Essentials of Stochastic Processes

by Richard Durrett

Overview

This book is for a first course in stochastic processes taken by undergraduates or master’s students who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and mathematical finance. 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

 

The book has undergone a thorough revision since the first edition. There are many new examples and problems with solutions that use the TI-83 to eliminate the tedious details of solving linear equations by hand. Some material that was too advanced for the level has been eliminated while the 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 seen in the treatment of mathematical finance.

 

Richard Durrett received his Ph.D. in Operations Research from Stanford in 1976. He taught at the UCLA math department for nine years and at Cornell for twenty-five before moving to Duke in 2010. He is the author of 8 books and almost 200 journal articles, and has supervised more that 40 Ph.D. students. Most of his current research concerns the applications of probability to biology: ecology, genetics, and most recently cancer.

Product Details

ISBN-13:	9781461436157
Publisher:	Springer-Verlag New York, LLC
Publication date:	05/19/2012
Series:	Springer Texts in Statistics
Sold by:	Barnes & Noble
Format:	eBook
File size:	6 MB

About the Author

Richard Durrett received his Ph.D. in Operations Research from Stanford in 1976. He taught at the UCLA math department for nine years and at Cornell for twenty-five before moving to Duke in 2010. He is the author of 8 books and almost 200 journal articles, and has supervised more that 40 Ph.D. students. Most of his current research concerns the applications of probability to biology: ecology, genetics, and most recently cancer.

Table of Contents

Review of Probability
--Probabilities, Independence
--Random Variables, Distributions
--Expected Value, Moments

1   Markov Chains
--Definitions and Examples
--Multi-step Transition Probabilities
--Classification of States
--Limit Behavior
--Some Special Examples
--One Step Calculations
--Infinite State Spaces
--Proofs of Convergence Theorems
--Exercises

2   Martingales
--Conditional Expectation
--Examples of Martingales
--Optional Stopping Theorem
--Applications
--Exercises

3   Poisson Processes
--Exponential Distribution
--Defining the Poisson Process
--Compund Poisson Processes
--Thinning and Superposition
--Conditioning
--Spatial Poisson Processes
--Exercises

4   Continuous-time Markov Chains
--Definitions and Examples
--Computing the Transition Probability
--Limiting Behavior
--Queueing Chains
--Reversibility
--Queueing Networks
--Closed Queueing Networks
--Exercises

5   Renewal Theory
--Basic Definitions
--Laws of Large Numbers
--Applications to Queueing Theory
--Age and Residual Life
--Exercises

6   Brownian Motion
--Basic Definitions
--Markov Property, Reflection Principle
--Martingales, Hitting Times
--Option Pricing in Discrete Time
--The Black-Scholes Formula
--Exercises

Editorial Reviews

It is the 3rd edition of the textbook devoted to initial information and basic topics from the theory of shastic 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 shastic 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)

From the Publisher