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 ann-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.

Exponential Stability of Stochastic Differential Equation with Mixed Delay

“…To develop our theories and results, we need to introduce the following concepts and important inequalities. For stochastic system, exponential stability in mean square and almost surely exponential stability are generally used [13]. Definition 1.…”

“…For example, Cong [1] and Li et al [2] obtained exponential stability conditions of linear stochastic neutral delay systems. Mao [8], Mao and Shah [9], Zhu and Hu [10], Zhu and Hu [11], S. Xie and L. Xie [12], and Zhu et al [13] established some stability criteria of the stochastic system with discrete delays. An improved delay-dependent stability criterion is derived for stochastic delay systems by a strict LMI in [14].…”

Exponential Stability of Stochastic Systems with Delay and Poisson Jumps

“…Currently, the study of analysis and synthesis of stochastic time delay systems, described by stochastic delayed differential equations (SDDE for short), is a popular topic in the field of control theory [2][3][4][5][6][7][8]. Delays in the dynamics can represent memory or inertia in the financial system [9].…”

“…So far, these topics have received a lot of attention and there are so many references about them. For instance, [2][3][4][5][6][7][8] established some stability criteria of the stochastic systems with delay by using Lyapunov function method or Razumikhin technique or inequality technique and so on. By using the fixed point theory and Borel-Cantelli lemma, Guo and Zhu [13] studied that the solution to a class of stochastic Volterra-Levin equations with Poisson jumps is not only existent and unique but also th moment exponentially stable.…”

Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

“…They can cause the stable equilibrium of a system to become unstable and make a system bifurcate periodic solutions. Dynamical systems with delay have been studied by many scholars [11][12][13][14][15][16][17][18][19][20]. Motivated by the works above and considering that the antivirus software may use a period to clean the worms in one node and that the recovered nodes may have a temporary immunity period due to the antivirus software, we consider the following delayed system in this paper:…”