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.
Highlights
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
Based on Lyapunov stability theory, Itoformula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system
The time delay upper limit is solved by using our theoretical results when the system is exponentially stable
Summary
By (H1.3), the stochastic nonlinear dynamical price system can be described by stochastic differential equation (SDE for short) as follows [5]: dx. The stability and the optimal control of stochastic nonlinear dynamical price model has been studied in [5]. 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 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.
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