EXPONENTIAL STABILITY OF SOLUTIONS FOR RETARDED STOCHASTIC DIFFERENTIAL EQUATIONS WITHOUT DISSIPATIVITY

. 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 expo- nential 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 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 R n be an n-dimensional Euclidean space endowed with the inner product u, v := n i=1 u i v i and the Euclidean norm |u| := u, u 1 2 for u, v ∈ R n . Denote R n ⊗ R m by the set of all n × m matrices A endowed with Hilbert-Schmidt norm A := trace(A T A), in which A T is the transpose of A. Let (Ω, P, F ) be a probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., F t = F t+ := ∩ s>t F s and F 0 contains all P-null sets). Let {W (t)} t≥0 be an m-dimensional Brownian motion defined on (Ω, P, F , {F t } t≥0 ). Fix τ ∈ (0, ∞), which will be referred to as the delay or time lag. Let C = C([−τ, 0]; R n ) be the family of all continuous functions ξ : [−τ, 0] → R n equipped with the uniform norm ξ ∞ := sup −τ ≤θ≤0 |ξ(θ)| for ξ ∈ C . For X(·) ∈ C([−τ, ∞]; R n ), define the segment process X t ∈ C by X t (θ) := X(t + θ), θ ∈ [−τ, 0], t ≥ 0. Let µ(·) and ρ(·) be R n ⊗ R n -valued finite signed measure on [−τ, 0], ν(·) a measure on [−τ, 0], C the set of all complex numbers and Re(z) the real part of z ∈ C. Let |ρ | = sup 1≤k≤n 1≤j≤n ρ kj 2 var , where ρ kj var is the total variation of ρ kj .
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 E|X(t; ξ)| p ≤ K ξ p ∞ e −κpt . 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 lim sup Then, for all p ≥ 1 and t > 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 where σ : C → R n ⊗ R m such that σ(0) = 0 n×m is Borel measurable. Throughout this section, we assume that, for any φ, ϕ ∈ C , there exists an L > 0 such that Indeed, for fixed t ≥ 0, by means of the chain rule, we have Integrating from 0 to t, together with (2.1) and (3.1), leads to Next, we shall investigate the p-th moment exponential stability and almost sure exponential stability for the solution X(t) to Eq. (3.1).
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 andα > 0 such that Proof. In what follows, for notation simplicity, we write X(t) in lieu of X(t; ξ). For In the sequel, we fix γ 2 > 1 such that (3.5). Then there exists a γ 1 > 1 such that By the elementary inequality: it follows from (3.3) and (3.6) that In what follows, we estimate the terms J 1 , J 2 and J 3 , one-by-one. For the first term, by virtue of (2.2), one has For the term J 2 , by the Hölder inequality and (2.2), it is readily seen that Now we estimate the last term of (3.8). By applying Lemma 2.3 and the Hölder inequality, it follows from (2.2) and (3.2) that where in the fourth step we have used Minkovskii's inequality. Substituting the previous estimates into (3.8), it gives that  Multiplying by e 2αt on both side of (3.10) gives that So, the Gronwall inequality (see, e.g., [18,Theorem 8.1,p.45]) leads to . This further gives 1 p , f or 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.
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 andα > 0 such that E|X(t; ξ)| p ≤ K ξ p ∞ e −pαt , 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 andα such that E|X(t; ξ) − X(t; η)| p ≤ K ξ − η p ∞ e −pαt , t ≥ 0, p ≥ 2, whereα 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 andα such that 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. Proof. To show this assertion, it is sufficient to show that there exists a constant δ > 0 such that Indeed, if (3.11) is true, using the Chebyshev inequality, we have for any γ < δ P ( sup n−1≤t≤n Since Σ ∞ n=1 e −2(δ−γ)n < ∞, in view of Borel-Cantelli lemma, there exists an Ω 0 ∈ Ω with P (Ω 0 ) = 1 such that for any ω ∈ Ω 0 there exists an integer n 0 (ω), for n ≥ n 0 (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 By the elementary inequality: (a + b + c) 2 ≤ 3a 2 + 3b 2 + 3c 2 , it follows from (3.4) that (3.12) Observe from Hölder's inequality and (3.4) that

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
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, a 2 ). Then for any p > 0 However, there is a minor problem in the proof. Let Y ∼ N (0, a 2 ). Then, by means of the characteristic function, it follows that And by a close scrutiny of the argument, this derivation in [20] should be revised to 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].
Proof. For any integer n ≥ 1, using the Doob martingale inequality and Hölder's inequality, together with (3.2), (4.1)and (4.2), we have For any ε ∈ (0,γ), using the Chebyshev inequality, we have The Borel-Cantelli lemma yields that for almost all ω ∈ Ω, there exists an integer n 0 (ω) such that Consequently, for almost all ω ∈ Ω, if t ≥ n 0 τ , On the other hand, set eγ τ |ρ | 2 < ε < 1. For t ≥ 0, note that Hence, we see that Also, for each T > 0, Through a straightforward mathematical computation, we get which implies that lim sup Let ε → 0. The required result is obtained.

Remark 4.
In the beginning of this paper, we suppose that σ(0) = 0 n×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 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 = 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 Lévy noise; Zhou and Yang [25] gave the criterion of mean square exponential stability for delayed neural networks with Lévy noise. 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 theories established in the previous section.
Example 2. Consider a semi-linear retarded SDE d X(t) + 1 3 where a ∈ R and W (t) is a real-valued Brownian motion. It is easy to see that the corresponding characteristic equation is λ + (1 + 1 3 λ)e −λ = 0, λ ∈ 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.