内容简介
本书主要介绍了It型非线性随机微分方程，包括时滞随机微分方程和中立型随机微分方程的基本理论，深入讨论了非线性随机微分方程的稳定性、稳定化及其数值方法的收敛性及稳定性等。此外，本书还综述了近年来国内外非线性随机微分方程研究成果。
目录
1 Stochastic Integral(1)
1.1 Variation(2)
1.2 Random Variable(3)
1.3 Stochastic Processes(9)
1.4 Brownian Motions(15)
1.5 Stochastic Integrals(21)
1.6 It Formula (28)
1.7 important Inequalities(32)
2 Stochastic Differential Equations(35)
2.1 Global Solution(35)
2.2 Almost Surely Asymptotic Estimates(52)
2.3 Stability(54)
2.4 Stabilization(63)
2.5 Convergence of Numerical Methods(73)
3 Stochastic Differential Delay Equations (84)
3.1 Global Solution(84)
3.2 Stability(94)
3.3 Stabilization(103)
3.4 Strong Convergence(114)
3.5 Stability of Numerical Method(124)
3.6 Stochastic Pantograph Equations(134)
4 Stochastic Functional Differential Equations(144)
4.1 Global Solution(144)
4.2 Boundedness and Moment Stability(154)
4.3 SFDE with Infinite Delay(165)
4.4 Stabilization(181)
4.5 Stability of Numerical Method (189)
4.6 Stochastic Differential Equations with Variable Delay (200)
5 Neutral Stochastic Functional Differential Equations(215)
5.1 Global Solution (215)
5.2 Boundedness and Moment Stability (224)
5.3 NSFDEs with Infinite Delay (236)
5.4 Exponential Stability of Numerical Solution (246)
5.5 Neutral Stochastic Differential Delay Equation (256)
6 Stochastic KolmogorovType Systems (265)
6.1 Global Positive Solution (266)
6.2 Moment Boundedness (271)
6.3 Asymptotic Properties(274)
6.4 Stochastic Kolmogorovtype System with Infinite Delay(278)
7 Stochastic Differential Equations with Markovian Switching (286)
7.1 Basic Markov Switching (287)
7.2 Polynomial Growth of Switching SDE (289)
7.3 Polynomial Growth of Switching Neutral Type Equations (300)
References (306)
摘要与插图
序言Nonlinear stochastic system has come to play an important role in many branches of science and industry.More and more researches have involved nonlinear stochastic differential equations.Recently,many research efforts have been devoted to deal with nonlinear stochastic differential systems.Many papers on them were published in different journals,which is not convenient for readers to understand the theory systematically.This book is therefore written.The main aim of this book is to explore systematically all various of nonlinear stochastic differential systems.Some important features of this text are as follows:
The text will be the first systematic presentation of the basic principles of various types of nonlinear stochastic systems,including stochastic differential equations,stochastic functional differential equations,stochastic equations of neutral type.It will emphasize the current research trends in the field of nonlinear systems at an advanced level,in which the local Lipschitz and onesided polynomial growth conditions will replace the classical uniform Lipschitz and linear growth conditions.
This text emphasizes the analysis of stability which is vital in the automatic control of stochastic systems.The Lyapunov method can be adopted to study all various of stable properties of stochastic systems.Especially,this text demonstrates that the Khasminskiitype criteria on stability is very effective for highly nonlinear stochastic systems with delay.
The text explains systematically the use of the Razumikhin technique in the study of exponential stability for stochastic functional differential equations and functional equations of neutraltype with finite or infinite delays as well as stability of the discrete EulerMaruyama approximate solution.
The text will be the first systematic presentation of the basic theory of nonlinear stochastic functional differential equations with infinite delays.It discusses the existence and exponential stability of stochastic functional differential equations on special and general measure spaces.
The text demonstrates systematically the stabilization of nonlinear deterministic system.It indicates a nonlinear Brownian noise feedback to suppress the potential explosion of the system,and a linear Brownian noise feedback to stabilize exponentially this system.
This text discusses new developments of the EulerMaruyama approximation schemes under the local and onesided Lipschitz conditions as well as onesided polynomial growth condition.This text studies linear stability of the EulerMaruyama approximate schemes and nonlinear stability of the backward EulerMaruyama approximate schemes.The advantage of the backward EulerMaruyama approximate schemes is that the approximate solution converges to the accurate solution under the local Lipschitz and onesided polynomial growth conditions.
This text is mainly based on the papers of Professor Fuke Wu
