Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay

Based on Lyapunov stability theory, Itô formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylor’s theorem, the stochastic nonlinear system with delay is reduced to an n-dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.


Introduction
Let us make the following assumptions.
(H1.1) Demand for product is quadratic function with respect to price. (H1. 2) The price is not very sensitive to the change of inventory.That is, damping of nonlinear dynamical system  is a -order infinitesimal ( > 0 is small enough).
(H1.3) Stochastic noise is related to price.That is, it can be treated as Gaussian white noise, and the excitation coefficient is √-order infinitesimal.
The price system can be described by linear equations because of their convenience in mathematical treatment.Therefore linear equations play an important role in theory and their applications.However, they can not perfectly describe the process of the price fluctuation in nonlinear version.Then the nonlinear equations should be employed, for their virtues that can deeply reflect the rules of price fluctuation.
For many real-world systems, there always exist random disturbances such as the measurement error and the control input of the system [2][3][4][5].The basic source of random disturbance is Gaussian white noise, which represents the joint effects of a large number of independent random forces acting on the systems, and the influence of individual is not significant.By (H1.3), the stochastic nonlinear dynamical price system can be described by stochastic differential equation (SDE for short) as follows [5]: where {(),  ≥ 0} is 1-dimensional Brownian motion.The above system also can be rewritten as the following matrix form: where  : , and ℎ(, ()) = (√, 0)  .Supply is not only influenced by price and demand but also influenced by production management, information feedback, transportation, and so forth.Therefore, () not only depends on the situation at  but also on the certain period  −  ( > 0 is a given time delay) in the past [6][7][8][9].Furthermore, the parameter perturbation of the system's internal structure should also be taken into account in this paper.
Stability is a very important dynamical feature for the stochastic price system with delay, and it is one of the main purposes of system designing [5,6].Keeping the price system steady within the cycle as long as possible to avoid inflation or deflation has the vital significance for the healthy development of the economy of the country.There is a rich literature on time delay system and stochastic system.Stability of stochastic system has been studied.See, for example, Liu and Feng [2], Liu and Deng [3], Yong and Zhou [4], Li and Xu [5], and Mao [10].Stability of time delay system has been studied.See, for example, Kazmerchuk et al. [6], Lv and Liu [7], Lv and Zhou [8], Zhu et al. [11], Zhu and Yi [12], and Trinh and Aldeen [13].Mao [14], Mao and Shah [15], Zhu and Hu [16], Zhu and Hu [17], and S. Xie and L. Xie [18] established some stability criteria of the stochastic system with delay by using an LMI approach.The Hopf bifurcation of price Rayleigh delayed equation on deterministic case has been studied extensively in recent years.See, for example, [6][7][8].The stability and the optimal control of stochastic nonlinear dynamical price model has been studied in [5].Unfortunately, there is a little previous literature on stochastic nonlinear dynamical price system with delay.Thus, we aim to fill this gap in this paper.We plug the time delay, the parameter perturbation, and the stochastic item into nonlinear dynamical price system (2).Such models may be identified as stochastic differential delayed equations (SDDEs for short).Our target in this paper is to derive some sufficient conditions of exponential stability for SDDEs.
Li and Xu only analyzed the stability for SDEs (4) in virtue of the marginal probability density about  in [5] but did not give the sufficient condition for stability.In this paper, using Taylor's theorem, the n-dimensional nonlinear SDDE ( 5) is reduced to an n-dimensional semilinear SDDE correspondingly.Some sufficient conditions of exponential stability and corollaries for such price system are established by using Lyapunov function.The time delay upper limit is solved by using our theoretical results when the system is exponentially stable.Thus, [5] is promoted and improved.Our theoretical results show that if the classical price Rayleigh equation ( 2) is exponentially stable, so is its perturbed system (5) with delay provided that both the time delay and the intensity of perturbations are small enough.Those results will help our government make a macrocontrol for price system and timely adjust their pricing strategies.
The rest of this paper is organized as follows.In Section 2, we introduce the definition of the exponential stability of SDDEs.Section 3 is devoted to the sufficient conditions for exponential stability and almost surely exponential stability of price system.Section 4 presents two simple examples to illustrate our results.Finally, Section 5 concludes the paper.
Then, the Itô integral of ℎ(, ()) (from  to ) is defined by where ℎ is a stochastic process with value in  × and and lim MS be a limit in mean square sense.
It is proved directly from the definition of Itô integral that where (⋅) is 1-dimensional Brownian motion.The extra term −( − )/2 shows that the Itô stochastic integral does not behave like ordinary integrals.It leads to Itô-type stochastic system being different from non-Itô-type.See [19] for the details.Now, let us present an existence and uniqueness result for system (5).First, we make the following assumptions for the coefficients of (5).
For stochastic system, exponential stability in mean square and almost surely exponential stability are generally used [2].Definition 2. The trivial solution of system ( 5) is said to be pth moment exponentially stable, if there exists a positive constant  such that lim sup for any  ∈   F 0 (−, 0;   ), where − is called pth moment Lyapunov exponent of the trivial solution.
In particular,  = 2; it is called mean square exponentially stable.
Definition 3. The trivial solution of system (5)
See Mao [10] for the proof of Theorem 4. In the study of mean square exponential stability, it is often to use a quadratic function as the Lyapunov function; that is, (, ) =   ()(), where  is a symmetric positive definite  ×  matrix (see [11,20]).Theorem 5. Let (H3.1) holds, and then the trivial solution of system (13) is exponentially stable in the mean square.Assume that there exists a pair of symmetric positive definite  ×  matrices  and  such that min () where  min () > 0 is the smallest eigenvalue of .
Remark 6.In the proof we gave, the estimate for the second moment Lyapunov exponent should not be greater than −.
By (39), we easily get Substituting the above two into (43) yields where  4 = (3 Then, (52) is exponentially stable in the mean square.
steady and rapid economic development.It is also an important guiding significance that our government can timely adjust their pricing strategies.Two examples are presented to illustrate our theoretical results, which are the same as [5].
Another challenging problem is to study a type of stochastic nonlinear dynamical price system with variable delay.We hope to study these problems in forthcoming papers.
is said to be almost surely exponentially stable, if there exists a positive constant  such that .