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- Table of Contents
Journal of Applied Mathematics
Volume 2014 (2014), Article ID 458306, 11 pages
A Class of Stochastic Nonlinear Delay System with Jumps
1College of Mathematics, Jilin University, Changchun 130061, China
2Florida State University, Department of Scientific Computing, Tallahassee, FL 32306, USA
Received 30 August 2013; Revised 25 November 2013; Accepted 9 December 2013; Published 30 January 2014
Academic Editor: Chong Lin
Copyright © 2014 Ling Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.
Usually, for a given nonlinear system satisfies the polynomial growth condition; the solutions may explode in finite time. It is well know that noises can stabilize the given unstable system or make this system even more stable if it is already stable. To stabilize the system, the authors in  introduced a polynomial Brownian noise to suppress potential explosion and then considered another linear Brownian noise to stabilize the suppressed equation. For more details, refer to [1–3].
For example, considering a simple logistic equation with initial value , the expression of the solution is , and there has been only a local solution for ; that is, will explode and the explosion time is . To suppress explosion and stabilize system (2), the dependent scalar Brownian noises are introduced in  and they disturbed system (2) into the theoretical proof and simulations show that noises may not only suppress explosion of solution (3), but also stabilize unstable system (2).
However, all of the papers mentioned above only considered the perturbation by Brownian noises. Recently, there has been increasing attention devoted to the effect of different noises. Non-Gaussian random processes also play an important role in modelling stochastic dynamics. Typical examples of non-Gaussian stochastic processes are Lévy processes and processes arising from Poisson random measures. Further we assume that the different stochastic processes are independent.
Motivated by the previous articles [1–3], we will study the following delay SDE with jumps: with initial data Here () is defined on the probability space, being a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all P-null sets). is real-valued Poisson counting measure with characteristic measure on a measurable subset of with , is the 1-dimensional standard Brownian motion with . We assume that and Poisson process are independent. Let and denote the family of continuous function from to , which is a Banach space with the norm . Denote by the family of all bounded, -measurable, -valued random variable.
We will impose the following assumptions on as local Lipschitz continuous and satisfy the polynomial growth condition.
Assumption 1. For any , , , and . For each integer , there is a positive number such that for all and those with .
Assumption 2. There are some nonnegative numbers , , , and such that for all .
In reference to the existing results in the literature, our contributions are as follows.(i)Use stochastic delay differential equation with jump diffusion to model the evolutions of nonlinear dynamics.(ii)Discuss the global solution of the stochastic equation under polynomial growth condition; in particular, we compare the effect of suppression solution of different type of noises.(iii)For stabilization of noise, we can verify that the linear jump process has an effect on a.s. exponential stability.
To the best of our knowledge, under the assumption of polynomial growth condition, analysis of nonlinear stochastic system with jumps has not been fully investigated and few results have been available so far. We aim to discuss the question in this work.
The organization of the paper is as follows. Section 2 shows the global solution of system (4), which indicates that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. The results of Section 3 guarantee that SDE (4) is stochastically ultimate boundedness. Our main results emerge in Section 4. We discuss the sample Lyapunov exponent is nonpositive and reveal the stabilization effect of Lévy noise. Conclusions and extensions are made in Section 5.
2. Global Solution
Now we first prove the existence of the global solution to (4).
Proof. Since both the drift term and the diffusion terms of (4) are locally Lipschitz, by similar standard truncated argument [4, 5], there is a unique local maximal solution on , where is the explosion time defined by
To show that this solution is actually global, we need to prove that a.s.
Let be a sufficiently large positive number such that . For each , we define the stopping time Clearly is increasing as and ; if we can obtain that a.s., then a.s.; that is, to complete the proof, all we need to show is that a.s.; this is also equivalent to proving that, for any , as . For , define a -function: by If, for any and , one can apply formula to compute that where and is defined as therefore we get from Dynkin’s formula , taking expectation from two sides of (11)
Using the inequality , , , we have because .
Combining (11) and (14) and recalling the well-known Young inequality, we therefore have then we have from (15) that where
Noting that , and , by the boundedness of polynomial function , there is a constant such that . Consider where is independent of .
By the definition of , let for sufficient large and for every such that equals ; hence, Letting implies that So we must obtain a.s., as required. The proof is complete.
The key of this proof in Theorem 3 is the boundedness of , under the conditions of Assumptions 1 and 2 and , which imply that the Brownian noise plays the important role to suppress potential explosion of the solution and guarantees the existence of the global solution.
In what follows, in order to highlight the explosive suppression by jump processes, we will consider the following suitable condition.
Assumption 4. For any , and . If , there exists and such that We still obtain the existence of global solution of (4).
Proof. Since the proof is similar to that of Theorem 3, we here only sketch the proof to point out the variation between them. Noting (12), (15), and (16), we obtain from Assumption 4 that for , that
Therefore, we have
Noting that and , by the boundedness of polynomial functions, there is a constant such that , where is independent of .
By and , the boundedness of follows; thus, the desired assertion can be carried out the same procedure as the last part of Theorem 3.
3. Stochastic Ultimate Boundedness
Theorems 3 and 5 show that the solution of SDE (4) with a given initial data will not explode. This nice property provides us with a great opportunity to discuss how the solution varies in in more details. In this section, we will give two definitions and then give sufficient conditions which guarantee SDE (4) is stochastically ultimate boundedness.
Definition 7. The solutions of SDE (4) are said to be asymptotically bounded in pth moment if there is a positive constant such that the solution of SDE (4) with a given initial value has the property that for any initial data .
In the light of Markov inequality, it is obvious that if a stochastic equation is pth moment boundedness, its solutions must be stochastically ultimately bounded. So we will begin with the following lemma and make use of it to obtain the stochastically ultimate boundedness of SDE (4).
Lemma 9. Under the conditions of Assumptions 1 and 2, for , if , there exists a constant such that the global solution of SDE (4) with any initial data has the property that where is dependent on and independent of the initial data.
Proof. First, Theorem 3 indicates that the solution of (4) will exist for with probability 1. For any and , by virtue of Dynkin’s formula to , we obtain
Here is defined as (12). Therefore, by Assumptions 1 and 2, (11)–(15), we have
Similar to the proof of Theorem 3, we have from (29) and (30) that
Notice that ; if and , (32) has upper boundedness , which means that, combined with (31), Letting gives
Lemma 9′. Under the conditions of Assumptions 1, 2, and 4, supposing that and , there exists a constant such that the global solution of SDE (4) with any given positive initial value has the property that where is dependent on and independent of the initial value.
In order to complete the proof, we only show the key step, from (29), combined with (22), which was rewritten as where
Because of , , the coefficient of polynomial in (38) is negative, it follows boundedness of . Repeating the procedure of Lemma 9, we have from (37) and (38) that the rest of proof can be completed. Omit it here.
This means that the solution is bounded in the pth moment; stochastically ultimate boundedness will follow directly. It shows that the solution trajectory is bounded with large probability.
Theorem 10. The solution of (4) is stochastically ultimately bounded under the condition of Lemma 9 or Lemma 9′; that is, for any , there is a positive constant such that for any positive initial value, the solution of (4) has the property that
Proof. This can be easily verified by Chebyshev’s inequality and Lemma 9 or Lemma 9′, by choosing to be sufficiently large because of the following Hence, as required.
4. Stabilization of Noises
From Sections 2 and 3, we know that both the Brownian noises and jump processes can suppress the potential explosion of the solution and guarantee this global solution to be bounded in the sense of the pth moment. This section is devoted to considering the stable effect of noises under some appropriate conditions; we show that jump processes may stabilize the given unstable nonlinear delay system .
Theorem 11. Suppose that for the initial data satisfying , the solution of SDE (4) is . Under Assumptions 1, 2, and 4, if, for , the solution of SDE (4) with any initial data satisfying has the property
Proof. By Theorem 3, suppose that almost all sample paths of of (4) starting from a nonzero state will never reach the origin for all . Applying the Itô formula to the function leads to
By virtue of exponential martingale inequality with jumps [7, Theorem , page 291].
Choose , and in the above equation. Since , by the Borel-Cantelli lemma, there exists with such that for any , there exists an integer that can be found satisfying whenever , .
Next, for any and , we still make use of exponential martingale inequality, (choose Thus, for all and , , we also have In the following, we divide two cases.(1)If , noting inequality , , , and for and , , substituting (47)-(49) into (45) gives where Also, means that there exists a constant such that , which leads to It follows that Using the limit and , we can deduce that Otherwise, we consider that(2)if and , we obtain where Noting that for and , , we know that is bounded by a polynomial with negative leading term. Note that We conclude that Thus, assertion (44) follows.
In order to highlight the stable effect of Lévy noise, let we will discuss almost sure exponentially stability of the following SDE with jumps:
Assumption 12. For any , there exists a constant such that
Theorem 13. Suppose the initial data satisfy ; then the solution of SDE (60) is such that a.s. for all . Under Assumptions 1, 2, and 12, if for and , the solution of SDE (60) has the property
In particular, nonlinear delay system (60) is a.s. exponentially stable if .
Proof. By Theorem 3 and supposing a.s., if the initial data satisfying , the solution with a given initial value will exist a.s. for all . Applying the Itô formula to the function leads to
By virtue of exponential martingale inequality [4, Theorem 7.4, page 44].
Choose , and , , and in the above equation. Since , by the Borel-Cantelli lemma, there exists an with such that, for any , there exists an integer that can be found such that Whenever a.s. Similarly, we can obtain Whenever a.s.
In other words, we have shown that where
Next, noting Assumption 12, for any , and with , Then the strong law of large numbers yields Thus, for and , it follows from (69) and (72) that Clearly, if , system (60) is a.s. exponentially stable; the proof is completed.
Remark 14. Compared with results in , authors discussed the SDE: the results show that polynomial Brownian noise can suppress this potential explosion and another linear Brownian noise has an effect to stabilize the suppressed equation.
Theorem 13 shows that under Assumptions 1, 2, and 12, choosing appropriate function , together with , , Brownian noises may suppress the given deterministic equation . Linear jump process has stable effect on system (60); it has the same role as linear Brownian noise.
Example 1. Consider stochastic differential equation where , , , , , , , Figure 2 shows the differential equation Simulation indicates that the solution of (78) is explosive in finite time. See Figure 2.
5. Conclusions and Extensions
In this paper, we consider stochastic nonlinear delay differential system with jumps, where satisfies the one-side polynomial growth condition. Our main results include the following theoretic analysis. Theorems 3 and 5 show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. Using the exponential martingale inequality with jumps, we discuss that the sample Lyapunov exponent is nonpositive in Theorem 11. Lévy noise is sudden and severe environmental perturbations; if we propose linear Lévy processes in (60), by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stable are investigated. Theorem 13 reveals the stabilization property of Lévy noise.
Two interesting questions deserve further consideration and investigation.
If we use the continuous time Markov chain with finite state space to model abrupt changes in their structure and parameters, another complex hybrid system with jumps can be studied: with initial data .
We can also consider stochastic functional equation: