0)|2 + |φ(θ) − ϕ(θ)|2ν(dθ) . (3.2) [−τ,0] Under (3.2), by [19, Theorem 2.1, p.36], Eq.(3.1) admits a unique strong solution {X(t; ξ)}t≥−τ with the initial value X0 = ξ ∈ C . It can be represented explicitly by Z Z 0 X(t; ξ) =r(t)ξ(0) + µ(dθ) r(t + θ − s)ξ(s)ds [−τ,0] θ (3.3) Z t + r(t − s)σ(Xs(ξ))dW (s). 0 Indeed, for ﬁxed t ≥ 0, by means of the chain rule, we have d(r(t − s)X(s)) = d(r(t − s))X(s) + r(t − s)dX(s), s ∈ [0, t]. Integrating from 0 to t, together with (2.1) and (3.1), leads to X(t) − r(t)ξ(0) Z t Z = − µ(dθ)r(t − s + θ) X(s)ds 0 [−τ,0] Z t Z Z t + r(t − s) µ(dθ)X(s + θ) ds + r(t − s)σ(Xs)dW (s) 0 [−τ,0] 0 Z Z t Z Z t+θ = − µ(dθ) r(t − s + θ)X(s)ds + µ(dθ) r(t − s + θ)X(s)ds [−τ,0] 0 [−τ,0] θ Z t + r(t − s)σ(Xs)dW (s) 0 STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2927 Z Z 0 Z Z t = µ(dθ) r(t − s + θ)X(s)ds − µ(dθ) r(t − s + θ)X(s)ds [−τ,0] θ [−τ,0] t+θ Z t + r(t − s)σ(Xs)dW (s) 0 Z Z 0 Z t = µ(dθ) r(t − s + θ)ξ(s)ds + r(t − s)σ(Xs)dW (s). [−τ,0] θ 0 Next, we shall investigate the p-th moment exponential stability and almost sure exponential stability for the solution X(t) to Eq. (3.1).

Theorem 3.1. Let p ≥ 2 and υ1 < 0. Assume further that (3.2) holds with L > 0 2 2ατ such that p(p − 1)LCα(1 + kνkvar)e ) < 4α, for α ∈ (0, −υ1), where Cα > 0 is introduced in (2.2). Then, for any initial value ξ ∈ C , the solution of (3.1) is p-th moment exponentially stable, i.e. there exist constants K > 0 and α˜ > 0 such that

p p −pαt˜ E|X(t; ξ)| ≤ Kkξk∞e , t ≥ 0. (3.4) Proof. In what follows, for notation simplicity, we write X(t) in lieu of X(t; ξ). For 2 2ατ any p ≥ 2, due to p(p − 1)LCα(1 + kνkvar)e ) < 4α, we choose γ2 > 1 such that

2 p 2 2ατ p(p − 1)γ2 LCα(1 + kνkvar)e ) < 4α. (3.5)

In the sequel, we ﬁx γ2 > 1 such that (3.5). Then there exists a γ1 > 1 such that q q q q + (a + b + c) ≤ γ1(a + b ) + γ2c , q > 1, a, b, c ∈ R . (3.6) By the elementary inequality:

θ θ θ + (a + b) ≤ a + b , 0 < θ < 1, a, b ∈ R , (3.7) it follows from (3.3) and (3.6) that

2 2 0 p Z Z 2 p p 2 E|X(t)| ≤γ1 Ekr(t)ξ(0)k + E µ(dθ) r(t + θ − s)ξ(s)ds [−τ,0] θ

2 t 2 Z p p p + γ2 E r(t − s)σ(Xs)dW (s) (3.8) 0 3 X =: Ji. i=1

In what follows, we estimate the terms J1, J2 and J3, one-by-one. For the ﬁrst term, by virtue of (2.2), one has

2 p 2 −2αt 2 J1 ≤ γ1 Cαe kξk∞.

For the term J2, by the H¨olderinequality and (2.2), it is readily seen that

2 Z Z 0 2 p J2 ≤γ1 E k|µk| µ(dθ) r(t + θ − s)ξ(s)ds [−τ,0] θ 2 Z Z 0 p 2 ≤γ1 E k|µk|τ µ(dθ) kr(t + θ − s)ξ(s)k ds [−τ,0] θ 2 Z τ p 2 2 2αs 2 −2αt ≤γ1 k|µk| Cατ e dskξk∞e . 0 2928 MIN ZHU, PANPAN REN AND JUNPING LI

Now we estimate the last term of (3.8). By applying Lemma 2.3 and the H¨older inequality, it follows from (2.2) and (3.2) that

t 2 Z p p

E r(t − s)σ(Xs)dW (s) 0 Z t 2 p p p ≤ (p − 1) Ekr(t − s)σ(Xs)k ds 2 0 Z t 2 p 2 −2α(t−s) p p ≤ (p − 1)Cα e Ekσ(Xs)k ds 2 0 Z t Z p 2 p 2 −2α(t−s)h 2 2 2 i p ≤L (p − 1)Cα e E |X(s)| + |X(s + θ)| ν(dθ) ds 2 0 [−τ,0] Z t 2 p 2 −2α(t−s)h p p ≤L (p − 1)Cα e E|X(s)| 2 0 Z 2 p p i + E|X(s + θ)| ν(dθ) ds [−τ,0] (3.9) t p Z 2 p 2 −2αt 2αs p p 2 −2αt ≤L (p − 1)Cαe e (E|X(s)| ) ds + L (p − 1)Cαe 2 0 2 Z t Z 2 2αs p p × e ds E|X(s + θ)| ν(dθ) 0 [−τ,0] Z t 2 p 2 −2αt 2αs p p p 2 −2αt ≤L (p − 1)Cαe e E|X(s)| ds + L (p − 1)Cαe 2 0 2 Z t 2ατ 2αs p 2 × kνkvare e (E|X(s)| ) p ds −τ t p Z 2 2 2ατ −2αt 2αs p p ≤L (p − 1)Cα(1 + e kνkvar)e e (E|X(s)| ) ds 2 0 p + L (p − 1)C2 e−2αtkνkvare2ατ (2α)−1kξk2 , 2 α ∞ where in the fourth step we have used Minkovskii’s inequality. Substituting the previous estimates into (3.8), it gives that

2 2 2 Z τ p p p 2 −2αt 2 p 2 2 2αs 2 −2αt E|X(t)| ≤γ1 Cαe kξk∞ + γ1 k|µk| Cατ e dskξk∞e 0

2 p + γ p L (p − 1)C2 e−2αtkνkvare2ατ (2α)−1kξk2 2 2 α ∞ 2 p + γ p L (p − 1)C2 (1 + e2ατ kνkvar)e−2αt 2 2 α (3.10) Z t 2αs p 2 × e (E|X(s)| ) p ds. 0 Multiplying by e2αt on both side of (3.10) gives that

2 2 2αt p p 2 p p 2 2ατ e |X(t)| ≤C1kξk + γ L (p − 1)C (1 + e kνkvar) E ∞ 2 2 α Z t 2αs p 2 × e (E|X(s)| ) p ds, 0 STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2929 where 2 2 Z τ 2 p 2 p 2 2 2αs p p 2 2ατ −1 C1 := γ1 Cα + γ1 k|µk| Cατ e ds + γ2 L (p − 1)Cαkνkvare (2α) > 0. 0 2 So, the Gronwall inequality (see, e.g., [18, Theorem 8.1, p.45]) leads to

2 p p 2 −2˜αt (E|X(t)| ) ≤ C1kξk∞e , t ≥ 0, 2 p p 2 2ατ where 2˜α := 2α − γ2 L 2 (p − 1)Cα(1 + e kνkvar). This further gives p 2m −pαt˜ E|X(t)| ≤ Kkξk∞ e , t ≥ 0. Note that (3.5) impliesα ˜ > 0, therefore the result (3.4) is established in the case of p ≥ 2. Noting p¯ 1 p 1 (E|X(t)| ) p¯ ≤ (E|X(t)| ) p , for 0 < p¯ < 2, p ≥ 2, together with (3.4), we see that the p-th moment exponential stability implies thep ¯-th moment exponential stability. Taking p = 2 yields to the estimate for E|X(t)|p¯(0 < p¯ < 2). Therefore, we have the following corollary.

Corollary 3.2. Let 0 < p < 2 and υ1 < 0. Assume further that (3.2) holds with 2 2ατ L > 0 such that LCα(1 + kνkvare ) < 2α, for α ∈ (0, −υ1), where Cα > 0 is introduced in (2.2). Then, for any initial value ξ ∈ C , the solution of (3.1) is p-th moment exponentially stable, i.e. there exist constants K > 0 and α˜ > 0 such that p p −pαt˜ E|X(t; ξ)| ≤ Kkξk∞e , t ≥ 0. Carrying out similar arguments as Theorem 3.1 and Corollary 3.2 respectively, we can show that the solution X(t) of Eq.(3.1) has the properties as follows: Theorem 3.3. Let the conditions of theorem 3.1 hold. Then for any different initial values ξ, η ∈ C , there exists a pair of positive constants K and α˜ such that p p −pαt˜ E|X(t; ξ) − X(t; η)| ≤ Kkξ − ηk∞e , t ≥ 0, p ≥ 2, where α˜ is given in Theorem 3.1. Corollary 3.4. Let the conditions corollary 3.2 hold. Then for any different initial values ξ, η ∈ C , there exists a pair of positive constants K and α˜ such that p p −pαt˜ E|X(t; ξ) − X(t; η)| ≤ Kkξ − ηk∞e , t ≥ 0, 0 < p < 2, where α˜ is given in Theorem 3.1. In a stable system, by virtue of the results of Theorem 3.3 and Corollary 3.4, trajectories of solutions corresponding to different initial values become closer in the p-th moment after a long time. In general, moment exponential stability and almost sure exponential stability do not imply each other. However, if some conditions are required, moment exponential stability implies almost sure exponential stability. The following result demonstrates this point. Theorem 3.5. Let the conditions of theorem 3.1 hold. Then for any initial values ξ ∈ C , there exists a constant γ > 0 such that 1 lim sup ln |X(t)| ≤ −γ. t→∞ t 2930 MIN ZHU, PANPAN REN AND JUNPING LI

Proof. To show this assertion, it is suﬃcient to show that there exists a constant δ > 0 such that 2 −2δn E( sup |X(t)| ) ≤ Ke , ∀ n ≥ 1. (3.11) n−1≤t≤n Indeed, if (3.11) is true, using the Chebyshev inequality, we have for any γ < δ 2 −2γn 2γn 2 −2(δ−γ)n P ( sup |X(t)| > e ) ≤ e E( sup |X(t)| ) ≤ Ke . n−1≤t≤n n−1≤t≤n ∞ −2(δ−γ)n Since Σn=1e < ∞, in view of Borel-Cantelli lemma, there exists an Ω0 ∈ Ω with P (Ω0) = 1 such that for any ω ∈ Ω0 there exists an integer n0(ω), for n ≥ n0(ω) and n − 1 ≤ t ≤ n, |X(t)|2 ≤ e−2γn ≤ e−2γt, which implies the desired conclusion. The remainder of the proof is to check (3.11). For n − 1 ≤ t ≤ n, n ≥ 1 + τ, X(t) can be represented as Z t Z Z t X(t) = X(n − 1) + µ(dθ)X(s + θ) ds + σ(Xs)dW (s). n−1 [−τ,0] n−1 By the elementary inequality: (a + b + c)2 ≤ 3a2 + 3b2 + 3c2, it follows from (3.4) that 2 2 E( sup |X(t)| ) ≤3E|X(n − 1)| n−1≤t≤n Z n Z 2 + 3E µ(dθ)X(s + θ) ds n−1 [−τ,0] (3.12) Z t 2

+ 3E sup σ(Xs)dW (s) n−1≤t≤n n−1 2 −2˜α(n−1) ≤3Kkξk∞e + 3N2 + 3N3. Observe from H¨older’sinequality and (3.4) that Z n Z 2 N2 ≤E µ(dθ)X(s + θ) ds n−1 [−τ,0] Z n Z 2 ≤k|µk|E |X(s + θ)| µ(dθ)ds n−1 [−τ,0] (3.13) Z n 2 2 ≤k|µk| E|X(s)| ds n−1−τ 2 2 −1 2˜ατ −2˜α(n−1) ≤k|µk| Kkξk∞(2˜α) e e . In terms of the Burkholder-Davis-Gundy inequality and (3.2), one obtains that Z n 2 N3 ≤4E kσ(Xs)k ds n−1 Z n Z 2 2 ≤4LE |X(s)| + |X(s + θ)| ν(dθ) ds n−1 [−τ,0] Z n Z n 2 2 (3.14) ≤4L E|X(s)| ds + 4Lkνkvar E|X(s)| ds n−1 n−1−τ Z n 2 ≤4L(1 + kνkvar) E|X(s)| ds n−1−τ 2 −1 2˜ατ −2˜α(n−1) ≤4L(1 + kνkvar)Kkξk∞(2˜α) e e . STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2931

Thus, combining (3.12) with (3.13) and (3.14) leads to

2 2 −2˜α(n−1) 2 2 −1 2˜ατ −2˜α(n−1) E( sup |X(t)| ) ≤3Kkξk∞e + 3k|µk| Kkξk∞(2˜α) e e n−1≤t≤n 2 −1 2˜ατ −2˜α(n−1) + 12L(1 + kνkvar)Kkξk∞(2˜α) e e ≤Ke−2˜αn. Let δ =α ˜, then the required assertion follows.

Remark 1. It is worth pointing out that the right-hand sides of (3.1) do not involve information on current time. The techniques used in [23] and [5] do not work. To overcome this difficulty, by use of the variation-of-constants formula we can deduce both p-th moment and almost sure exponential stability of solution. From Theorem 3.5, under some conditions we obtain that p-th moment exponential stability implies almost sure exponential stability. Remark 2. Due to the fact that Z t r(t − s)σ(Xs)dW (s) 0 is not a martingale, when p > 2 the estimate of this term makes the analysis more difficult. It cannot be obtained directly from [7, Lemma 7.7, p.194]. And some ideas of the aforementioned reference cannot be used. Lemma 2.3 overcome this difficulty. The established method in the estimate of (3.8) can be extended to study the p-th moment exponential stability for a wide range of SDEs. Remark 3. D. Nguyen (2014) in [20] gave a method for the estimate for diffusion process. They used the following derivation: Let Y denote a random variable following an N(0, a2). Then for any p > 0 p p+1 2 p 2 Γ( 2 ) p E|Y | = 1 a . Γ( 2 ) However, there is a minor problem in the proof. Let Y ∼ N(0, a2). Then, by means of the characteristic function, it follows that

( 2k p a (2k − 1)!!, p = 2k, k = 1, 2, ... E|Y | = 0, p = 2k − 1, k = 1, 2, .... And by a close scrutiny of the argument, this derivation in [20] should be revised to p p+1 2 p 2 Γ( 2 ) p E|Y | ≤ 1 a . Γ( 2 ) If the diffusion coefficient σ(·) is a deterministic function of time t, the approach of D. Nguyen [20] can be successfully used in investigating the p-th moment exponential stability for the solution. However, this approach seems hard to work for the diffusion coefficient σ(·) involving the retarded elements. This difficulty from the diffusion coefficient can be resolved by using the above method applied in Theorem 3.1. Therefore, the established method in proof of Theorem 3.1 and Corollary 3.2 can be extend to study the stability of a class of SDEs with the diffusion coefficient involving the retarded element. So it can be used to improve those results given in [20]. 2932 MIN ZHU, PANPAN REN AND JUNPING LI

4. Exponential stability for retarded SDEs of neutral type. Next we pro- ceed to extend Eq.(3.1) to retarded SDE of neutral type in the form Z Z d X(t) − ρ(dθ)X(t + θ) = µ(dθ)X(t + θ) dt + σ(Xt)dW (t). (4.1) [−τ,0] [−τ,0]

Theorem 4.1. Let p ≥ 2 and υ2 < 0. Assume further that (3.2) holds with L > 0 2 2βτ such that p(p − 1)LCβ(1 + e kνkvar) < 4β, for β ∈ (0, −υ2), where Cβ > 0 is introduced in (2.5). Then, for any initial value ξ ∈ W 1,2([−τ, 0]; Rn), the solution of (4.1) is p-th moment exponentially stable, i.e. there exist constants K > 0 and β˜ > 0 such that p p −pβt˜ E|X(t; ξ)| ≤ Kkξk∞e , t ≥ 0. (4.2) Proof. By (2.4) and [2, Theorem 1], for any ξ ∈ W 1,2([−τ, 0]; Rn), Eq.(4.1) can be written as Z t X(t) =Y (t; ξ) + G(t − s)σ(Xs)dW (s) 0 Z Z Z 0 =G(t)ξ(0) − ρ(dθ)G(t + θ)ξ(0) + µ(dθ) G(t + θ − s)ξ(s)ds [−τ,0] [−τ,0] θ Z Z 0 Z t 0 + ρ(dθ) G(t + θ − s)ξ (s)ds + G(t − s)σ(Xs)dW (s), [−τ,0] θ 0 which, in addition to (3.6), further implies that for any p ≥ 2 there exist constants γ¯1 > 1 andγ ¯2 > 1 such that Z p p p E|X(t)| ≤γ¯1E|G(t)ξ(0)| +γ ¯1E ρ(dθ)G(t + θ)ξ(0) [−τ,0] Z Z 0 p

+γ ¯1E µ(dθ) G(t + θ − s)ξ(s)ds [−τ,0] θ (4.3) Z Z 0 p 0 +γ ¯1E ρ(dθ) G(t + θ − s)ξ (s)ds [−τ,0] θ Z t p

+γ ¯2E G(t − s)σ(Xs)dW (s) . 0 By carrying out a similar argument of (3.9), one ﬁnds that t 2 Z p p

E G(t − s)σ(Xs)dW (s) 0 t p Z 2 2 2βτ −2βt 2βs p p (4.4) ≤L (p − 1)Cβ(1 + e kνkvar)e e (E|X(s)| ) ds 2 0 p + L (p − 1)C2e−2βtkνkvare2βτ (2β)−1kξk2 . 2 β ∞ And following a similar argument for estimate J2 in the proof of Theorem 3.1, we obtain that Z Z 0 2 Z τ 2 2 2 −2βt 2βs E µ(dθ) G(t + θ − s)ξ(s)ds ≤k|µk| τCβkξk∞e e ds (4.5) [−τ,0] θ 0 and Z Z 0 2 Z τ 0 2 2 −2βt 0 2 2βs E ρ(dθ) G(t + θ − s)ξ (s)ds ≤k|ρk| Cβτe kξ k∞ e ds. (4.6) [−τ,0] θ 0 STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2933

Combining (4.4) and (4.5) with (4.6) and taking (2.5) and (3.7) into consideration, the following inequality is obtained: p 2 (E|X(t)| ) p 2 2 p 2 −2βt 2 p 2 2βτ 2 −2βt 2 ≤γ¯1 Cβe kξk∞ +γ ¯1 k|ρk| e Cβe kξk∞ 2 Z τ 2 Z τ p 2 2 2 −2βt 2βs p 2 2 −2βt 0 2 2βs +γ ¯1 k|µk| τCβkξk∞e e ds +γ ¯1 k|ρk| Cβτe kξ k∞ e ds 0 0 2 p +γ ¯ p L (p − 1)C2e−2βtkνkvare2βτ (2β)−1kξk2 2 2 β ∞ t 2 p Z 2 p 2 2βτ −2βt 2βs p p +γ ¯2 L (p − 1)Cβ(1 + e kνkvar)e e (E|X(s)| ) ds, 2 0 which further implies that t 2 2 p Z 2 2βt p p p 2 2βτ 2βs p p e (E|X(t)| ) ≤ K1 +γ ¯2 L (p − 1)Cβ(1 + e kνkvar) e (E|X(s)| ) ds, 2 0 where 2 2 Z τ p 2 2 p 2 2βτ 2 2 2 2 2 2βs K1 =¯γ1 Cβkξk∞ +γ ¯1 k|ρk| e Cβkξk∞ +γ ¯1k|µk| τCβkξk∞ e ds 0 2 Z τ 2 p 2 2 0 2 2βs p p 2 2βτ −1 2 +γ ¯1 k|ρk| Cβτkξ k∞ e ds +γ ¯2 L (p − 1)Cβkνkvare (2β) kξk∞. 0 2

As K1 is a positive constant, by the Gronwall inequality, we have p 2 −2βt˜ (E|X(t)| ) p ≤ K1e , t ≥ 0, 2 ˜ p p 2 2βτ where 2β := 2β − γ¯2 L 2 (p − 1)Cβ(1 + e kνkvar). This further gives that for p ≥ 2 p −pβt˜ E|X(t)| ≤ Ke , t ≥ 0. 2 2βτ For any p ≥ 2, due to p(p−1)LCβ(1+kνkvare ) < 4β, we choose L > 0 suﬃciently small andγ ¯2 > 1 such that 2 p 2 2βτ p(p − 1)¯γ2 LCβ(1 + kνkvare ) < 4β. Therefore the result (4.2) is established in the case of p ≥ 2. Moreover, the higher moment can estimate the lower moment, so we can obtain the following results.

Theorem 4.2. Let 0 < p < 2 and υ2 < 0. Assume further that (3.2) holds with 2 2βτ L > 0 such that LCβ(1 + e kνkvar) < 2β, for β ∈ (0, −υ2), where Cβ > 0 is introduced in (2.5). Then, for any initial value ξ ∈ W 1,2([−τ, 0]; Rn), the solution of (4.1) is p-th moment exponentially stable, i.e. there exist constants K > 0 and β˜ > 0 such that p p −pβt˜ E|X(t; ξ)| ≤ Kkξk∞e , t ≥ 0. Theorem 4.3. Let the conditions of Theorem 4.1 ( resp. Theorem 4.2) hold. Then for any diﬀerent initial values ξ, η ∈ W 1,2([−τ, 0]; Rn), there exists a constant K such that p p −pβt˜ E|X(t; ξ) − X(t; η)| ≤ Kkξ − ηk∞e , t ≥ 0, p ≥ 2 (resp. 0 < p < 2), where β˜ is given in Theorem 4.1. 2934 MIN ZHU, PANPAN REN AND JUNPING LI

Theorem 4.4. Let the conditions of theorem 4.1 hold. Then for any initial values ξ ∈ W 1,2([−τ, 0]; Rn), there exists a constant γ¯ > 0 such that

1 lim sup ln |X(t)| ≤ −γ.¯ t→∞ t

˜ −1 1 ˜ where γ¯ = min{2β, τ ln k|ρk|2 }, β is given in Theorem 4.1.

Proof. For any integer n ≥ 1, using the Doob martingale inequality and H¨older’s inequality, together with (3.2), (4.1)and (4.2), we have

Z 2 ¯ ¯ E sup X(nτ + θ) − ρ(dθ)X(nτ + θ + θ) 0≤θ¯≤τ [−τ,0] Z 2 Z (n+1)τ Z 2

≤3E X(nτ) − ρ(dθ)X(nτ + θ) + 3E µ(dθ)X(s + θ) ds [−τ,0] nτ [−τ,0] ¯ Z nτ+θ 2

+ 3E sup σ(Xs)dW (s) 0≤θ¯≤τ nτ Z 2 2 ≤6E|X(nτ)| + 6E ρ(dθ)X(nτ + θ) [−τ,0] Z (n+1)τ Z 2 Z (n+1)τ 2 + 3τE µ(dθ)X(s + θ) ds + 12 Ekσ(X(s + θ))k ds nτ [−τ,0] nτ Z 2 2 ≤6E|X(nτ)| + 6k|ρk| E|X(nτ + θ)| ρ(dθ) + 3τk|µk| [−τ,0] Z (n+1)τ Z 2 × E |X(s + θ)| µ(dθ)ds nτ [−τ,0] Z (n+1)τ Z 2 2 + 12LE |X(s)| + |X(s + θ)| ν(dθ) ds nτ [−τ,0] 2 2 2 ≤6E|X(nτ)| + 6k|ρk| sup E|X(nτ + θ)| + 3τk|µk| −τ≤θ≤0 Z Z (n+1)τ 2 × µ(dθ) E|X(s + θ)| ds [−τ,0] nτ Z (n+1)τ Z (n+1)τ Z 2 2 + 12L E|X(s)| ds + 12L E|X(s + θ)| ν(dθ) ds nτ nτ [−τ,0] Z (n+1)τ 2 2 2 2 2 ≤6E|X(nτ)| + 6k|ρk| sup E|X(nτ + θ)| + 3τk|µk| E|X(s)| ds −τ≤θ≤0 nτ−τ Z (n+1)τ Z (n+1)τ 2 2 + 12L E|X(s)| ds + 12Lkνkvar E|X(s)| ds nτ nτ−τ Z (n+1)τ 2 −γnτ¯ 2 2 −γ¯(nτ−τ) 2 2 −γs¯ ≤6Kkξk∞e + 6k|ρk| Kkξk∞e + 3τk|µk| Kkξk∞ e ds nτ−τ Z (n+1)τ Z (n+1)τ 2 −γs¯ 2 −γs¯ + 12LKkξk∞ e ds + 12LkνkvarKkξk∞ e ds nτ nτ−τ STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2935

2 −γnτ¯ 2 2 γτ¯ −γnτ¯ 2 2 −1 γτ¯ −γnτ¯ ≤6Kkξk∞e + 6k|ρk| Kkξk∞e e + 3τk|µk| Kkξk∞γ¯ e e 2 −1 −γnτ¯ 2 −1 γτ¯ −γnτ¯ + 12LKkξk∞γ¯ e + 12LkνkvarKkξk∞γ¯ e e =Ce−γnτ¯ , 2 2 γτ¯ 2 −1 γτ¯ −1 −1 γτ¯ where C = Kkξk∞[6+6k|ρk| e +3τk|µk| γ¯ e )+12Lγ¯ +12Lkνkvarγ¯ e ]. For any ε ∈ (0, γ¯), using the Chebyshev inequality, we have Z 2 ¯ ¯ −(¯γ−ε)nτ −εnτ P ω : sup X(nτ + θ) − ρ(dθ)X(nτ + θ + θ) > e ≤ Ce . 0≤θ¯≤τ [−τ,0] The Borel-Cantelli lemma yields that for almost all ω ∈ Ω, there exists an integer n0(ω) such that Z 2 ¯ ¯ −(¯γ−ε)nτ sup X(nτ + θ) − ρ(dθ)X(nτ + θ + θ) ≤ e , n ≥ n0. 0≤θ¯≤τ [−τ,0]

Consequently, for almost all ω ∈ Ω, if t ≥ n0τ, Z 2 −(¯γ−ε)(t−τ) X(t) − ρ(dθ)X(t + θ) ≤ e . [−τ,0] 2 Moreover, for t ∈ [0, n τ], X(t) − R ρ(dθ)X(t + θ) is ﬁnite. So, for almost all 0 [−τ,0] ω ∈ Ω, there exists a ﬁnite constant H = H(ω), if t ≥ 0, Z 2 −(¯γ−ε)t X(t) − ρ(dθ)X(t + θ) ≤ He . [−τ,0] On the other hand, set eγτ¯ k|ρk|2 < ε < 1. For t ≥ 0, note that Z 2

X(t) − ρ(dθ)X(t + θ) [−τ,0] Z Z 2 2 ≥|X(t)| − 2|X(t)| ρ(dθ)X(t + θ) + ρ(dθ)X(t + θ) [−τ,0] [−τ,0] Z 2 2 −1 ≥(1 − ε)|X(t)| + (ε − 1) ρ(dθ)X(t + θ) . [−τ,0] Hence, we see that 1 Z 2 1 Z 2 2 |X(t)| ≤ X(t) − ρ(dθ)X(t + θ) + ρ(dθ)X(t + θ) . 1 − ε [−τ,0] ε [−τ,0] Also, for each T > 0, H 1 h Z 2i (¯γ−ε)t 2 (¯γ−ε)t sup [e |X(t)| ] ≤ + sup e ρ(dθ)X(t + θ) 0≤t≤T 1 − ε ε 0≤t≤T [−τ,0] H e(¯γ−ε)τ k|ρk|2 ≤ + sup [e(¯γ−ε)t|X(t)|2]. 1 − ε ε −τ≤t≤T Through a straightforward mathematical computation, we get (¯γ−ε)τ 2 (¯γ−ε)τ 2 e k|ρk| (¯γ−ε)t 2 H e k|ρk| 2 1 − sup [e |X(t)| ] ≤ + kξk∞. ε 0≤t≤T 1 − ε ε which implies that 1 γ¯ − ε lim sup ln |X(t)| ≤ − a.s. t→∞ t 2 Let ε → 0. The required result is obtained. 2936 MIN ZHU, PANPAN REN AND JUNPING LI

Remark 4. In the beginning of this paper, we suppose that σ(0) = 0n×m. This assumption plays a key role in our stability analysis as above. From the assumption (3.2) on the diffusion coefficient we deduce that for any φ ∈ C there exists an L > 0 such that Z kσ(φ)k2 ≤ L |φ(0)|2 + |φ(θ)|2ν(dθ) , [−τ,0] which has been used in the proof of stability theory. It guarantees that X(t) admits the property as the form E|X(t)|p ≤ ae−bt. Otherwise, we arrive at E|X(t)|p ≤ c + ae−bt (a, b, c ∈ R), and the constants c 6= 0 cannot be dumped. Moreover, this method is widely applied in the stability analysis. For example, Zhu [26] studied asymptotic stability in the pth moment for SDEs with Lévynoise; Zhou and Yang [25] gave the criterion of mean square exponential stability for delayed neural networks with Lévynoise. However, if we only seek convergence of solutions from different initial value, this assumption can be removed.

5. Examples. In this section, we consider a couple of examples to verify the the- ories established in the previous section.

Example 1. Consider a semi-linear retarded SDE

dX(t) = −X(t − 1)dt + σ(X(t − 1))dW (t),X0 = ξ ∈ C . (5.1)

It is impossible to choose constants b1 > b2 > 0 such that

2 2 2 −2xy + σ (y) ≤ c − b1|x| + b2|y| , x, y ∈ R. So, (5.1) does not satisfy a dissipative condition. In view of the corresponding characteristic equation λ+e−λ = 0, we deduce that the unique root is λ = −0.3181+ 1.3372i. So, we have υ = −0.3181. Taking p = 2 and by Theorem 3.1, when the 2 2ατ Lipschitz constant L of σ such that LCα(1 + e ) < 2α for α ∈ (0, 0.3181), the solution X(t) of (5.1)is almost surely exponentially stable and exponentially stable in mean square.

Example 2. Consider a semi-linear retarded SDE

1 Z 0 d X(t) + X(t − 1) = −X(t − 1)dt + a X(t + θ)dθdW (t),X0 = ξ ∈ C , (5.2) 3 −1 where a ∈ R and W (t) is a real-valued Brownian motion. It is easy to see that the corresponding characteristic equation is 1 λ + (1 + λ)e−λ = 0, λ ∈ . (5.3) 3 C A simple calculation by Matlab yields that the unique root of (5.3) is λ = −2.313474269. Then, by the results in section 4 we deduce that the solution X(t) of (5.2) is almost surely exponentially stable and moment exponentially stable if a ∈ R is suﬃciently small.

Acknowledgments. The authors wish to thank the referees for their detailed com- ments and helpful suggestions. STABILITY OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS 2937

REFERENCES

[1] J. A. D. Appleby, X. Mao and H. Wu, On the almost sure running maxima of solutions of affine stochastic functional differential equations, SIAM Journal on Mathematical Analysis, 42 (2010), 646–678. [2] J. A. D. Appleby, H. Wu and X. Mao, On the almost sure running maxima of solutions of affine neutral stochastic functional differential equations, preprint, arXiv:1310.2349 (2013). [3] J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society,, 465 (2009), 2111– 2134. [4] J. Bao and C. Yuan, Stationary distributions for retarded stochastic differential equations without dissipativity, Stochastic, (2016), 1–20. [5] J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Analysis: Theory, Methods and Applications, 98 (2014), 66–82. [6] J. Bao, G. Yin and C. Yuan, Asymptotic Analysis for Functional Stochastic Equations, Sp- inger, 2016. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1992. [8] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993. [9] Z. Hou, J. Bao and C. Yuan, Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps, Journal of Mathematical Analysis and Applications, 366 (2010), 44–54. [10] S. Janković,J. Randjelovic and M. Jovanović,Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 355 (2009), 811–820. [11] K. Liu, The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Applied Mathematics and Optimization, 60 (2009), 1–38. [12] K. Liu and Y. Shi, Razumikhin-type theorems of infinite dimensional stochastic functional differential equations, in IFIP Conference on System Modeling and Optimization, Springer US, 202 (2006), 237—247. [13] X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems & Control Letters, 26 (1995), 245–251. [14] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Processes and their Applications, 65 (1996), 233–250. [15] X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM Journal on Mathematical Analysis, 28 (1997), 389–401. [16] X. Mao, Y. Shen and C. Yuan, Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications, 118 (2008), 1385–1406. [17] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, Journal of Mathematical Analysis and Applications, 268 (2002), 125–142. [18] X. Mao, Stochastic Differential Equationa and Applications, 2nd edition, Horwood Publishing Limited, Chichester, 2008. [19] S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. [20] D. Nguyen, Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 1–7. [21] M. Reiß, M. Riedle and O. Gaans, Delay differential equations driven by Lévyprocesses: stationarity and Feller properties, Stochastic Processes and their Applications, 116 (2006), 1409–1432. [22] L. Tan, W. Jin and Y. Suo, Stability in distribution of neutral stochastic functional differential equations, Stochastics and Probability Letters, 107 (2015), 27–36. [23] F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete and Continuous Dynamical System Series-B, 18 (2013), 1715–1734. [24] C. Yuan, J. Zou and X. Mao, Stability in distribution of stochastic differential delay equations with Markovian switching. Systems and Control Letters, 50 (2003), 195–207. 2938 MIN ZHU, PANPAN REN AND JUNPING LI

[25] W. Zhou, J. Yang and X. Yang, p-th moment exponential stability of stochastic delayed hybrid systems with Lévynoise, Applied Mathematical Modelling, 39 (2015), 5650–5658. [26] Q. Zhu, Asymptotic stability in the p-th moment for stochastic differential equations with Lévynoise, Journal of Mathematical Analysis and Applications, 416 (2014), 126–142. [27] X. Zong and F. Wu, Exponential stability of the exact and numerical solutions for neutral stochastic delay differential equations, Applied Mathematical Modelling, 40 (2016), 19–30. Received July 2016; revised April 2017. E-mail address: Min Zhu: [email protected] E-mail address: Panpan Ren: [email protected] E-mail address: Junping Li: [email protected]