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Summary
Summary
This book presents a self-contained introduction to stochastic processes with emphasis on their applications in science, engineering, finance, computer science, and operations research. It provides theoretical foundations for modeling time-dependent random phenomena in these areas and illustrates their application by analyzing numerous practical examples.
The treatment assumes few prerequisites, requiring only the standard mathematical maturity acquired by undergraduate applied science students. It includes an introductory chapter that summarizes the basic probability theory needed as background. Numerous exercises reinforce the concepts and techniques discussed and allow readers to assess their grasp of the subject. Solutions to most of the exercises are provided in an appendix. While focused primarily on practical aspects, the presentation includes some important proofs along with more challenging examples and exercises for those more theoretically inclined.
Mastering the contents of this book prepares readers to apply stochastic modeling in their own fields and enables them to work more creatively with software designed for dealing with the data analysis aspects of stochastic processes.
Author Notes
Frank Beichelt is Professor of Operations Research at the University of the Witwatersrand, Johannesburg, South Africa
Reviews 1
Choice Review
Beichelt (Univ. of Witwatersrand, South Africa) offers a well-written, entertaining work on stochastic processes that emphasizes practical applications. It takes only a few lines for the author to arrive at the first real-life examples after a definition is made. Remarkably, this is achieved without compromising theoretical accuracy and clarity. The level of the exercises ranges widely, so even the best student will be challenged from time to time. The introductory chapter on probability provides sufficient background for any talented upper-level undergraduate to read the rest of the book. Although the book serves the practically inclined student, the theoretically inclined instructor will also enjoy it, as Beichelt offers a rare opportunity to understand the practical implications of the theory involved. Though the material covered in each chapter can be found elsewhere, the tandem of practical focus and theoretical clarity leads to a product that makes the book indispensable. ^BSumming Up: Essential. Upper-division undergraduates through faculty and researchers. M. Bona University of Florida
Table of Contents
| Preface | |
| Symbols and Abbreviations | |
| 1 Probability Theory | |
| 1.1 Random Events and Their Probabilities | p. 1 |
| 1.2 Random Variables | p. 6 |
| 1.2.1 Basic Concepts | p. 6 |
| 1.2.2 Discrete Random Variables | p. 9 |
| 1.2.2.1 Numerical Parameter | p. 9 |
| 1.2.2.2 Important Discrete Probability Distributions | p. 10 |
| 1.2.3 Continuous Random Variables | p. 14 |
| 1.2.3.1 Probability Density and Numerical Parameter | p. 14 |
| 1.2.3.2 Important Continuous Probability Distributions | p. 16 |
| 1.2.4 Mixtures of Random Variables | p. 22 |
| 1.2.5 Functions of Random Variables | p. 26 |
| 1.3 Transformation of Probability Distributions | p. 28 |
| 1.3.1 z-Transformation | p. 29 |
| 1.3.2 Laplace-Transformation | p. 31 |
| 1.4 Classes of Probability Distributions Based on Aging Behaviour | p. 35 |
| 1.5 Order Relations Between Random Variables | p. 43 |
| 1.6 Multidimensional Random Variables | p. 46 |
| 1.6.1 Basic Concepts | p. 46 |
| 1.6.2 Two-Dimensional Random Variables | p. 47 |
| 1.6.2.1 Discrete Components | p. 47 |
| 1.6.2.2 Continuous Components | p. 48 |
| 1.6.3 n-Dimensional Random Variables | p. 57 |
| 1.7 Sums of Random Variables | p. 62 |
| 1.7.1 Sums of Discrete Random Variables | p. 62 |
| 1.7.2 Sums of Continuous Random Variables | p. 63 |
| 1.7.3 Sums of a Random Number of Random Variables | p. 68 |
| 1.8 Inequalities in Probability Theory | p. 70 |
| 1.8.1 Inequalities for Probabilities | p. 70 |
| 1.8.2 Inequalities for Moments | p. 72 |
| 1.9 Limit Theorems | p. 73 |
| 1.9.1 Convergence Criteria for Sequences of Random Variables | p. 73 |
| 1.9.2 Laws of Large Numbers | p. 74 |
| 1.9.3 Central Limit Theorem | p. 76 |
| 1.10 Exercises | p. 81 |
| 2 Basics of Stochastic Processes | |
| 2.1 Motivation and Terminology | p. 91 |
| 2.2 Characteristics and Examples | p. 95 |
| 2.3 Classification of Stochastic Processes | p. 99 |
| 2.4 Exercises | p. 105 |
| 3 Random Point Processes | |
| 3.1 Basic Concepts | p. 107 |
| 3.2 Poisson Processes | p. 113 |
| 3.2.1 Homogeneous Poisson Processes | p. 113 |
| 3.2.1.1 Definition and Properties | p. 113 |
| 3.2.1.2 Homogeneous Poisson Process and Uniform Distribution | p. 119 |
| 3.2.2 Nonhomogeneous Poisson Processes | p. 126 |
| 3.2.3 Mixed Poisson Processes | p. 130 |
| 3.2.4 Superposition and Thinning of Poisson Processes | p. 136 |
| 3.2.4.1 Superposition | p. 136 |
| 3.2.4.2 Thinning | p. 137 |
| 3.2.5 Compound Poisson Processes | p. 140 |
| 3.2.6 Applications to Maintenance | p. 142 |
| 3.2.6.1 Nonhomogeneous Poisson Process and Minimal Repair | p. 142 |
| 3.2.6.2 Standard Replacement Policies with Minimal Repair | p. 144 |
| 3.2.6.3 Replacement Policies for Systems with Two Failure Types | p. 147 |
| 3.2.6.4 Repair Cost Limit Replacement Policies | p. 149 |
| 3.3 Renewal Processes | p. 155 |
| 3.3.1 Definitions and Examples | p. 155 |
| 3.3.2 Renewal Function | p. 158 |
| 3.3.2.1 Renewal Equations | p. 158 |
| 3.3.2.2 Bounds on the Renewal Function | p. 164 |
| 3.3.3 Asymptotic Behaviour | p. 166 |
| 3.3.4 Recurrence Times | p. 170 |
| 3.3.5 Stationary Renewal Processes | p. 173 |
| 3.3.6 Alternating Renewal Processes | p. 175 |
| 3.3.7 Compound Renewal Processes | p. 179 |
| 3.3.7.1 Definition and Properties | p. 179 |
| 3.3.7.2 First Passage Time | p. 183 |
| 3.3.8 Regenerative Stochastic Processes | p. 185 |
| 3.4 Applications to Actuarial Risk Analysis | p. 188 |
| 3.4.1 Basic Concepts | p. 188 |
| 3.4.2 Poisson Claim Arrival Process | p. 190 |
| 3.4.3 Renewal Claim Arrival Process | p. 196 |
| 3.4.4 Normal Approximations for Risk Processes | p. 198 |
| 3.5 Exercises | p. 200 |
| 4 Markov Chains in Discrete Time | |
| 4.1 Foundations and Examples | p. 207 |
| 4.2 Classification of States | p. 214 |
| 4.2.1 Closed Sets of States | p. 214 |
| 4.2.2 Equivalence Classes | p. 215 |
| 4.2.3 Periodicity | p. 218 |
| 4.2.4 Recurrence and Transience | p. 220 |
| 4.3 Limit Theorems and Stationary Distribution | p. 226 |
| 4.4 Birth-and Death Processes | p. 231 |
| 4.5 Exercises | p. 233 |
| 5 Markov Chains in Continuos Time | |
| 5.1 Basic Concepts and Examples | p. 239 |
| 5.2 Transition Probabilities and Rates | p. 243 |
| 5.3 Stationary State Probabilities | p. 252 |
| 5.4 Sojourn Times in Process States | p. 255 |
| 5.5 Construction of Markov Systems | p. 257 |
| 5.6 Birth-and Death Processes | p. 261 |
| 5.6.1 Birth Processes | p. 261 |
| 5.6.2 Death Processes | p. 264 |
| 5.6.3 Birth and Death Processes | p. 266 |
| 5.6.3.1 Time-Dependent State Probabilities | p. 266 |
| 5.6.3.2 Stationary State Probabilities | p. 274 |
| 5.6.3.3 Inhomogeneous Birth-and Death Processes | p. 277 |
| 5.7 Applications to Queueing Models | p. 281 |
| 5.7.1 Basic Concepts | p. 281 |
| 5.7.2 Loss Systems | p. 283 |
| 5.7.2.1 M/M/[infin]-Systems | p. 283 |
| 5.7.2.2 M/M/s/O-Systems | p. 284 |
| 5.7.2.3 Engset's Loss System | p. 286 |
| 5.7.3 Waiting Systems | p. 287 |
| 5.7.3.1 M/M/s/[infin]-Systems | p. 287 |
| 5.7.3.2 M/G/1/[infin]-Systems | p. 290 |
| 5.7.3.3 G/M/1/[infin]-Systems | p. 293 |
| 5.7.4 Waiting-Loss-Systems | p. 294 |
| 5.7.4.1 M/M/s/m-System | p. 294 |
| 5.7.4.2 M/M/s/[infinity]-System with Impatient Customers | p. 296 |
| 5.7.5 Special Single-Server Systems | p. 298 |
| 5.7.5.1 System with Priorities | p. 298 |
| 5.7.5.2 M/M/1/m-System with Unreliable Server | p. 300 |
| 5.7.6 Networks of Queueing Systems | p. 303 |
| 5.7.6.1 Introduction | p. 303 |
| 5.7.6.2 Open Queueing Networks | p. 303 |
| 5.7.6.3 Closed Queueing Networks | p. 310 |
| 5.8 Semi-Markov Chains | p. 314 |
| 5.9 Exercises | p. 321 |
| 6 Martingales | |
| 6.1 Discrete-Time Martingales | p. 331 |
| 6.1.1 Definition and Examples | p. 331 |
| 6.1.2 Doob-Type Martingales | p. 336 |
| 6.1.3 Martingale Stopping Theorem and Applications | p. 340 |
| 6.1.4 Inequalities for Discrete-Time Martingales | p. 344 |
| 6.2 Continuous-Time Martingales | p. 345 |
| 6.3 Exercises | p. 349 |
| 7 Brownian Motion | |
| 7.1 Introduction | p. 351 |
| 7.2 Properties of the Brownian Motion | p. 353 |
| 7.3 Multidimensional and Conditional Distributions | p. 357 |
| 7.4 First Passage Times | p. 359 |
| 7.5 Transformations of the Brownian Motion | p. 366 |
| 7.5.1 Identical Transformations | p. 366 |
| 7.5.2 Reflected Brownian Motion | p. 367 |
| 7.5.3 Geometric Brownian Motion | p. 368 |
| 7.5.4 Ornstein-Uhlenbeck Process | p. 369 |
| 7.5.5 Brownian Motion with Drift | p. 370 |
| 7.5.5.1 Definitions and First Passage Times | p. 370 |
| 7.5.5.2 Application to Option Pricing | p. 374 |
| 7.5.5.3 Application to Maintenance | p. 379 |
| 7.5.5.4 Point Estimation for Brownian Motion with Drift | p. 384 |
| 7.5.6 Integral Transformations | p. 387 |
| 7.5.6.1 Integrated Brownian Motion | p. 387 |
| 7.5.6.2 White Noise | p. 389 |
| 7.6 Exercises | p. 392 |
| Answers to Selected Exercises | p. 397 |
| References | p. 405 |
| Index | p. 411 |
