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 and Professor Xuerong Mao as well as some recent research papers,for example,Wu and Hu (2009a),Wu and Hu(2010a,b),Wu,Hu and Mao(2011),Wu and Hu(2011e),Mao and Szpruch(2012),Mao and Szpruch(2013),Zhou and Xie(2014),Zhou(2014a),Zhou(2014b).It hence discusses many hot topics including the discrete  Razumikhintype theorem,numerical convergence and stability,population dynamics and stochastic stabilization.

The text is suitable for advanced undergraduate students,graduate students and teachers in colleges and universities as well as research workers involving stochastic dynamic systems.

I have to thank Professor Shigeng Hu for his constant support and kind assistance and encouragement.I have to thank Professor Fuke Wu,who has provided me a great deal of material during the writing process,for his support and help.I also wish to thank Ph.D.Yangzi Hu for her support and encouragement.Moreover,I should thank my family for their constant support and understanding.I also would like to thank the Teaching Material Foundation of HUST,the National Natural Science Foundation of China (Grant No.11301198 and 11422110) as well as National Excellent Young Foundation of China (61473125) for their financial supports.