周少波， 湖北浠水人，华中科技大学数学与统计学院概率统计专业博士、讲师、硕士生导师，主要研究领域为非线性随机微分动力系统的稳定性、稳定化、数值方法的收敛性及其稳定性。在《Journal of Mathematics Analysis and Application》等国内外重要学术期刊上发表论文３0余篇。
目錄：
1 Stochastic Integral1
1.1 Variation2
1.2 Random Variable3
1.3 Stochastic Processes9
1.4 Brownian Motions15
1.5 Stochastic Integrals21
1.6 It Formula 28
1.7 Important Inequalities32
2 Stochastic Differential Equations35
2.1 Global Solution35
2.2 Almost Surely Asymptotic Estimates52
2.3 Stability54
2.4 Stabilization63
2.5 Convergence of Numerical Methods73
3 Stochastic Differential Delay Equations 84
3.1 Global Solution84
3.2 Stability94
3.3 Stabilization103
3.4 Strong Convergence114
3.5 Stability of Numerical Method124
3.6 Stochastic Pantograph Equations134
4 Stochastic Functional Differential Equations144
4.1 Global Solution144
4.2 Boundedness and Moment Stability154
4.3 SFDE with Infinite Delay165
4.4 Stabilization181
4.5 Stability of Numerical Method 189
4.6 Stochastic Differential Equations with Variable Delay 200
5 Neutral Stochastic Functional Differential Equations215
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 Properties274
6.4 Stochastic Kolmogorovtype System with Infinite Delay278
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
內容試閱：
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 Hu2010a,b,Wu,Hu and Mao2011,Wu and Hu2011e,Mao and Szpruch2012,Mao and Szpruch2013,Zhou and Xie2014,Zhou2014a,Zhou2014b.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.
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