Abstract
In this paper, we consider the random periodic solution to a neutral stochastic functional differential equation driven by Brownian motion. We obtain the existence and uniqueness of the random periodic solution by Banach fixed point theorem. Moreover, we introduce two examples to illustrate our results.
1. Introduction
In the last two decades, the theory of stochastic dynamical systems has attracted much attention due to its wide applications in the fields such as financial market, insurance, biology, medical science, population dynamic, and control (see [1–8]). The idea to regard stochastic differential equations (SDEs) as random dynamical systems can be traced back to late 1970s and early 1980s (see [9–11]). The study of periodic solutions has occupied a central role in the theory of dynamical system since Poincaré’s seminal work [12]. Periodic solutions of stochastic differential equations or stochastic partial differential equations (SPDEs) have also been studied by a number of authors such as Chojnowska-Michalik [13, 14], Fife [15], Vejvoda [16], and Zhang and Zhao [17]. On the other hand, the theory of stochastic functional differential equations (SFDEs) has received a great deal of attention in the last two decades. More recently researchers have given special interest to the study of equations in which the variable delay argument occurs in the derivative of the state variable, or so-called neutral stochastic functional differential equations (NSFDEs). Many well-known theorems in SFDEs are successfully extended to NSFDEs; see, for example, [18–21]. However, contrast to the extensive studies on the periodicity of usual stochastic differential equations, there has been little systematic investigation on stochastic periodic solutions of NSFDEs. The main reason for this is the complexity of dependence structures such equations. In this paper, we consider the random periodic solutions of such stochastic equations.
Random periodic behaviour is more accurate to describe the real world problems, e.g., in biological, environmental, and economic systems. But these problems are often subject to random perturbations or under the influence of noises. In our opinion, it is greatly important to study the pathwise random periodic solutions of random dynamical systems. The definition of the pathwise random periodic solutions was firstly given by Zhao and Zheng [17]. Later the definition of random periodic solutions and their existence for semi-flows generated by nonautonomous SDEs and SPDEs with additive noise were given by Feng and Zhao in [22, 23].
It is well-known that the most powerful method to prove the existence of the periodic solution in the deterministic case is, for example, fixed point theorem, monotone semi-flow, topological degree theorem, and Lyapunov method (see, for example, [24–28]). But the existence of the periodic solutions of the stochastic differential systems is more complex than the deterministic case. Recently, there are some works studying existence random periodic solutions for SDEs, such as Feng et al. [29–31]. In [30], Feng et al. study the existence of random periodic solutions to semilinear stochastic differential equations. In [31], an ergodic theory of random dynamical systems has been built under the stationary regime, in which stationary solutions and stationary measures, which are “equivalent,” are fundamental objects. In [29], random periodic solutions and their approximating by the Euler-Maruyama scheme were studied.
Motivated by the above, in this paper we consider the existence of random periodic solution to NSFDEs of the formFor the equation, we introduce some assumptions which are very natural. Let and be two given constants and let denote the space of continuous functions from to with the usual norm . Moreover, we also claim that (i) and are Borel measurable functions and there is a constant such that for any , i.e., and are -periodic in time;(ii) is Borel measurable with ;(iii) is a symmetric and negative-definite matrix;(iv) is a two-sided Wiener process in defined on a probability space . The filtration is defined as follows: (v) is regarded as a -valued stochastic process;(vi)the initial data is -measurable -valued random variable such that .
By the variation of constant formula, we can get the solution of (1) as follows:which is equivalent to Denote the standard -preserving ergodic Wiener shift by , and for all . We then see that the solution of (1) satisfies the following semi-flow property and periodicity: for all .
For convenience, if there is no confusion, usually initial value or will be omitted. Let be the solution of (1) starting from time .
The paper is organized as follows. In Section 2, we briefly present some basic notations and preliminaries. In Section 3, we give some basic lemmas and existence of random periodic solution of (1). In Section 4, some examples are given to illustrate our result.
2. Preliminaries
In this section, we will give the definition of the random periodic solution and present briefly some basic notations.
Definition 1 (see [22, 23]). A random periodic path of period of the semi-flow is an -measurable map such that for any and almost every , where , is a metric dynamical system.
We introduce the next notations.(i)Let denote eigenvalues of a negative-definite -symmetric matrix such that (ii) For , we denote the -norm of a random process by (iii) For a matrix , we denote by the Hilbert-Schmidt norm (Frobenius norm)
We introduce some assumptions about (1).
Condition 1. Assume that there exist two constants such thatMoreover, assume also that there are the other two constants such thatfor any and .
Condition 2. There exists a constant , such that
Lemma 2 (see [32]). If (14) holds, we then have for all and .
Lemma 3 (see [32]). If (14) holds, we have for all .
Lemma 4. Denote by two solutions of system (1) with different initial values , respectively. If (14) holds, we have for all .
Proof. For any , we have that Let yield that Therefore and the required assertion follows immediately.
Lemma 5 (see [33]). Let be an eigenvalue of . If then there exist and such that
3. Existence and Uniqueness of Random Periodic Solution
In this section, we consider the random periodicity of (1).
Lemma 6. Let the matrix be as stated above and let Conditions 1 and 2 hold. Then there exists a constant such that, for any , the solution of (1) satisfies
Proof. Using Itô’s formula to , we have thatfor any . We need to estimate the six terms in the right hand side of (25).
For the first term, by Lemma 2 we have For the third term, we have By Lemma 2 and (15), we derive that Notice that since the matrix is non-positive definite. We get that Similarly, by Lemma 2 and (13), we can show that It follows from (25) that where .
On the other hand, by the Burkholder-Davis-Gundy inequalities and (13), we derive the following:for all . Substituting this into (32) yields that where . By Lemma 3 we have that where . It follows that for all . Thus, the Gronwall inequality implies that by Condition 2 and the lemma follows.
Theorem 7. Assume Condition 1 holds, andand then problem (1) has a unique random periodic solution.
Proof. From Lemma 6, we have known that is bounded in . Define the operator on as follows: We will prove that is a contraction. Assume that and are two solutions of (1), and , for all . It follows that for each , which means that is a contraction if (38) hold. By the contraction principle, there exists a unique fixed point for in , such that , that is, (1) has a unique solution.
We now show the random periodicity of . Notice that by the continuity of and , where . On the other hand, we have for any . It follows from pathwise uniqueness of the solution of (1) thatfor any and almost every . That is to say, the unique solution is random periodic solution.
The following result shows that the solution of (1) is continuously dependent on initial value.
Theorem 8. Denote two solutions of system (1) with different initial values , respectively, starting with the time . Assume that Conditions 1 and 2 hold for both initial values; we then havefor any , where and .
Proof. Let and denote the solutions of system (1) starting from with initial values and , respectively. According to (4), one has for any . Denote . According to the method used in Lemma 6, we get the following result similar to (32): by Lemma 4, where . On the other hand, by the Burkholder-Davis-Gundy inequalities and the conditions in Section 2, similar to the proof of (33), we see that for all . Combining this with (47), we get for all . It follows that where . Applying Lemma 5, we see that where . Consequently for all . Finally, by the Gronwall inequality, we obtain that and the required assertion (45) follows by taking and .
4. Examples
By using the Euler-Maruyama method for NSFDEs established in [34], we can simulate the solution of (1).
Example 1. Consider the NSDE (1) with
We simulate two samples as shown in Figures 1 and 2. We can still see some periodic property of the solutions in these two figures though they are not really periodic.
Example 2. The noise in the above example is relevantly small. We consider relatively larger noise with and as follows:
We simulate two different samples as demonstrated in Figures 3 and 4. We can also see periodicity in Figures 3 and 4.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
L. Gao and L. Yan carried out the mathematical studies, participated in the sequence alignment, drafted the manuscript, and participated in the design of the study and performed proof of results. All authors read and approved the final manuscript.
Acknowledgments
The project was sponsored by National Natural Science Foundation of China (11571071) and the Fundamental Research Funds for the Central Universities (No. 18D310405).