Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

Wemake the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.


Introduction
Recently, stochastic functional differential equations with infinite delay (IDSFDEs) have attracted broad attention of many researchers.In the literature, there are two main lines of research on the IDSFDEs.On one hand, existence and uniqueness of solution are basic properties for equations.So a great number of authors have devoted themselves to this research, and thus many excellent results on the existence and uniqueness of the solutions to IDSFDEs and neutral IDSFDEs can be found in [1][2][3][4][5][6] and references cited therein.On the other hand, the study of stability and boundedness of solutions is one of the most attracting topics in the qualitative theory of differential equations because of its various applications in many areas such as physics and control theory [7,8].Hence, more and more researchers study them and especially focus on the stability of solutions.An important issue in stochastic analysis is whether or not random disturbance can change the qualitative properties of system, which is particularly important in control field.In most cases, people are interested in the performance of antidisturbance of system.So it is vital to seek some antidisturbance systems or present the intensity of stochastic perturbation that stable system can tolerate without losing the property of stability [9].In recent years, many meaningful works on this topic have come out; see, for example, [10][11][12][13][14][15][16][17][18][19][20][21].
Volterra integrodifferential equations (VIDEs) are widely applied in biology, ecology, medicine, physics, among other scientific areas and thus have been encountered by many researchers in numerical and theoretic analysis; see [7,[22][23][24][25].It is well known that concrete systems are inevitably affected by external perturbations usually modeled by stochastic noise.So a great deal of attention has been paid to the research of stochastic VIDEs [9,26,27].Additionally, time delay is always ubiquitous and infinite delay systems have wide applications in many fields.Hence, there is naturally an important kind of IDSFDEs, that is, stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs).In practice, many applications of IDSVIDEs are greatly dependent on the stability and boundedness of their solutions.However, to the best of the authors' knowledge, few research results mentioned above focus on the stability and boundedness of IDSVIDEs, which motivates the present study.Precisely, this paper investigates in detail the problem of stability and boundedness of solutions for the following IDSVIDE: (1) Both stochastic perturbation and infinite delay are considered in the IDSVIDEs.
(2) A new Lyapunov function is constructed to derive stability and boundedness criteria for IDSVIDEs efficiently.
(3) The problem of how much the stochastic noise VIDEs with infinite delay can tolerate without losing the properties of stability and boundedness has been solved.

Preliminaries
Let (Ω, F, F, P) be a complete probability space with a filtration F = {F  } ≥0 satisfying the usual conditions.As usual, (⋅) denotes a scalar Brownian motion defined on the space and E(⋅) is the mathematical expectation with respect to P. Write ((−∞, 0]; R) as the family of bounded continuous real-valued functions  defined on (−∞, 0] with the norm In this paper, we suppose there exists a constant  > 0 such that |()| ≤ || 2 .Then by Theorem 5.2.7 of [28], there exists a unique global solution () to (1) if () and () are bounded.For more details, readers can see [29,30].
Here and in the rest of the paper, write the solutions with the initial condition   0 =  ∈ ((−∞, 0]; R) as (;  0 , ).Throughout this paper, unless otherwise specified, we use the following Lyapunov function: For any  > 0, define two stopping times: In order to study the stability and boundedness of solutions to (1), we state the following assumptions: (K1) There exist nonnegative continuous functions ℎ : R − → R + and  : R → R, such that (K2) For any  ∈ (0, 1), there exists  > 1 such that  () + where () is defined in (K2).
Remark 1. Conditions (K2) and (K3) are the conditions about the intensity of perturbations.Note that constant  in equality ( 2) is the same as the one which appeared in conditions (K2) and (K3).

Stability of Solutions of IDSVIDEs
In order to consider the stability of ( 1), without loss of generality, we suppose that () =  1 () = 0 and (0) = 0 hold in this section.Hence, (1) has the trivial solution () ≡ 0. First, we introduce three kinds of definitions about stability of solutions to (1).It is easy to see that these definitions are a strict generalization of deterministic cases.
(1) The trivial solution of ( 1) is said to be stochastically stable if for every pair  ∈ (0, 1) and  > 0, there exists a  = (, ,  0 ) > 0 such that whenever ‖‖ < .Additionally, it is said to be stochastically uniformly stable if  is independent of  0 .
(3) The trivial solution of ( 1) is said to be stochastically globally asymptotically stable if it is stochastically stable and, moreover, for all  ∈ ((−∞, 0]; R), it follows that In the following, we will apply the Lyapunov-Krasovskii functional approach to delve into some sufficient criteria, under which the trivial solution to (1) is stochastically stable, stochastically asymptotically stable, and stochastically globally asymptotically stable, respectively.The Itô formula used in this paper can be seen in [28].Theorem 3. Suppose that (K1) and (K2) hold.Then the trivial solution of (1) is stochastically uniformly stable.
Remark 4. If () = (, ) = 0 in (1), then we can get the corresponding disturbance free system.Theorem 3 really tells us that stochastic perturbation cannot disturb the stability of original deterministic system if the noisy intensity satisfies (K2).
Remark 11.Notice that, when conditions (K1) and (K2) hold, the solutions of stochastic system (1) are stable and bounded.That is, environmental noise (in the sense of Itô) cannot disturb stability and boundedness of solutions for some systems if noisy intensity satisfies (K2).Hence, we can construct some anti-interference systems in practice.
Proof.Give  > 0. For any  ∈ (0, 1),  ∈ ((−∞, 0]; R).From Theorem 10, there exists  > 0, such that From here, we can show in the same way as in the proof of Theorem 6 that we could choose sufficiently small , such that 2 2 / 2 <  and (94) Then it follows that After letting  → ∞, we have Let (, ) =   .Then the above inequality is equivalent to The proof is complete.
Remark 13.Obviously, we can verify that the solutions of IDSVIDE (68) are stochastically bounded and stochastically equibounded.And the solutions of IDSVIDE (72) are stochastically ultimately bounded.

Conclusions
Throughout this paper, by combining the Lyapunov-Krasovskii method, we have obtained various kinds of sufficient stability and boundedness criteria, where the ranges of noisy intensity that stable and bounded systems can tolerate without losing the properties of stability and boundedness are presented, respectively.These sufficient conditions are very necessary for us to verify stability and boundedness of stochastic Volterra integrodifferential equations with infinite delays.Moreover, the conditions obtained in this paper can also help us to construct some antidisturbance systems in the applications.In addition, two examples have been given to illustrate our theoretical results.