应用随机过程的清华大学书

书名：应用随机过程

ISBN：9787302089407

作者：张波

定价：29元

出版日期：200-9

出版社：清华大学出版社 本书是现代应用随机过程教材，内容从入门知识到学术前沿，包括预备知识、随机过程的基本类型、Poisson过程、更新过程、Markov链、鞅、Brown运动、随机积分、随机微分方程及其应用和Levy过程等，本书配有大量与社会、经济、金融、生物等专业相关的例题和习题，并给出了参考答案，方便自学。

本书可以作为高等院校统计、经济、金融、管理专业的本科生教材，也可以作为其他相关专业的研究生教材和教学参考书，对广大从事与随机现象相关工作的实际工作者也极具参考价值。 本书的初稿曾在中国人民大学统计学系本科生的教学中多次使用，反映良好.此次出版，我们根据广大读者的反馈意见，对部分内容进行了适当调整，对Markov过程的讨论更加详尽，并增加了随机微分方程和Levy过程等新的内容.

几十年来，由于实际问题的需要和数学工作者的努力，随机过程无论在理论上还是在应用上都有了蓬勃的发展.它的基本知识和方法，不仅为数学、概率统计专业所必需，也为工程技术、生物信息及经济领域的应用与研究所需要.因此，随机分析的方法越来越受到人们的重视，高等院校的学生、工程技术人员、金融工作者，更迫切地需要学习和掌握随机过程的知识.本书是为适应这种需求，根据近年来讲授这门课的教学实践所积累的资料，参考国内外有关著作编写而成.由于随机过程这门学科发展十分迅速，其内容十分丰富，作为一本大学本科生用教科书，不可能包括其全部内容.因此，我们力图根据经济类和管理类本科生教学选择素材.为适应更广泛的读者, 本书着重于随机过程的基础知识和基本方法的介绍，特别注重实际应用，尽量回避测度论水平的严格证明，只有第6章的部分内容、第8章和第9章不可避免的用到一些测度论知识.这些内容初学者可以根据各自的基础进行取舍, 数学基础稍好、有测度论基础或对数理金融有兴趣的读者可以选学.为了方便读者，我们在第1章中用很小的篇幅，对概率测度和积分进行了初步介绍，希望对读者有所帮助.一般读者只要具有高等数学及概率论的基础知识便可阅读和理解本书的大部分内容.我们建议对大学本科生以54学时讲授本书前7章的内容.如果课程设置为60学时以上，则可以讲授前8章的全部内容，并对第9章做简单介绍.如果课时比较少，教师可根据授课对象适当选择教学内容.

全书大体可分为3个部分.第1部分是预备知识和随机过程最基本的内容，一般教科书都包含这部分内容（第1，2，3，5章）.第2部分是更新过程，这一内容在许多教科书中没有单独讨论.考虑到它在应用中的重要性，特别是在人口学和保险论中的应用，故将它放在第4章讲授.第3部分包括第6，7，8，9章，鉴于在经济和金融领域非常广泛的应用，分别介绍鞅、Brown运动、随机微分方程及其应用和Levy过程.考虑到实际问题的需要，本书第一次将Levy过程写入随机过程的教科书中.

本书配有一些与社会、经济、金融、管理以及生物等领域相关的例题和习题，以帮助学生加深理解，提高应用随机过程理论解决问题的能力.为了便于自学，书末给出了大部分习题的答案，供自学者参考.为了便于有兴趣的读者进一步学习，我们对主要内容增加了一个文献评注,同时书后列出较多的参考书目，为这些读者提供线索.因此，虽然我们强调主要着眼于经济管理类本科学生，但是对于这些专业的研究生以及某些应用数学和其他理工科的本科生、研究生来说，也不难发现使用本书的方便之处.

本书的编写得到吴喜之、张尧廷、易丹辉、顾岚、肖争艳等许多同仁的鼓励、支持和帮助；宋士吉教授和刘立新博士分别在清华大学、北京大学和对外经贸大学使用过本书的初稿，并对本书提出了许多宝贵的修改意见；薛芳，李晓明，刘晓华，吴孟书，何艳青，胡威等同学提供了习题参考答案.在此谨表衷心谢意！

同时也要感谢中国人民大学统计学院，使得笔者有机会在教学实践中完成本书的写作和修改.还要感谢教育部的支持，将本书列为普通高等教育“十五”国家级规划教材，使得本书得以顺利出版.

由于编者水平所限，书中的缺点错误在所难免，敬请读者批评指正.

编者2004年2月 第1章预备知识1

1.1概率空间1

1.2随机变量和分布函数3

1.3数字特征、矩母函数与特征函数7

1.3.1数字特征7

1.3.2Riemann?Stieltjes积分8

1.3.3关于概率测度的积分9

1.3.4矩母函数和特征函数11

1.4条件概率、条件期望和独立性13

1.4.1条件概率13

1.4.2条件期望14

1.4.3独立性15

1.4.4独立随机变量和的分布16

1.5收敛性17

第2章随机过程的基本概念和基本类型20

2.1基本概念20

2.2有限维分布与Kolmogorov定理21

2.3随机过程的基本类型24

2.3.1平稳过程24

2.3.2独立增量过程30

习题31

第3章Poisson过程32

3.1Poisson过程32

3.2与Poisson过程相联系的若干分布37

3.2.1Xn和Tn的分布37

3.2.2事件发生时刻的条件分布39

3.3Poisson过程的推广42

3.3.1非齐次Poisson过程42

3.3.2复合Poisson过程45

3.3.3条件Poisson过程46

习题48

第4章更新过程50

4.1更新过程定义及若干分布50

4.1.1更新过程的定义50

4.1.2N(t)的分布及E[N(t)]的一些性质51

4.2更新方程及其应用54

4.2.1更新方程54

4.2.2更新方程在人口学中的一个应用57

4.3更新定理59

4.4Lundberg?Cramèr破产论64

4.5更新过程的推广69

4.5.1延迟更新过程69

4.5.2更新回报过程69

4.5.3交替更新过程71

习题73

第5章Markov链74

5.1基本概念74

5.1.1Markov链的定义74

5.1.2转移概率75

5.1.3一些例子76

5.1.4n步转移概率C?K方程81

5.2停时与强Markov性84

5.3状态的分类及性质85

5.4极限定理及不变分布92

5.4.1极限定理92

5.4.2不变分布与极限分布100

5.5Markov链的大数定律与中心极限定理104

5.5.1大数定律与不变分布104

5.5.2Markov链的中心极限定理108

5.6群体消失模型与人口模型110

5.6.1群体消失模型（分支过程）110

5.6.2人口结构变化的Markov链模型113

5.7连续时间Markov链115

5.7.1连续时间Markov链115

5.7.2转移概率pij(t)和Kolmogorov微分方程119

5.8应用——数据压缩与熵126

习题130

第6章鞅133

6.1基本概念133

6.2鞅的停时定理138

6.2.1停时定理138

6.2.2Doob极大不等式144

6.2.3停时定理的应用——关于期权值的界146

6.3一致可积性149

6.4鞅收敛定理151

6.5连续鞅154

习题156

第7章Brown运动159

7.1基本概念与性质159

7.2Gauss过程163

7.3Brown运动的鞅性质165

7.4Brown运动的Markov性166

7.5Brown运动的最大值变量及反正弦律168

7.6Brown运动的几种变化172

7.6.1Brown桥172

7.6.2有吸收值的Brown运动173

7.6.3在原点反射的Brown运动174

7.6.4几何Brown运动174

7.6.5有漂移的Brown运动175

习题176

第8章随机积分与随机微分方程178

8.1关于随机游动的积分178

8.2关于Brown运动的积分179

8.3It-积分过程183

8.4It-公式187

8.5随机微分方程191

8.5.1解的存在惟一性定理191

8.5.2扩散过程192

8.5.3简单例子196

8.6应用——金融衍生产品定价197

8.6.1Black?Scholes模型197

8.6.2等价鞅测度199

习题207

第9章Levy过程与关于点过程的随机积分简介209

9.1Levy过程209

9.2关于Poisson点过程的随机积分210

习题参考答案216

文献评注247

参考文献248

1.Keyou You and Lihua Xie, “Quantized Kalman Filtering of Linear Stochastic Systems”, Editor: Joaquín M. Gomez, Book Chapter in Book Title: Kalman Filtering, Nova Science Publishers, 2010

2.Keyou You, Nan Xiao and Lihua Xie, “Analysis and Design of Networked Control Systems”, Communications and Control Engineering Series, Springer-Verlag, London, 2012 (accepted). 1.Tianju Sui, Keyou You, and Minyue Fu, “Stability of Kalman Filtering with Multiple Sensors Involving Lossy Communications”，submitted to IFAC world congress, November 2013

2.Tianju Sui, Keyou You, and Minyue Fu, “Stability Conditions for Distributed State Estimation Over Lossy Network”，Automatica, submitted as a regular paper, November 2013

3.Keyou You, Shiji Song and Li Qiu, “Randomized Incremental Least Squares for Distributed Estimation Over Sensor Networks, submitted to IFAC world congress, October 2013

4.Zhenghua Wang, Keyou You, Juanjuan Xu, Huanshui Zhang, “Consensus Design for Unstable Multi-agent Systems with Communication Delay”, IET Control Theory and Applications, submitted, September 2013

5.Keyou You, Tianju Sui, Minyue Fu and Shiji Song, “Kalman Filtering over Lossy Networks Under Switching Sensors”, Asia Journal of Control, submitted, September 2013

6.Tianju Sui, Keyou You and Minyue Fu, “Stability of LMMSE Estimation over Lossy Networks Using Linear Coding, Automatica, submitted, August 2013 1.Zhixin Fu, Keyou You, and Yue Yuan, “Performance evaluation of WSN based Condition Monitoring System for Offshore Wind Turbine, Transactions of the Institute of Measurement and Control, accepted for publication, 2013

2.Keyou You, Zhongkui Li and Lihua Xie, “Consensus Condition for Linear Multi-agent Systems over Randomly Switching Topologies, Automatica, vol. 49, no. 10, pp. 3125–3132, 2013.

3.Keyou You, Lihua Xie, and Shiji Song, “Asymptotically Optimal Parameter Estimation with Scheduled Measurements”, IEEE Transactions on Signal Processing, vol. 61, no. 14, 2013 (regular paper)

4.Zhixin Fu and Keyou You, Optimal Mobile Sensor Scheduling for a Guaranteed Coverage Ratio in Hybrid Wireless Sensor Networks”, International Journal of Distributed Sensor Networks, volume 2013, Article ID 740841, 11 pages, April 2013.

5.Keyou You and Lihua Xie, “Kalman Filtering with Scheduled Measurements,” IEEE Transactions on Signal Processing, vol. 61, no. 6, pp. 1520-1530, 2013 (regular paper).

6.Keyou You and Lihua Xie, “Survey of Recent Progress in Networked Control Systems”, Acta Automatica Sinica, vol. 39, no. 2, pp. 97-114, 2013 （invited by Academician HUANG Lin）.

7.游科友,谢立华 ,“网络控制系统的最新研究综述”,自动化学报, vol. 39, no. 2, pp. 97-114, 2013(黄琳院士约稿译稿, in Chinese).

8.Damian Marelli, Keyou You and Minyue Fu, “Identification of ARMA Models Using Intermittent and Quantized Output Observations”, Automatica, vol. 49, no. 2, pp. 360-369, 2013 (regular paper).

9.Kristian Hengster-Movric, Keyou You, Frank L. Lewis and Lihua Xie, “Synchronization of Discrete-Time Multi-agent Systems on Graphs Using Riccati Design ”, Automatica, vol. 49, no. 2, pp. 414-423, 2013 (regular paper).

10.Gao Huang, Shiji Song, Cheng Wu and Keyou You, “Robust Support Vector Regression for Uncertain Input and Output Data”, IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 11, pp. 1690-1700, 2012 (regular paper)

11.Keyou You, Minyue Fu and Lihua Xie, “Mean Square Stability for Kalman Filtering with Markovian Packet Losses”, Automatica, vol. 47, no. 12, pp. 1247-1257, 2011 (regular paper)

12.Keyou You and Lihua Xie, “Network Topology and Communication Data Rate for Consensusability of Discrete-time Multi-agent Systems”, IEEE Transactions on Automatic Control, vol. 56, no.10, pp. 2262-2275, 2011 (regular paper)

13.Keyou You and Lihua Xie, “Minimum Data Rate for Mean Square Stabilizability of Linear Systems with Markovian Packet Losses”, IEEE Transactions on Automatic Control, vol. 56, no.4, pp. 772-785, 2011 (regular paper)

14.Keyou You, Weizhou Su, Minyue Fu and Lihua Xie, “Attainability of the Minimum Data Rate for Stabilization of Linear Systems via Logarithmic Quantization”, Automatica, vol. 47, no.1, pp.170-176, 2011

15.Keyou You and Lihua Xie, “Coordination of Discrete-time Multi-agent Systems via Relative Output Feedback”, International Journal of Robust and Nonlinear Control, vol. 21, no.13, pp.1587-1605, 2011

16.Keyou You, Lihua Xie, Shuli Sun and Wendong Xiao, “Quantized Filtering of Linear Stochastic Systems”, Transactions of the Institute of Measurement and Control, vol. 33, no. 6, pp. 683-698, 2011

17.Keyou You and Lihua Xie, “Linear Quadratic Gaussian Control with Quantised Innovations Kalman Filter over a Symmetric Channel”, IET Control Theory and applications, vol. 5, no.3, pp. 437-446, 2011

18.Keyou You and Lihua Xie, “Minimum Data Rate for Mean Square Stabilization of Discrete LTI Systems over Lossy Channels”, IEEE Transactions on Automatic Control, vol. 55, no. 10, pp.2373-2378, 2010 1.Keyou You and Shiji Song, “Observability of Linear Systems for Kalman Filtering with Packet Losses”, 32nd Chinese Control Conference, Xi’an, China

2.Keyou You, Tianju Sui, Minyue Fu and Shiji Song, “Stability of the Kalman Filtering with Two Periodically Switching Sensors over Lossy Networks”, 10th IEEE International Conference on Control & Automation, Hang Zhou, China

3.Tianju Sui, Keyou You and Minyue Fu, “Kalman Filtering with Intermittent Observations Using Measurements Coding”,10th IEEE International Conference on Control & Automation, Hang Zhou, China

4.Nan Xiao, Xuehe Wang, Tichakorn Wongpiromsarn, Keyou You, et al, “Average Strategy Fictitious Play with Application to Road Pricing”, 2013 American Control Conference, Washington.

5.Tichakorn Wongpiromsarn, Nan Xiao, Keyou You, et al “Road Pricing for Spreading Peak Travel: Modeling and Design”, 17th International conference of Hongkong Society for Transportation Studies, December 2012

6.Kristian Hengster-Movric, Keyou You, Frank L. Lewis and Lihua Xie, “Synchronization of Discrete-Time Multi-agent Systems on Graphs Using Riccati Design”, IEEE Multi-Conference on Systems and Control, Dubrovnik, Croatia, 2012

7.Keyou You and Lihua Xie, “Kalman filtering with scheduled measurements – Part II: stability and performance analysis”, 31st Chinese Control Conference, Hefei, China, January 2012

8.Keyou You and Lihua Xie, “Kalman filtering with scheduled measurements – Part I: estimation framework”, 9th World Congress on Intelligent Control and Automation, Beijing, China, January 2012

9.Keyou You, Zhongkui Li and Lihua Xie, Consensus for General Multi-agent Systems over Random Graphs, 9th IEEE International Conference on Control & Automation, Santiago, Chile, December, 2011

10.Keyou You, Minyue Fu and Lihua Xie, “Necessary and Sufficient Conditions for Stability of Kalman Filtering with Markovian Packet Losses”, 18th World Congress of the International Federation of Automatic Control, Milan, Italy, August 2011

11.Keyou You and Lihua Xie, Consensusability of Discrete-time Multi-agent Systems over Directed Graphs, Proc. 30th Chinese Control Conference, Yantai, China, July 2011.

12.Damian Marelli, Keyou You and Minyue Fu, “Identification of ARMA Models Using Intermittent and Quantized Output Observations ”, Proc. 36th International Conference on Acoustics, Speech and Signal Processing, Prague, Czech Republic, May 2011

13.Keyou You and Lihua Xie, “Consensusability of Discrete-time Multi-agent Systems via Relative Output Feedback”, Proc. 11th International Conference on Control, Automation, Robotics and Vision, Singapore, December 2010

14.Keyou You and Lihua Xie, Consensusability of Discrete-time Multi-agent Systems over Directed Graphs, Postgraduate student conference on the 2010 Asia-Oceania Top University League on Engineering, Dandung, Indonesia, November 2010.

15.Keyou You and Lihua Xie, “Necessary and Sufficient Conditions for Consensusability of Discrete-time Multi-agent Systems”, Proc. 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems, Annecy, France, September 2010

16.Keyou You and Lihua Xie, “Mean Square Stabilizability of Linear Systems with Limited Feedback Data Rates and Markovian Packet Losses”, Proc. 29th Chinese Control Conference, Beijing, China, July 2010 (Guan Zhao-zhi best paper award)

17.Tichakorn Wongpiromsarn, Keyou You and Lihua Xie, “A Consensus Approach to the Assignment Problem: Application to Mobile Sensor Dispatch”, Proc. 8th IEEE International Conference on Control & Automation, Xiamen, China, June 2010

18.Keyou You and Lihua Xie, “Minimum Data Rate for Mean Square Stabilization of Single Input Discrete Systems over Lossy Channels”, Proc. 7th IEEE International Conference on Control & Automation, Christchurch, New Zealand, December 2009

19.Keyou You, Weizhou Su, Minyue Fu and Lihua Xie, “Optimality of the Logarithmic Quantizer for Stabilization of Linear Systems: Achieving the Minimum Data Rate”, Proc. 48th IEEE Conference on Decision and Control & 28th Chinese Control Conference, Shanghai, China, December 2009

20.Keyou You, Yanlong Zhao and Lihua Xie, “Recursive Quantized State Estimation of Discrete-Time Linear Stochastic Systems”, Proc. 7th Asian Control Conference, Hong Kong, China, August 2009

21.Keyou You, Lihua Xie, Shuli Sun and Wendong Xiao, “Multiple-Level Quantized Innovation Kalman Filter”, Proc. 17th International Federation of Automatic Control, Seoul, Korea, July 2008

22.Keyou You and Lihua Xie, “LQG Control with Quantized Innovation Kalman Filter”, Proc. 27th Chinese Control Conference, Kunming, China, July 2008 。 10/2013, 香港科技大学电子与计算机工程系，邀请人Li Qiu教授

10/2013, 香港中文大学信息工程系, 邀请人Wong Wing Shing教授

06/2013, 香港大学机械工程系, 邀请人Michael Z.Q. Chen 教授

06/2013, 香港城市大学机械与生物医学工程系, 邀请人 Gang Feng 教授

06/2013, 香港科技大学电子与计算机工程系，邀请人Li Qiu教授

11/2012, 山东大学控制科学与工程学院，邀请人张焕水教授

09/2012, 中国科学院自动化所，邀请人侯增广教授

08/1012, 华东理工大学自动化系，邀请人严怀成教授

08/2012, 上海交通大学自动化系, 邀请人杨博教授

07/2012, 10th World Congress on Intelligent Control and Automation, Beijing, China, January 2012

07/2012, 31st Chinese Control Conference, Hefei, China

06/2012, 浙江大学控制科学与工程学系，邀请人陈积明教授

05/2012, 香港大学机械工程系, 邀请人Michael Z.Q. Chen 教授

05/2012, 香港科技大学电子与计算机工程系，邀请人Ling Shi教授

04/2012, 北航自动化科学与电气工程学院; 邀请人郭雷教授

03/2012, 中国科学院系统科学研究所，邀请人李韬博士

03/2012, 北京大学力学与空天技术系， 邀请人段志生教授

10/2011, 清华大学自动化系, 邀请人宋士吉教授

09/2011, 山东大学控制科学与工程学院, 邀请人张焕水教授

09/2011, 北京理工大学自动化学院，邀请人夏元清教授

09/2011, 7th IEEE Singapore Control Systems Chapter Graduate Student Workshop on Control and Automation, Singapore

07/2011, 30th Chinese Control Conference, Yantai, China

12/2010, 11th International Conference on Control, Automation, Robotics and Vision, Singapore

11/2010, Postgraduate Student Conference on 5th AOTULE, Institut Teknologi Bandung, Indonesia

08/2010, 山东大学控制科学与工程学院, 邀请人张焕水教授

07/2010, 29th Chinese control conference, Beijing, China

07/2010, Workshop on networked control systems at Zhejiang University, Hangzhou, China; 邀请人付敏跃教授

06/2010, School of Electrical Engineering and Computer Science, University of Newcastle, Australia; 邀请人付敏跃教授

12/2009, Joint 48th IEEE conference on control and decision & 28th Chinese control conference, Shanghai, China

10/2009, 华南理工大学自动化科学与工程学院，邀请人苏为洲教授

06/2009, 华南理工大学自动化科学与工程学院，邀请人苏为洲教授