Stochastic processes in science, engineering, and finance

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. Front Matter Title Page 2 Preface 5 Acknowledgment 6 Short Biography of Frank E. Beichelt 7 Symbols and Abbreviations 8 Probability Theory 8 Stochastic Processes 8 Table of Contents 10 1 – Probability Theory 14 1.1 Random Events and Their Probabilities 14 1.2 Random Variables 19 1.2.1 Basic Concepts 19 1.2.2 Discrete Random Variables 22 1.2.2.1 Numerical Parameters 22 1.2.2.2 Important Discrete Probability Distributions 23 1.2.3 Continuous Random Variables 27 1.2.3.1 Probability Density and Numerical Parameters 27 1.2.3.2 Important Continuous Probability Distributions 29 1.2.4 Mixtures of Random Variables 35 1.2.5 Functions of a Random Variable 39 1.3 Transformation of Probability Distributions 41 1.3.1 z-Transformation 42 1.3.2 Laplace Transformation 44 1.4 Classes of Probability Distributions Based on Aging Behaviour 48 1.5 Order Relations between Randon Variables 56 1.6 Multidimensional Random Variables 59 1.6.1 Basic Concepts 59 1.6.2 Two-Dimensional Random Vectors 60 1.6.2.1 Discrete Components 60 1.6.2.2 Continuous Components 61 1.6.3 n-Dimensional Random Variables 70 1.7 Sums of Random Variables 75 1.7.1 Sums of Discrete Random Variables 75 1.7.2 Sums of Continuous Random Variables 76 1.7.3 Sums of a Random Number of Random Variables 81 1.8 Inequalities in Probability Theory 83 1.8.1 Inequalities for Probabilities 83 1.8.2 Inequalities for Moments 85 1.9 Limit Theorems 86 1.9.1 Convergence Criteria for Sequences of Random Variables 86 1.9.2 Laws of Large Numbers 87 1.9.3 Central Limit Theorem 89 1.10 Exercises 94 Sections 1.1 to section 1.3 94 Sections 1.4 and 1.5 100 Sections 1.6 to 1.9 101 2 – Basics of Stochastic Processes 104 2.1 Motivations and Terminology 104 2.2 Characteristics and Examples 108 2.3 Classification of Stochastic Processes 112 2.4 Exercises 118 3 – Random Point Processes 120 3.1 Basic Concepts 120 3.2 Poisson Processes 126 3.2.1 Homogeneous Poisson Processes 126 3.2.1.1 Definition and Properties 126 3.2.1.2 Homogeneous Poisson Process and Uniform Distribution 132 3.2.2 Nonhomogeneous Poisson Processses 139 3.2.3 Mixed Poisson Processes 143 3.2.4 Superposition and Thinning of Poisson Processes 149 3.2.4.1 Superposition 149 3.2.4.2 Thinning 150 3.2.5 Compound Poisson Processes 153 3.2.6 Applications to Maintenance 155 3.2.6.1 Nonhomogeneous Poisson Process and Minimal Repair 155 3.2.6.2 Standard Replacement Policies with Minimal Repair 157 3.2.6.3 Replacement Policies for Systems with two Failure Types 160 3.2.6.4 Repair Cost Limit Replacement Policies with Minimal Repair 162 3.3 Renewal Processes 168 3.3.1 Definitions and Examples 168 3.3.2 Renewal Function 171 3.3.2.1 Renewal Equations 171 3.3.2.2 Bounds on the Renewal Function 177 3.3.3 Asymptotic Behaviour 179 3.3.4 Recurrence Times 183 3.3.5 Stationary Renewal Processes 186 3.3.6 Alternating Renewal Processes 188 3.3.7 Compound Renewal Processes 192 3.3.7.1 Definition and Properties 192 3.3.7.2 First Passage Time 196 3.3.8 Regenerative Stochastic Processes 198 3.4 Applications to Actuarial Risk Analysis 201 3.4.1 Basic Concepts 201 3.4.2 Poisson Claim Arrival Process 203 3.4.3 Renewal Claim Arrival Process 209 3.4.4 Normal Approximations for Risk Processes 211 3.5 Exercises 213 Sections 3.1 and 3.2 213 Sections 3.3 and 3.4 216 4 – Discrete-Time Markov Chains 220 4.1 Foundations and Examples 220 4.2 Classification of States 227 4.2.1 Closed Sets of States 227 4.2.2 Equivalence Classes 228 4.2.3 Periodicity 231 4.2.4 Recurrence and Transience 233 4.3 Limit Theorems and Stationary Distribution 239 4.4 Birth and Death Processes 244 4.5 Exercises 246 5 – Continuous-Time Markov Chains 252 5.1 Basic Concepts and Examples 252 5.2 Transition Probabilities and Rates 256 5.3 Stationary State Probabilities 265 5.4 Sojourn Times in Process States 268 5.5 Construction of Markov Systems 270 5.6 Birth and Death Processes 274 5.6.1 Birth Processes 274 5.6.2 Death Processes 277 5.6.3 Birth and Death Processes 279 5.6.3.1 Time-Dependent State Probabilities 279 5.6.3.2 Stationary State Probabilities 287 5.6.3.3 Nonhomogeneous Birth and Death Processes 290 5.7 Applications to Queueing Models 294 5.7.1 Basic Concepts 294 5.7.2 Loss Systems 296 5.7.2.1 M/M/∞-System 296 5.7.2.2 M/M/s/0-System 297 5.7.2.3 Engset's Loss System 299 5.7.3 Waiting Systems 300 5.7.3.1 M/M/s/∞ - System 300 5.7.3.2 M/G/1/∞ - System 303 5.7.3.3 GI/M/1/∞ - System 306 5.7.4 Waiting-Loss Systems 307 5.7.4.1 M/M/s/m - System 307 5.7.4.2 M/M/s/∞-System with Impatient Customers 309 5.7.5 Special Single-Server Queueing Systems 311 5.7.5.1 System with Priorities 311 5.7.5.2 M/M/1/m-System with Unreliable Server 313 5.7.6 Networks of Queueing Systems 316 5.7.6.1 Introduction 316 5.7.6.2 Open Queueing Networks 316 5.7.6.3 Closed Queueing Networks 323 5.8 Semi-Markov Chains 327 5.9 Exercises 334 6 – Martingales 344 6.1 Discrete-Time Martingales 344 6.1.1 Definition and Examples 344 6.1.2 Doob-Martingales 349 6.1.3 Martingale Stopping Theorem and Applications 353 6.1.4 Inequalities for Discrete-Time Martingales 357 6.2 Continous-Time Martingales 358 6.3 Exercises 362 7 – Brownian Motion 364 7.1 Introduction 364 7.2 Properties of the Brownian Motion 366 7.3 Multidimensional and Conditional Distributions 370 7.4 First Passage Times 372 7.5 Transformations of the Brownian Motion 379 7.5.1 Identical Transformations 379 7.5.2 Reflected Brownian Motion 380 7.5.3 Geometric Brownian Motion 381 7.5.4 Ornstein-Uhlenbeck Process 382 7.5.5 Brownian Motion with Drift 383 7.5.5.1 Definitions and First Passage Times 383 7.5.5.2 Application to Option Pricing 387 7.5.5.3 Application to Maintenance 392 7.5.5.4 Point Estimation for the Brownian Motion with Drift 397 7.5.6 Integral Transformations 400 7.5.6.1 Integrated Brownian Motion 400 7.5.6.2 White Noise 402 7.6 Exercises 405 Answers to Selected Exercises 409 Chapter 1 409 Chapter 2 411 Chapter 3 411 Chapter 4 412 Chapter 5 413 Chapter 6 416 Chapter 7 416 References 417 "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." "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."--Jacket. Introduces stochastic processes with emphasis on their applications in science, engineering, finance, computer science, and operations research. This book includes exercises that reinforce the concepts and techniques discussed and allow readers to assess their understanding.