Stability And Applications Of Stochastic Differential Equations

It has great theoretical significance and a wide range of application background on the stability problems on stochastic differential equation.In 1902, Gibbs studied the integration problem of the differential systems whose initial state is random when he was discussing the issues of statistical mechanics. Then in 1951 Ito published "On stochastic differential equations" at the first time. For half a century, various scholars have made many efforts to study stochastic differential equations. Stochastic differential equation is the combination of probability theory and ordinary differential equations which developed a door edge disciplines. It is not only play the role of effective coupling in many branches in the field of mathematics, but also widely used in financial economics, systems engineering, physical sciences, systems biology and other fields.So study the stochastic differential equations on stability and application is of vital importance. In this paper we studied on the stability of the stochastic differential equation and the paper consists of five following parts.In the first chapter, we outlined the history of the development of stochastic differential equations.In the second chapter we gave prior knowledge of stochastic differential equations.In the third chapter, we studied the stability of the distributed delay singular random system, by constructed a Lyapunov function, and applied the Ito differentia. We got linear matrix inequalities, and then studied a distribution of random delay singular system stability index. Finally gave the system stability index-judgment of algebra.In the fourth chapter, we studied Markov distributed delay uncertainties stabilization of stochastic differential equations. Based on the Lyapunov stability theory, by using Ito formulas, Shure supplement matrix inequalities, and Functional methods such as Lyapunov-Krasovskii.we studied a random delay system, and proved that nominal system of optimal controllers can robust stabilization, and also it has the Markov jump parameters as well as distribution delays and uncertainties of stochastic differential system parameters. Finally gave the system stability index-judgment of algebra. And based on the above conclusions we can get some Deductions whose known conditions are more concise. Chapter 5 was the application of the stability of stochastic differential equations, first choose a random time delay and variable delay Hopfield neural network model, use the space average of Lyapunov function variables and Ito formula, we discussed at the random delay Hopfield neural network stability problems. Not only has we get of almost complete the exponential stability, but also get an estimate of convergence.