Essentials of Stochastic Processes

ISBN-13: 9783319456133 / Angielski / Twarda / 2016 / 275 str.

Richard Durrett

Opis

This test is designed for a Master's Level course in stochastic processes. It features the introduction and use of martingales, which allow one to do much more with Brownian motion, e.g., option pricing, and queueing theory is integrated into the Continuous Time Markov Chain and Renewal Theory chapters as examples.

Recenzje

"It is the 3rd edition of the textbook devoted to initial information and basic topics from the theory of stochastic processes. ... The book is very useful for anyone who is interested in probability theory and its ramifications and applications. It can be recommended both for students and postgraduates, teachers and practitioners. ... The book contains a lot of examples which contribute to a better understanding of the text." (Yuliya S. Mishura, zbMATH 1378.60001, 2018)

"This is the third edition of a popular textbook on stochastic processes. It is intended for advanced undergraduates and beginning graduate students and aimed at an intermediate level between an undergraduate course in probability and the first graduate course that uses measure theory." (William J. Satzer, MAA Reviews, maa.org, February, 2017)

Spis treści

1) Markov Chains

1.1 Definitions and Examples

1.2 Multistep Transition Probabilities

1.3 Classification of States 

1.4 Stationary Distributions

1.4.1 Doubly stochastic chains

1.5 Detailed balance condition

1.5.1 Reversibility 

1.5.2 The Metropolis-Hastings algorithm

1.5.3 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 Times

1.11 Infinite State Spaces* 

1.12 Chapter Summary

1.13 Exercises

2) Poisson Processes 

2.1 Exponential Distribution 

2.2 Defining the Poisson Process

2.2.1 Constructing the Poisson Process

2.2.2 More realistic models

2.3 Compound Poisson Processes 

2.4 Transformations

2.4.1 Thinning 

2.4.2 Superposition

2.4.3 Conditioning

2.5 Chapter Summary

2.6 Exercises 

3) Renewal Processes

3.1 Laws of Large Numbers

3.2 Applications to Queueing Theory

3.2.1 GI/G/1 queue

3.2.2 Cost equations 

3.2.3 M/G/1 queue

3.3 Age and Residual Life*

3.3.1 Discrete case

3.3.2 General case 

3.4 Chapter Summary 

3.5 Exercises

4) Continuous Time Markov Chains 

4.1 Definitions and Examples

4.2 Computing the Transition Probability

4.2.1 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 queues

4.5.2 Multiple servers

4.5.3 Departure Processes 

4.6 Queueing Networks*

4.7 Chapter Summary

4.8 Exercises 

5) Martingales 

5.1 Conditional Expectation 

5.2 Examples

5.3 Gambling Strategies, Stopping Times 

5.4 Applications 

5.4.1 Exit distributions

5.4.2 Exit times 

5.4.3 Extinction and ruin probabilities

5.4.4 Positive recurrence of the GI/G/1 queue*

5.5 Exercises

6) Mathematical Finance

6.1 Two Simple Examples

6.2 Binomial Model 

6.3 Concrete Examples 

6.4 American Options

6.5 Black-Scholes formula

6.6 Calls and Puts

6.7 Exercises

A) Review of Probability 

A.1 Probabilities, Independence 

A.2 Random Variables, Distributions 

A.3 Expected Value, Moments

A.4 Integration to the Limit 

O autorze

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.

Blurb

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.
 • A concise treatment and textbook on the most important topics in Stochastic Processes
 • Illustrates all concepts with examples and presents more than 300 carefully chosen exercises for effective learning
 • New edition includes added and revised exercises, including many biological exercises, in addition to restructured and rewritten sections with a goal toward clarity and simplicity

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 cancer modeling.