Building upon the previous editions, this textbook is a first course in stochastic processes taken by undergraduate and graduate students (MS and PhD students from math, statistics, economics, computer science, engineering, and finance departments) who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and option pricing. One can only learn a subject by seeing it in action, so there are a large number of examples and more than 300 carefully chosen exercises to deepen the reader's understanding.
Drawing from teaching experience and student feedback, there are many new examples and problems with solutions that use TI-83 to eliminate the tedious details of solving linear equations by hand, and the collection of exercises is much improved, with many more biological examples. Originally included in previous editions, material too advanced for this first course in stochastic processes has been eliminated while treatment of other topics useful for applications has been expanded. In addition, the ordering of topics has been improved; for example, the difficult subject of martingales is delayed until its usefulness can be applied in the treatment of mathematical finance.
Richard Durrett received his Ph.D. in Operations Research from Stanford in 1976. He taught at the UCLA mathematics department for 9 years and at Cornell for 25 years before moving to Duke in 2010. He is author of 8 books and more than 200 journal articles and has supervised more that 45 Ph.D. students. He is a member of the National Academy of Science. Most of his current research concerns the applications of probability to biology: ecology, genetics, and most recently cancer.
1) Markov Chains1.1 Definitions and Examples1.2 Multistep Transition Probabilities1.3 Classification of States 1.4 Stationary Distributions1.4.1 Doubly stochastic chains1.5 Detailed balance condition1.5.1 Reversibility 1.5.2 The Metropolis-Hastings algorithm1.5.3 Kolmogorow cycle condition 1.6 Limit Behavior 1.7 Returns to a fixed state 1.8 Proof of the convergence theorem*1.9 Exit Distributions 1.10 Exit Times1.11 Infinite State Spaces* 1.12 Chapter Summary1.13 Exercises 2) Poisson Processes 2.1 Exponential Distribution 2.2 Defining the Poisson Process2.2.1 Constructing the Poisson Process2.2.2 More realistic models2.3 Compound Poisson Processes 2.4 Transformations2.4.1 Thinning 2.4.2 Superposition2.4.3 Conditioning2.5 Chapter Summary2.6 Exercises 3) Renewal Processes3.1 Laws of Large Numbers3.2 Applications to Queueing Theory3.2.1 GI/G/1 queue3.2.2 Cost equations 3.2.3 M/G/1 queue3.3 Age and Residual Life*3.3.1 Discrete case3.3.2 General case 3.4 Chapter Summary 3.5 Exercises 4) Continuous Time Markov Chains 4.1 Definitions and Examples4.2 Computing the Transition Probability4.2.1 Branching Processes 4.3 Limiting Behavior 4.3.1 Detailed balance condition 4.4 Exit Distributions and Exit Times 4.5 Markovian Queues 4.5.1 Single server queues4.5.2 Multiple servers4.5.3 Departure Processes 4.6 Queueing Networks*4.7 Chapter Summary4.8 Exercises 5) Martingales 5.1 Conditional Expectation 5.2 Examples5.3 Gambling Strategies, Stopping Times 5.4 Applications 5.4.1 Exit distributions5.4.2 Exit times 5.4.3 Extinction and ruin probabilities5.4.4 Positive recurrence of the GI/G/1 queue*5.5 Exercises 6) Mathematical Finance6.1 Two Simple Examples6.2 Binomial Model 6.3 Concrete Examples 6.4 American Options6.5 Black-Scholes formula6.6 Calls and Puts6.7 Exercises A) Review of Probability A.1 Probabilities, Independence A.2 Random Variables, Distributions A.3 Expected Value, MomentsA.4 Integration to the Limit