Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay

Abstract

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.

1. 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 𝜆 is a 𝜀-order infinitesimal (𝜀>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 √𝜀-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.

Suppose 𝑆(𝑡),𝐷(𝑡),𝑃(𝑡), and  𝑄(𝑡) as supply, demand, price, and inventory at time 𝑡, respectively. 𝑆₀ is initial supply, 𝑎,𝑏,𝑐,𝜆,𝛼 and 𝛾 are constants, and alphabet with a bar is equilibrium value. Reference [1] gives a deterministic nonlinear price model as follows: 𝑑²𝑃𝑑𝑡²=𝜆𝑎𝑃²+𝑏𝑃+𝑐𝑑𝑃𝑆𝑑𝑡−𝜆𝛼𝑃−𝜆₀−𝐷. (1)

If there exists an equilibrium point in the above price system, denoted by 𝑃, then 𝑃=(𝐷−𝑆₀)/𝛼 and 𝐷=𝑆₀+𝛼𝑃. Let 𝑥(𝑡)=𝑃(𝑡)−𝑃 and 𝑡=𝜂/√𝛼𝜆,𝜇=𝜆/𝛼. Applying Lienard transformation, we obtain the classical price Rayleigh equation as follows: 𝑑𝑥𝜇1𝑑𝜂=−𝑦+𝜀3𝑎𝑥³+12𝑏₀𝑥²+𝑐₀𝑥,𝑑𝑦𝑑𝜂=𝑥, (2) where 𝑏₀=2𝑎𝑃+𝑏, 𝑐₀=𝑎𝑃²+𝑏𝑃+𝑐, 𝜇≜𝜀𝜇.

For many real-world systems, there always exist random disturbances such as the measurement error and the control input of the system [2–5]. 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]: 𝑑𝑥=√𝜆𝛼−𝑦+𝜀𝜇13𝑎𝑥³+12𝑏₀𝑥²+𝑐₀𝑥𝑑𝑡+𝜀^1/2(𝑡),√𝛾𝑑𝑊𝑑𝑦=𝜆𝛼𝑥𝑑𝑡, (3) where {𝑊(𝑡),𝑡≥0} is 1-dimensional Brownian motion. The above system also can be rewritten as the following matrix form: 𝑑𝑧(𝑡)=𝑓(𝑡,𝑧(𝑡))𝑑𝑡+ℎ(𝑡,𝑧(𝑡))𝑑𝑊(𝑡), (4) where 𝑓∶𝑅⁺×𝑅²→𝑅², ℎ∶𝑅⁺×𝑅²→𝑅², 𝑓(𝑡,𝑧(𝑡))=√𝜆𝛼(−𝑦+𝜀𝜇(𝑎𝑥³/3+𝑏₀𝑥²/2+𝑐₀𝑥),𝑥)^𝑇,  𝑧(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇∈𝑅², and ℎ(𝑡,𝑧(𝑡))=(√𝜀𝛾,0)^𝑇.

Supply is not only influenced by price and demand but also influenced by production management, information feedback, transportation, and so forth. Therefore, 𝑆(𝑡) not only depends on the situation at 𝑡 but also on the certain period 𝑡−𝜏 (𝜏>0 is a given time delay) in the past [6–9]. Furthermore, the parameter perturbation of the system’s internal structure should also be taken into account in this paper.

Based on the abovementioned, the price system (4) can be extended to more general 𝑛-dimensional stochastic nonlinear price systems with delay as follows: 𝑑𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))+𝑓((𝑡)(((𝑡)((𝑡)𝑥(𝑡)(𝑡)𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑡+ℎ𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑊,𝑡≥0,=𝜉,−𝜏≤𝑡≤0, (5) where 𝜏 is a given time delay. The maps 𝑓,𝑓∈𝐶(𝑅⁺×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛), ℎ∈𝐶(𝑅⁺×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛×𝑚). 𝑓 represents the uncertainty. {𝑊(𝑡),𝑡≥0} is an 𝑚-dimensional Brownian motion, and the term ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑊(𝑡) represents the stochastic disturbance. Furthermore, we always assume that 𝑓(𝑡,0,0)=𝑓(𝑡,0,0)=ℎ(𝑡,0,0)≡0 for the stability purpose of 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, [6–8]. 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 𝑥 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.

2. Preliminaries

Throughout this paper and unless specified, we let 𝑊(𝑡)=(𝑊₁(𝑡),…,𝑊_𝑚(𝑡))^𝑇 be an 𝑚-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with a natural filtration {ℱ_𝑡}_𝑡≥0 (i.e., ℱ_𝑡=𝜎{𝑊(𝑠)∶0≤𝑠≤𝑡} and augmented by all the 𝑃-null sets in ℱ). Denote by |⋅| the Euclidean norm. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. If 𝐴 is a matrix, denote by ‖𝐴‖ the operator norm of 𝐴, that is, ‖𝐴‖=sup{|𝐴𝑥|∶|𝑥|=1}. 𝜉(⋅)∈𝐶[−𝜏,0] is the initial path of 𝑥, where 𝜏>0 is a given finite time delay and 𝐶[−𝜏,0] is the set of continuous functions from [−𝜏,0] into 𝑅^𝑛. Moreover, denote by 𝐿^𝑝_ℱ0(−𝜏,0;𝑅^𝑛) the family of 𝑅^𝑛-valued adapted stochastic processes 𝜉(𝑠),−𝜏≤𝑠≤0 such that 𝜉(𝑠) is ℱ₀-measurable and ∫⁰_−𝜏𝐸|𝜉(𝑠)|^𝑝𝑑𝑠<+∞(𝑝>1).

We define a norm mean square in 𝑆^𝑛 as follows: |𝑥|_MS=𝐸|𝑥|^21/2,forany𝑥∈𝑆^𝑛, (6) where 𝑆^𝑛 is the set of random variable in probability space (Ω,ℱ,𝑃). Similarly, we define ‖𝐴‖_MS=sup{|𝐴𝑥|_MS∶|𝑥|_MS=1}.

Then, the Itô integral of ℎ(𝑡,𝑥(𝑡)) (from 𝑎 to 𝑏) is defined by (𝐼)^𝑏_𝑎ℎ(𝑡,𝑥(𝑡))𝑑𝑊(𝑡)=lim_MS^𝑁_{𝜆→0𝑖=0}ℎ𝑡_𝑖,𝑥𝑡_𝑖𝑊𝑡_𝑖+1−𝑊𝑡_𝑖limitin𝐿²(𝑃), (7) where ℎ is a stochastic process with value in 𝑆^𝑛×𝑚 and ∫^𝑏_𝑎‖ℎ(𝑡,𝑥(𝑡))‖²_MS𝑑𝑡<∞(0≤𝑎<𝑏). Let 𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑏, 𝜆=max{Δ𝑡_𝑖,𝑖=0,1,…,𝑁−1}, Δ𝑡_𝑖=𝑡_𝑖+1−𝑡_𝑖, and lim_MS be a limit in mean square sense.

It is proved directly from the definition of Itô integral that ^𝑏_𝑎𝑊(𝑡)(𝑡)=1𝑑𝑊2𝑊²(𝑏)−𝑊²(𝑎−1)2(𝑏−𝑎), (8) where 𝑊(⋅) is 1-dimensional Brownian motion. The extra term −(𝑏−𝑎)/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).(H2.1) The maps 𝑓,𝑓, and ℎ are locally Lipschitz continuous.(H2.2) The maps 𝑓,𝑓, and ℎ satisfy the linear growth condition.

Theorem 1 (see [4]). Let (H2.1) and (H2.2) hold. Then, for any 𝜉(𝑡)∈𝐿²_ℱ0(−𝜏,0;𝑅^𝑛), (5) has a unique strong solution which is denoted by 𝑥(𝑡;𝜉), and 𝑥(𝑡;𝜉) is square integrable. So, (5) has a trivial solution 𝑥(𝑡;0)=0.

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 𝜀 such that limsup_𝑡→∞1𝑡ln𝐸𝑥(𝑡;𝜉)^𝑝≤−𝜀 (9) for any 𝜉∈𝐿^𝑝_ℱ0(−𝜏,0;𝑅^𝑛), where −𝜀 is called 𝑝th moment Lyapunov exponent of the trivial solution.

In particular, 𝑝=2; it is called mean square exponentially stable.

Definition 3. The trivial solution of system (5) is said to be almost surely exponentially stable, if there exists a positive constant 𝜆 such that limsup_𝑡→∞1𝑡ln𝑥(𝑡;𝜉)≤−𝜆a.s. (10) for any 𝜉∈𝐿^𝑝_ℱ0(−𝜏,0;𝑅^𝑛), where −𝜆 is called almost surely Lyapunov exponent of the trivial solution.

3. Exponential Stability for Stochastic Price System with Delay

To obtain sufficient condition of the exponential stability of system (5), we assume that the functions 𝑓(𝑡,𝑥,𝑦),𝑓(𝑡,𝑥,𝑦) are 1-order continuously differentiable in the neighbourhood of (𝑡,0,0) with respect to (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛. According to Taylor expansion, for 0<𝜃<1𝑓𝑡,𝑥,𝑦=𝑓_𝑥𝑡,𝜃𝑥,𝜃𝑦𝑥+𝑓_𝑦𝑡,𝜃𝑥,𝜃𝑦𝑦,𝑓𝑡,𝑥,𝑦=𝑓_𝑥𝑡,𝜃𝑥,𝜃𝑦𝑥+𝑓_𝑦𝑡,𝜃𝑥,𝜃𝑦𝑦. (11) Thus, 𝑓𝑡,𝑥,𝑦+𝑓=𝑓𝑡,𝑥,𝑦_𝑥𝑡,𝜃𝑥,𝜃𝑦+𝑓_𝑥𝑥+𝑓𝑡,𝜃𝑥,𝜃𝑦_𝑦𝑡,𝜃𝑥,𝜃𝑦+𝑓_𝑦𝑡,𝜃𝑥,𝜃𝑦𝑦. (12)

Therefore, system (5) can be reduced to an 𝑛-dimensional semilinear stochastic differential delayed equation as follows: 𝑑𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))+𝑓((𝑡)(((𝑡)((𝑡)𝑥(𝑡)(𝑡)𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑡+ℎ𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑊,𝑡≥0,=𝜉,−𝜏≤𝑡≤0, (13) where 𝐴₁,𝐵₁ are 𝑛×𝑛 matrices. 𝐴₁(𝑡), 𝐴₂(𝑡−𝜏),𝐵₁(𝑡), and 𝐵₂(𝑡−𝜏) represent the uncertainties. They are bounded 𝑛×𝑛 matrix-valued functions. Here, 𝜏,  𝑊(𝑡),  ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏)), and 𝜉(𝑡) are the same as in the previous section.

We make the following assumption for the coefficients of system (13).(H3.1) There exist nonnegative constants 𝛼_𝑖,𝛽_𝑖(𝑖=1,2,3), for any 𝑡≥0 such that𝐴₁(𝑡)≤𝛼₁,𝐴₂(𝑡−𝜏)≤𝛼₂,𝐵₁(𝑡)≤𝛽₁,𝐵₂(𝑡−𝜏)≤𝛽₂ (14)

and for any (𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))∈𝑅⁺×𝑅^𝑛×𝑅^𝑛 such that trℎ^𝑇(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))≤𝛼₃|𝑥(𝑡)|²+𝛽₃|𝑥(𝑡−𝜏)|². (15)

Theorem 4. Let ℎ satisfy (H2.1)-(H2.2), and condition (14) hold. Then, for any 𝜉(𝑡)∈𝐿²_ℱ0(−𝜏,0;𝑅^𝑛), system (13) has a unique strong solution which is denoted by 𝑥(𝑡;𝜉), and 𝑥(𝑡;𝜉) is square integrable. So, (13) has a trivial solution 𝑥(𝑡;0)=0.

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, 𝑉(𝑡,𝑥)=𝑥^𝑇(𝑡)𝐺𝑥(𝑡), where 𝐺 is a symmetric positive definite 𝑛×𝑛 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 𝑛×𝑛 matrices 𝐺 and 𝑄 such that 𝐺(𝐴+𝐵)+(𝐴+𝐵)^𝑇𝜆𝐺=−𝑄,_min(𝑄)>‖𝐺‖2𝛼₁+2𝛼₂+2𝛽₁+2𝛽₂+𝛼₃+𝛽₃‖‖⋅+2𝐺𝐵2𝜏6𝜏‖𝐴‖²+‖𝐵‖²+𝛼²₁+𝛽²₁+𝛼²₂+𝛽²₂+𝛼₃+𝛽₃, (16)(17) where 𝜆_min(𝑄)>0 is the smallest eigenvalue of 𝑄.

Proof. Fix the initial data 𝜉(𝑡) arbitrarily, and write 𝑥(𝑡;𝜉)=𝑥(𝑡) simply. Applying Itô’s formula to 𝑥^𝑇(𝑡)𝐺𝑥(𝑡), we have 𝑑𝑥^𝑇(𝑡)𝐺𝑥(𝑡)=2𝑥^𝑇(𝑡)𝐺[𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)]𝑑𝑡+2𝑥^𝑇(𝑡)𝐺𝐴₁(𝑡)𝑥(𝑡)+𝐵₁(𝑡)𝑥(𝑡−𝜏)𝑑𝑡+2𝑥^𝑇(𝑡)𝐺𝐴₂(𝑡−𝜏)𝑥(𝑡)+𝐵₂(𝑡−𝜏)𝑥(𝑡−𝜏)𝑑𝑡+2𝑥^𝑇((𝑡)((𝑡)ℎ𝐺ℎ𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑊+tr^𝑇(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝐺ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑡. (18) By (16), we can estimate the first item of (18) as follows: 2𝑥^𝑇(𝑡)𝐺[𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)]=2𝑥^𝑇(𝑡)𝐺[𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)]+2𝑥^𝑇(𝑡)𝐺𝐵𝑥(𝑡)−2𝑥^𝑇(𝑡)𝐺𝐵𝑥(𝑡)≤−𝜆_min(𝑄)|𝑥(𝑡)|²+𝛽|𝑥(𝑡)|²+1𝛽‖𝐺𝐵‖²⋅|𝑥(𝑡)−𝑥(𝑡−𝜏)|², (19) where ‖‖⋅𝛽=𝐺𝐵2𝜏6𝜏‖𝐴‖²+‖𝐵‖²+𝛼²₁+𝛽²₁+𝛼²₂+𝛽²₂+𝛼₃+𝛽₃. (20)
By (14), we can estimate the second item and the third item of (18), respectively, 2𝑥^𝑇(𝑡)𝐺𝐴₁(𝑡)𝑥(𝑡)+𝐵₁(𝑡)𝑥(≤‖𝐺𝑡−𝜏)‖2𝛼₁+𝛽₁|𝑥(𝑡)|²+𝛽₁|𝑥(𝑡−𝜏)|²,2𝑥^𝑇(𝑡)𝐺𝐴₂(𝑡−𝜏)𝑥(𝑡)+𝐵₂(𝑡)𝑥(≤‖𝐺𝑡−𝜏)‖2𝛼₂+𝛽₂|𝑥(𝑡)|²+𝛽₂|𝑥(𝑡−𝜏)|². (21) By (15), the last item of (18) yields trℎ^𝑇((𝑡)(((𝑡)(≤‖𝐺𝛼𝑡,𝑥,𝑥𝑡−𝜏))𝐺ℎ𝑡,𝑥,𝑥𝑡−𝜏))‖₃|𝑥(𝑡)|²+𝛽₃|𝑥(𝑡−𝜏)|². (22) Substituting (21) and (22) into (18), we get 𝑑𝑥^𝑇(𝑡)(𝑡𝜆𝐺𝑥)≤−_min(𝑄)−‖𝐺‖2𝛼₁+𝛽₁+2𝛼₂+𝛽₂+𝛼₃⋅|𝑥(𝑡−𝛽)|²𝑑𝑡+‖𝐺‖𝛽₁+𝛽₂+𝛽₃|𝑥(𝑡−𝜏)|²+1𝑑𝑡𝛽‖𝐺𝐵‖²⋅|𝑥(𝑡)−𝑥(𝑡−𝜏)|²𝑑𝑡+2𝑥^𝑇(𝑡)𝐺ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑊(𝑡). (23) If (17) holds, then we can choose 𝜀>0 small enough such that 𝜆_min(𝑄)=‖𝐺‖2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝛼₃+‖𝐺𝛽+𝜀‖₁+𝛽₂+𝛽₃𝑒^𝜀𝜏+1+𝛽𝛽‖𝐺𝐵‖²2𝜏6𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃𝑒^𝜀𝜏+2𝜏6𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃𝑒^2𝜀𝜏. (24) Applying Itô’s formula to 𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡), we have 𝑑𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡)=𝜀𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡)𝑑𝑡+𝑒^𝜀𝑡𝑑𝑥^𝑇(𝑡)𝐺𝑥(𝑡),forany𝑡≥0. (25) Substituting (23) into (25) yields 𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡)≤𝜉^𝑇(0)𝐺𝜉(0)+𝜀^𝑡₀𝑒^𝜀𝑠𝑥^𝑇(𝑠)(𝑠)−𝐺𝑥𝑑𝑠_𝑡0𝑒^𝜀𝑠𝜆_min(𝑄)−‖𝐺‖2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝛼₃⋅|𝑥(𝑠−𝛽)|²𝑑𝑠+^𝑡₀𝑒^𝜀𝑠⋅‖𝐺‖𝛽₁+𝛽₂+𝛽₃|𝑥(𝑠−𝜏)|²+𝑑𝑠_𝑡0𝑒^𝜀𝑠⋅1𝛽‖𝐺𝐵‖²⋅|𝑥(𝑠)−𝑥(𝑠−𝜏)|²+𝑑𝑠_𝑡0𝑒^𝜀𝑠⋅2𝑥^𝑇(𝑠)𝐺ℎ(𝑠,𝑥(𝑠),𝑥(𝑠−𝜏))𝑑𝑊(𝑠). (26) Taking the expectation in (26), we have 𝐸𝑒^𝜀𝑡𝑥^𝑇(𝑡)(𝑡𝜉𝐺𝑥)≤𝐸^𝑇(0)(0)−𝜆𝐺𝜉_min(𝑄)−‖𝐺‖2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝛼₃⋅+𝜀−𝛽_𝑡0𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²+‖𝐺‖𝛽𝑑𝑠₁+𝛽₂+𝛽₃^𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠−𝜏)|²+1𝑑𝑠𝛽‖𝐺𝐵‖^2𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²𝑑𝑠. (27) In order to give an estimate of 𝐸(𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡)), we now estimate the last two terms on the right-hand side of (27). First of all, for any 𝑡≥𝜏, we have ^𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠−𝜏)|²=𝑑𝑠_𝜏0𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠+^𝑡_𝜏𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠≤𝑒^𝜀𝜏⋅^𝜏₀𝐸𝜉(𝑠−𝜏)²𝑑𝑠+𝑒^𝜀𝜏⋅^𝑡_𝜏𝑒^{𝜀(𝑠−𝜏)}⋅𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠=𝑒^𝜀𝜏⋅⁰_−𝜏𝐸𝜉(𝑢)²𝑑𝑢+𝑒^𝜀𝜏⋅₀^𝑡−𝜏𝑒^𝜀𝑢⋅𝐸|𝑥(𝑢)|²𝑑𝑢(𝑢=𝑠−𝜏)=𝑐₁𝑒^𝜀𝜏+𝑒^𝜀𝜏⋅₀^𝑡−𝜏𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²𝑑𝑠, (28) where 𝑐₁=∫⁰_−𝜏𝐸|𝑥(𝑠)|²𝑑𝑠.
Next, recalling (14) and (15), for 𝑠≥𝜏, we derive 𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²≤2𝜏⋅𝐸^𝑠_𝑠−𝜏6‖𝐴‖²+𝛼²₁+𝛼²₂|𝑥(𝑡)|²+‖𝐵‖²+𝛽²₁+𝛽²₂|𝑥(𝑡−𝜏)|²𝑑𝑡+2𝐸^𝑠_𝑠−𝜏𝛼₃|𝑥(𝑡)|²+𝛽₃𝑥(𝑡−𝜏)²‖𝐴‖𝑑𝑡≤26𝜏²+𝛼²₁+𝛼²₂+𝛼₃^𝑠_𝑠−𝜏𝐸|𝑥(𝑡)|²‖𝐵‖𝑑𝑡+26𝜏²+𝛽²₁+𝛽²₂+𝛽₃^𝑠_𝑠−𝜏𝐸|𝑥(𝑡−𝜏)|²𝑑𝑡. (29) Similar to (28), for 𝑡≥𝜏, we have ^𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²=𝑑𝑠_𝜏0𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²+𝑑𝑠_𝑡𝜏𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²𝑑𝑠≤𝑐₂+26𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃⋅^𝑡_𝜏𝑒^𝜀𝑠𝑠_𝑠−𝜏𝐸|𝑥(𝜈)|²‖𝐵‖𝑑𝜈𝑑𝑠+26𝜏²+𝛽²₁+𝛽²₂+𝛽₃⋅^𝑡_𝜏𝑒^𝜀𝑠𝑠_𝑠−𝜏𝐸|𝑥(𝜈−𝜏)|²𝑑𝜈𝑑𝑠, (30) where 𝑐₂=∫^𝜏₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²𝑑𝑠. Moreover, ^𝑡_𝜏𝑒^𝜀𝑠𝑠_𝑠−𝜏𝐸|𝑥(𝜈)|²𝑑𝜈𝑑𝑠=^𝑡₀𝐸|𝑥(𝜈)|^{2(𝜈+𝜏)∧𝑡}_𝜈∨𝜏𝑒^𝜀𝑠𝑑𝑠𝑑𝜈≤𝜏𝑒^𝜀𝜏𝑡₀𝑒^𝜀𝜈⋅𝐸|𝑥(𝜈)|²𝑑𝜈. (31) Similarly, ^𝑡_𝜏𝑒^𝜀𝑠𝑠_𝑠−𝜏𝐸|𝑥(𝜈−𝜏)|²𝑑𝜈𝑑𝑠=^𝑡₀𝐸|𝑥(𝜈−𝜏)|^{2(𝜈+𝜏)∧𝑡}_𝜈∨𝜏𝑒^𝜀𝑠𝑑𝑠𝑑𝜈≤𝜏𝑒^𝜀𝜏𝑡₀𝑒^𝜀𝜈⋅𝐸|𝑥(𝜈−𝜏)|²𝑑𝜈. (32) Substituting (28) into (32) yields ^𝑡_𝜏𝑒^𝜀𝑠𝑠_𝑠−𝜏𝐸|𝑥(𝜈−𝜏)|²𝑑𝜈𝑑𝑠≤𝜏𝑒^𝜀𝜏𝑐₁𝑒^𝜀𝜏+𝑒^𝜀𝜏⋅₀^𝑡−𝜏𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²𝑑𝑠<𝜏𝑐₁𝑒^2𝜀𝜏+𝜏𝑒^{2𝜀𝜏𝑡}₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²𝑑𝑠. (33) Substituting (31) and (33) into (30), for 𝑡≥𝜏, we get ^𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²𝑑𝑠≤𝑐₂+26𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃⋅𝜏𝑒^𝜀𝜏𝑡₀𝑒^𝜀𝜈⋅𝐸|𝑥(𝜈)|²‖𝐵‖𝑑𝜈+26𝜏²+𝛽²₁+𝛽²₂+𝛽₃⋅𝜏𝑐₁𝑒^2𝜀𝜏+𝜏𝑒^{2𝜀𝜏𝑡}₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²𝑑𝑠. (34) Substituting (28) and (34) into (27) and recalling (24), we obtain that for 𝑡≥𝜏𝐸𝑒^𝜀𝑡𝑥^𝑇(𝑡)(𝑡𝜉𝐺𝑥)≤𝐸^𝑇(0)(0)−𝜆𝐺𝜉_min(𝑄)−‖𝐺‖2𝛼₁+2𝛼₂+𝛽₁+𝛽₂+𝛼₃⋅+𝜀−𝛽_𝑡0𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²+‖𝐺‖𝛽𝑑𝑠₁+𝛽₂+𝛽₃𝑐₁𝑒^𝜀𝜏+𝑒^𝜀𝜏⋅₀^𝑡−𝜏𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²+1𝑑𝑠𝛽‖𝐺𝐵‖²𝑐₂+26𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃⋅𝜏𝑒^𝜀𝜏𝑡₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²‖𝐵‖𝑑𝑠+26𝜏²+𝛽²₁+𝛽²₂+𝛽₃⋅𝜏𝑐₁𝑒^2𝜀𝜏+𝜏𝑒^{2𝜀𝜏𝑡}₀𝑒^𝜀𝑠⋅𝐸|𝑥(𝑠)|²𝑑𝑠=𝑐₃, (35) where 𝑐₃=𝐸𝜉^𝑇(0)𝐺𝜉(0)+‖𝐺‖𝛽₁+𝛽₂+𝛽₃⋅𝑐₁𝑒^𝜀𝜏+1𝛽‖𝐺𝐵‖²𝑐₂+26𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃⋅𝜏𝑐₁𝑒^2𝜀𝜏. (36) Since 𝐺 is positive definite, 𝑥^𝑇(𝑡)𝐺𝑥(𝑡)≥𝜆_min(𝐺)|𝑥(𝑡)|², (37) where 𝜆_min(𝐺)>0 is the smallest eigenvalue of 𝐺.
Thus, 𝐸𝑒^𝜀𝑡𝑥^𝑇(𝑡)𝐺𝑥(𝑡)≥𝐸𝑒^𝜀𝑡𝜆_min(𝐺)|𝑥(𝑡)|². (38) It then follows from (35) that 𝐸|𝑥(𝑡)|²≤𝑐₃𝜆_min(𝐺)⋅𝑒^−𝜀𝑡forany𝑡≥𝜏. (39) Hence, 1𝑡ln𝐸|𝑥(𝑡)|²≤1𝑡𝑐ln³𝜆_min(𝐺)⋅𝑒^−𝜀𝑡1=−𝜀+𝑡𝑐ln³𝜆_min(𝐺). (40) This easily yields limsup_𝑡→∞1𝑡ln𝐸|𝑥(𝑡)|²≤−𝜀. (41) Then, (13) is exponentially stable in the mean square. The proof is complete.

Remark 6. In the proof we gave, the estimate for the second moment Lyapunov exponent should not be greater than −𝜀.

Theorem 7. The trivial solution of system (13) is also almost surely exponentially stable under the same assumption as Theorem 5.

Proof. For 𝑡∈[𝑘𝜏,(𝑘+1)𝜏],𝑘=2,3,…, we have |𝑥(𝑡)|²≤3|𝑥(𝑘𝜏)|²+3^(𝑘+1)𝜏_𝑘𝜏𝐴+𝐴₁(𝑡)+𝐴₂()𝑥(𝑡)+𝑡−𝜏𝐵+𝐵₁(𝑡)+𝐵₂(𝑡−𝜏)𝑥(𝑡−𝜏)𝑑𝑡²+3^(𝑘+1)𝜏_𝑘𝜏ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑊(𝑡)². (42) Recalling (14) and (15), we derive 𝐸sup_{()𝜏𝑘𝜏≤𝑡≤𝑘+1}|𝑥(𝑡)|²≤3𝐸|𝑥(𝑘𝜏)|²+18𝜏^(𝑘+1)𝜏_𝑘𝜏‖𝐴‖²+𝛼²₁+𝛼²₂𝐸|𝑥(𝑡)|²+‖𝐵‖²+𝛽²₁+𝛽²₂𝐸|𝑥(𝑡−𝜏)|²𝑑𝑡+3^(𝑘+1)𝜏_𝑘𝜏𝛼₃𝐸|𝑥(𝑡)|²+𝛽₃𝐸|𝑥(𝑡−𝜏)|²𝑑𝑡. (43) By (39), we easily get 𝐸|𝑥(𝑘𝜏)|²≤𝑐₃𝜆_min(𝐺)⋅𝑒^{−𝜀𝑘𝜏}𝐸|𝑥(for𝑘𝜏≥𝜏,𝑡−𝜏)|²≤𝑐₃𝜆_min(𝐺)⋅𝑒^{−𝜀(𝑡−𝜏)}for𝑡−𝜏≥𝜏. (44) Substituting the above two into (43) yields 𝐸sup_{()𝜏𝑘𝜏≤𝑡≤𝑘+1}|𝑥(𝑡)|²≤3𝑐₃𝜆_min(𝐺)𝑒^{−𝜀𝑘𝜏}+6𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃⋅^(𝑘+1)𝜏_𝑘𝜏𝑒^−𝜀𝑡𝑑𝑡+6𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃⋅𝑒^{𝜀𝜏(𝑘+1)𝜏}_𝑘𝜏𝑒^−𝜀𝑡≤𝑑𝑡3𝑐₃𝜆_min(𝐺)𝑒^{−𝜀𝑘𝜏}+6𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃+6𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃𝑒^𝜀𝜏⋅1𝜀𝑒^{−𝜀𝑘𝜏}1−𝑒^−𝜀𝜏≤3𝑐₃𝜆_min(𝐺)𝑒^{−𝜀𝑘𝜏}+6𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃+6𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃𝑒^𝜀𝜏𝜏𝑒^{−𝜀𝑘𝜏}=3𝜏𝑐₃𝜆_min(𝐺)1𝜏+6𝜏‖𝐴‖²+𝛼²₁+𝛼²₂+𝛼₃+6𝜏‖𝐵‖²+𝛽²₁+𝛽²₂+𝛽₃𝑒^𝜀𝜏⋅𝑒^{−𝜀𝑘𝜏}=𝑐₄𝑒^{−𝜀𝑘𝜏}, (45) where 𝑐₄=(3𝜏𝑐₃/𝜆_min(𝐺)){(1/𝜏)+[6𝜏(‖𝐴‖²+𝛼²₁+𝛼²₂)+𝛼₃]+[6𝜏(‖𝐵‖²+𝛽²₁+𝛽²₂)+𝛽₃]𝑒^𝜀𝜏}.
Let 𝜀₀∈(0,𝜀) be arbitrary. By Doob’s martingale inequality, it follows from (45) that 𝑃𝜔∶sup_{()𝜏𝑘𝜏≤𝑡≤𝑘+1}|𝑥(𝑡)|>𝑒^{−(𝜀−𝜀0)𝑘𝜏/2}≤𝑐₄𝑒^{−𝜀0𝑘𝜏}. (46) It then follows from the Borel-Cantelli lemma that for almost all 𝜔∈Ω, there exists a 𝑘₀(𝜔),𝑘≥𝑘₀(𝜔), and sup_{()𝜏𝑘𝜏≤𝑡≤𝑘+1}|𝑥(𝑡)|≤𝑒^{−(𝜀−𝜀0)𝑘𝜏/2} (47) holds in probability 1. Thus, for 𝑘𝜏≤𝑡≤(𝑘+1)𝜏, and 𝑘≥𝑘₀(𝜔), we have ln|𝑥(𝑡)|≤−𝜀−𝜀₀𝑘𝜏2a.s. (48) Hence, 1𝑡ln|𝑥(𝑡)|≤−𝜀−𝜀₀2a.s. (49) This easily yields limsup_𝑡→∞1𝑡ln|𝑥(𝑡)|≤−𝜀−𝜀₀2a.s. (50) Since 𝜀₀ is arbitrary, we must have limsup_𝑡→∞1𝑡|𝑥(𝑡𝜀ln)|≤−2a.s. (51) Then, (13) is almost surely exponentially stable. The proof is complete.

Remark 8. Again in the proof we gave, the estimate for the almost surely Lyapunov exponent should not be greater than −𝜀/2.

Let us single out three important special cases.

Case 1. If ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))≡0, then (13) reduces to a semilinear differential delay equation 𝑑𝑥(𝑡)=𝐴+𝐴₁(𝑡)+𝐴₂()𝑥(𝑡)+𝑡−𝜏𝐵+𝐵₁(𝑡)+𝐵₂(𝑡−𝜏)𝑥(𝑡−𝜏)𝑑𝑡. (52)

Corollary 9. Let condition (14) holds. Assume that there exists a pair of symmetric positive definite 𝑛×𝑛 matrices 𝐺 and 𝑄 such that 𝐺(𝐴+𝐵)+(𝐴+𝐵)^𝑇𝜆𝐺=−𝑄,_min(𝑄)>2‖𝐺‖𝛼₁+𝛼₂+𝛽₁+𝛽₂+4𝜏‖𝐺𝐵‖⋅3‖𝐴‖²+‖𝐵‖²+𝛼²₁+𝛽²₁+𝛼²₂+𝛽²₂. (53)(54) Then, (52) is exponentially stable in the mean square.

Case 2. Let us further assume that 𝐴₁(𝑡)=𝐴₂(𝑡−𝜏)=𝐵₁(𝑡)=𝐵₂(𝑡−𝜏)≡0, then (52) becomes an ordinary differential equation 𝑑𝑥(𝑡)=(𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏))𝑑𝑡. (55)

Corollary 10. Assume that there exists a pair of symmetric positive definite 𝑛×𝑛 matrices 𝐺 and 𝑄 such that 𝐺(𝐴+𝐵)+(𝐴+𝐵)^𝑇𝜆𝐺=−𝑄,_min(𝑄)>4𝜏‖𝐺𝐵‖⋅3‖𝐴‖²+‖𝐵‖². (56)(57) Then, (55) is exponentially stable in the mean square.

Remark 11. Corollary 10 clearly shows that if the price system 𝑥̇(𝑡)=(𝐴+𝐵)𝑥(𝑡) (the general case of the classical price Rayleigh equation (2)) is exponentially stable (this is guaranteed by condition (16) and 𝜆_min(𝑄)>0), then the corresponding delayed system (55) is also exponentially stable provided that the time delay 𝜏 is small enough (bounded by (57)).

Remark 12. Condition (17) controls the intensity of the uncertainties, that is, the parameters 𝛼_𝑖,𝛽_𝑖(𝑖=1,2,3), and the time delay 𝜏 should be small enough to have the stability of (13) which is regarded as the perturbed system of (55). In other words, Theorem 5 shows that if (55) is exponentially stable, so is its perturbed system (13) provided that the intensity of perturbations is small enough.

Case 3. Let us assume that 𝐵=𝐴₂(𝑡−𝜏)=𝐵₁(𝑡)=𝐵₂(𝑡−𝜏)≡0 and 𝜏≡0, then (13) reduces to a stochastic differential equation 𝑑𝑥(𝑡)=𝐴+𝐴₁(𝑡)𝑥(𝑡)𝑑𝑡+ℎ(𝑡,𝑥(𝑡))𝑑𝑊(𝑡). (58)

Corollary 13. Assume that there exists a pair of symmetric positive definite 𝑛×𝑛 matrices 𝐺 and 𝑄 such that 𝐺𝐴+𝐴^𝑇𝜆𝐺=−𝑄,_min(𝑄)>‖𝐺‖2𝛼₁+𝛼₃. (59)(60) Then, (58) is exponentially stable in the mean square and is also almost surely exponentially stable.

4. Example

Let us now present two simple examples to illustrate our results which can help us find the time delay upper limit.

Example 14. Let us start with (13), where 𝛼₁=𝛼₂=𝛼₃=𝛽₁=𝛽₂=𝛽₃=0.1, 𝐴=−10−1−1,𝐵=−201−1. (61) For convenience, let us choose 𝑄= the 2-order identity matrix, and then 𝜆_min(𝑄)=1. By plugging these into (16), it is easy to find that 1𝐺=60014. (62) We obtain via a simple calculation ‖𝐴‖²=2.618,‖𝐵‖²‖𝐺‖‖‖=5.2361,=0.25,𝐺𝐵=0.4488. (63) Substituting (63) into (17), we derive that if 𝜏<0.08380, then (13) is exponentially stable in the mean square and is also almost surely exponentially stable. If ℎ(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))≡0, by Corollary 9, we would conclude that (52) is exponentially stable provided that 𝜏<0.09156.

Example 15. Now, let us recall price system (4) with 𝑎=−0.5,𝑏₀=1.0,𝑐₀=1.0,𝛾=0.2,𝜀=0.1, √𝜆𝛼=1, and 𝜇=0.06. It is clear that 𝑧(𝑡)=(0,0)^𝑇 is the trivial solution of system (4). Applying Taylor expansion to 𝑓(𝑡,𝑧(𝑡)), we get 𝑓(𝑡,𝑧(𝑡))=𝑓(𝑡,0)+𝐴𝑧(𝑡)+𝐴₁(𝑡)𝑧(𝑡), (64) where 𝐴=^0.0061₋₁₀, tr[ℎ^𝑇(𝑡,𝑥(𝑡))ℎ(𝑡,𝑥(𝑡))]=0.004, and ‖𝐴₁(𝑡)‖≤0.001.

To compute conveniently, let us choose 𝛼₃=0.004 and 𝑄=^0.010_00.01, and then 𝜆_min(𝑄)=0.01. By Corollary 13, it is easy to find 𝐺=,‖𝐺‖1.66667−0.00500−0.005001.66670=0.5285. (65) Substituting (65) into (60), it is easy to verify that (60) holds. So, the price system (4) with 𝑎=−0.5,𝑏₀=1.0,𝑐₀=1.0,𝛾=0.2,𝜀=0.1,√𝜆𝛼=1, and 𝜇=0.06 is exponentially stable in the mean square and is also almost surely exponentially stable. This result is the same as [5].

5. Concluding Remarks

In this paper, 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 the 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, and [5] is promoted and improved. Remarks 11 and 12 show that if price Rayleigh equation (2) is exponentially stable, so is its perturbed system (13) provided that both the time delay and the intensity of perturbations are small enough. These results are very helpful for our government strengthening and improving macrocontrol and promoting 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.

Acknowledgments

This work is Supported by the Fundamental Research Funds for the Central Universities (JBK130213) and the Fundamental Research Funds for the Central Universities (JBK130401).