Essentials of Stochastic Processes

by Richard Durrett

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

Genres: Mathematics

288 pages, Paperback

First published December 1, 2010

About the author

Richard Timothy Durrett is a mathematician known for his research and books on mathematical probability theory, stochastic processes and their application to mathematical ecology and population genetics.

He received his BS and MS at Emory University in 1972 and 1973 and his Ph.D. at Stanford University in 1976 under advisor Donald Iglehart. From 1985 until 2010 was on the faculty at Cornell University. Since 2010, Durrett has been a professor at Duke University.

He was elected to the United States National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society.[1]

Durrett is the founder of the Cornell Probability Summer Schools, and he is still its scientific organizer.