Boundedness analysis of stochastic integro-differential systems with Lévy noise

ABSTRACT This paper is concerned with the pth moment globally exponential ultimate boundedness of stochastic integro-differential systems with Lévy noise. With the help of the Lyapunov function methods and the inequality techniques, several sufficient criteria on the exponential ultimate boundedness are presented for the systems. The results show that the boundedness was determined by the coefficients in the estimation of the Itô operator of the energy function V along the trajectories of the addressed systems.


Introduction
Integro-differential system is a kind of very important type of an ordinary differential equation which has not only integrals of unknown function but also derivatives of unknown function. Integro-differential system has received much attention from researchers since this system has a wide application in many fields, such as biological system [1], switched systems [2] and neural networks [3].
Stochastic systems with Lévy noise are used to describe the evolutionary processes in which the structures are subject to stochastic abrupt changes. Recently, these systems received remarkable attention from the researchers. Many important results can be found in the literature concerning the stability [23][24][25], the periodic solution [26] and the existence and uniqueness [27] of these systems. However, the problem of the boundedness of stochastic integro-differential systems with Lévy noise is more complicated and still open.
Inspired by the aforementioned discussions, the present paper is focused on the pth moment globally exponential ultimate boundedness of stochastic integro-differential systems with Lévy noise. With the help of the Lyapunov function methods and the inequality techniques, several sufficient conditions are presented for the pth moment globally exponential ultimate boundedness and the pth moment globally exponential stability of the systems. The main contributions of this paper are highlighted as follows: (i) concerned with the problem of the boundedness for a class of stochastic integro-differential systems and take fully into account the effects of Lévy noise and infinite time-delay; (ii) some sufficient conditions, including both Lyapunov type and coefficients type, are derived for the exponential ultimate boundedness; and (iii) the estimation of the ultimate bound sets is also given out.
In this paper, we study the following stochastic integro-differential systems with Lévy noise: where the initial value φ(s) the scalar c ∈ (0, ∞] stands for the maximum allowable jump size, N stands for a Poisson random measure defined on R t 0 × (R n \ {0}) with intensity measure v and compensatorÑ . Moreover, N is independent of B and v is a lévy measure such thatÑ ( dt, du) The pair (B, N ) generally referred to as a Lévy noise. We assume that for any φ(s) , there exists at least one solution of system (1).
Using the condition (i), we have This ends the proof of Theorem 3.1.

Assumption 3.1:
There exist a symmetric positivedefinite matrix S, a function h(s) ∈ L and several constants p > 0 and K i (i = 1, 2, . . . , 12) such that Theorem 3.2: Suppose that the Assumption 3.1 holds. If 3 >ˆ 4 > 0, then system (1) is p-GEUB, and the solutions y(t) of (1) will converge to the ultimate bound set where the scalar λ ∈ (0, λ 0 ) satisfies the following relation Proof: Consider the Lyapunov function candidate Obviously, we have Using (16) to (19), we arrive successively Applying Lemma 2.1 to (28) yields Since using the continuity and noting h(s) ∈ L e , there exist a scalar λ ∈ (0, λ 0 ) such that (25) holds. Consequently, from (27), (29) and Theorem 3.1, one derives the following estimate Remark 3.1: In [39], several exponential ultimate boundedness criteria have been obtained for stochastic integro-differential systems. However, the criteria proposed in [39] are unable to detect the exponential ultimate boundedness of system (1) because Lévy noise was ignored in [39].

Remark 3.2:
In [25], authors derived some interesting sufficient conditions for the pth moment exponential stability of stochastic delay differential systems with Lévy noise. Compared with [25], system (1) here is an infinite distributed delay system. Moreover, the results in [25] are limited to stability. Up till now, the boundedness problem has not been studied for stochastic integro-differential systems with Lévy noise, this indicates that our results are new.

An example
Example 4.1: Let us consider the one-dimensional stochastic integro-differential system with Lévy noise as follows where the Lévy measure v satisfies v( du) = du/1 + |u| 2 .
Here, we choose λ 0 = 0.9 and λ = 0.5 < λ 0 which satisfies the inequality (6). According to Theorem 3.1, system (33) is GEUB in the mean square, and the solutions y(t) will converge to the ultimate bound set

Conclusions
This paper has investigated the pth moment globally exponential ultimate boundedness issue of stochastic integro-differential systems with Lévy noise. Using the Lyapunov function methods and combining the inequality techniques, some sufficient criteria on the exponential ultimate boundedness have been presented for the systems. The method presented in this paper may be applied to some other kinds of stochastic systems such as stochastic systems with exogenous disturbances [40], semi-Markov switched stochastic systems [41] and stochastic systems with additive noise [42].

Funding
The work is supported by the National Natural Science Foundation of China (NNSF) (grant number 11501518).