本书作者——Samuel Karlin，为斯坦福大学荣休教授，国际著名的应用概率学家，美国科学院院士，数理统计学会会士。1987年获冯·诺伊曼奖。在生灭过程中计算平稳分布的Karlin-McGregor定理即以他的名字命名。

本书堪称随机过程教材的典范，非常透彻地讨论了所有重要的主题，应用丰富，习题非常具有挑战性。本书是通往华尔街的必备读物。书中讨论的随机过程知识将为你理解期权定价打下坚实基础。

本书是人民邮电出版社影印和翻译出版的《随机过程初级教程》的姊妹篇，内容包括马尔可夫链的代数方法、转移概率的比定理及应用、连续时间马尔可夫链、扩散过程、复合随机过程、独立同分布随机变理部分和波动理论、排队过程等很多主题。本书将理论与应用有机地结合在一起，取得了完美的平衡。

本书适用而广，可供数学、物理学、生物学、社会学、管理学和其他工程领域的理论研究者和实践者学习。

Chapter 10　ALGEBRAIC METHODS IN MARKOV CHAINS

1. Prelimmanes　1　

2. Relations of Eigenvalues and Recurrence Classes　3

3. Periodic Classes　6

4. Special Computational Methods in Markov Chains　10

5. Examples　14

6. Applications to Coin Tossing　18

　　Elementary Problems　23

　　Problems　25

　　Notes　30

　　References　30

Chapter 11　RATIO THEOREMS OF TRANSITION PROBABILITIES AND APPLICATIONS

1. Taboo Probabdiues　31

2. Ratio Theorems　33

3. Existence of Generalized Stationary Distributions　37

4. Interpretation of Generalized Stationary Distributions　42

5. Regular, Superregular, and Subregular Sequences for Markov Chains　44

6. Stopping Rule Problems　50

　　Elementary Problems　65

　　Problems　65

　　Notes　70

　　References　71

Chapter 12　SUMS OF INDEPENDENT RANDOM VARIABLES AS A MARKOV CHAIN

1. Recurrence Properties of Sums of Independent Random Variables　72

2. Local Limit Theorems　76

3. Right Regular Sequences for the Markov Chain {Sn} 83

4. The Discrete Renewal Theorem　93

　　Elementary Problems　95　Problems　96

　　Notes　99

　　References　99

Chapter 13　ORDER STATISTICS, POISSON PROCESSES, AND APPLICATIONS

 1. Order Statistics and Their Relation to Poisson Processes　100

 2. The Ballot Problem　107

 3. Empirical Distribution Functions and Order Statistics 113

 4. Some Limit Distributions for Empirical Distribution Functions　119

　　 Elementary Problems　124

　　 Problems　125

　　 Notes 137

　　 References　137

Chapter 14　CONTINUOUS TIME MARKOV CHAINS

 1. Differentiability Properties of Transition Probabilities　138

 2. Conservative Processes and the Forward and Backward Differential Equations　143

 3. Construction of a Continuous Time Markov Chain from Its Infinitesimal Parameters　145

 4. Strong Markov Property　149

　　 Problems　152

　　 Notes　156

　　 References　156

Chapter 15　DIFFUSION PROCESSES

 1. General Description　157

 2. Examples of Diffusion　169

 3. Differential Equations Associated with Certain Functionals　191

 4. Some Concrete Cases of the Functional Calculations 205

 5. The Nature of Backward and Forward Equations and Calculation of Stationary Measures　 213

 6. Boundary Classification for Regular Diffusion Processes 226

 7. Some Further Characterization of Boundary Behavior　242

 8. Some Constructions of Boundary Behavior of Diffusion Processes　251

 9. Conditioned Diffusion Processes　261

 10. Some Natural Diffusion Models with Killing　272

 11. Semigroup Formulation of Continuous Time Markov Processes　285

 12. Further Topics in the Semigroup Theory of Markov Processes and Applications to Diffusions　305

 13. The Spectral Representation of the Transition Density for a Diffusion　330

 14. The Concept of Stochastic Differential Equations 340

 15. Some Stochastic Differential Equation Models　 358

 16. A Preview of Stochastic Differential Equations and Stochastic Integrals　368

　　 Elementary Problems　377

　　 Problems　382

　　 Notes　395

　　 References　395

Chapter 16　COMPOUNDING STOCHASTIC PROCESSES

 1. Multidimensional Homogeneous Poisson Processes　398

 2. An Application of Multidimensional Poisson Processes to Astronomy　404 

 3. Immigration and Population Growth　405

 4. Stochastic Models of Mutation and Growth　408

 5. One-Dimensional Geometric Population Growth　413

 6. Stochastic Population Growth Model in Space and Time　416

 7. Deterministic Population Growth with Age Distribution　419

 8. A Discrete Aging Model　425

 9. Compound Poisson Processes　426

　　 Elementary Problems　441

　　 Problems　41

　　 Notes　450

　　 References　450

Chapter 17　FLUCTUATION THEORY OF PARTIAL SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES

 1. The Stochastic Process of Partial Sums 451

 2. An Equivalence Principle　453

 3. Some Fundamental Identities of Fluctuation Theory and Direct Applications　459

 4. The Important Concept of Ladder Random Variables 464

 5. Proof of the Main Fluctuation Theory Identities　468

 6. More Applications of Fluctuation Theory　473

　　 Problems　484

　　 Notes　488

　　 References　488

Chapter 18　QUEUEING PROCESSES　

1. General Description　489

2. The Simplest Queueing Processes(M/M/l)　490

3. Some General One-Server Queueing Models 492

4. Embedded Markov Chain Method Applied to the Queueing Model(M/GI/l)　497

5. Exponential Service Times(G/M/1)　504

6. Gamma Amval Dtstnbutlon and Generalizations(Ek/M/1)　506

7. Exponential Service with s Servers(GI/M/s)　511

8. The Virtual Waiting Time and the Busy Period　513

　　Problems　519

　　Notes　23

　　References　525