Essentials of Stochastic Processes / Edition 2

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Overview

This book is for a first course on stochastic processes to be taken by undergraduates or masters students who have had a course in probability theory, but who have not had a course in measure theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal theory, and Brownian motion and martingales. The last two topics are important for the brief treatment of option pricing.. "The book presents only the essentials of the subject, the parts of the theory most important for applications. To allow readers to choose their own level of detail, many of the proofs begin with a nonrigorous answer to the question "Why is this true?" followed by a proof that fills in the missing details.
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Editorial Reviews

From the Publisher

This book is an introduction to stochastic processes written for undergraduates or beginning grad. students who have had a previous course in probability. Durrett has sketched a 25-page review of probability at the beginning of the book, which includes many examples and some challenging exercises. The rest of the book covers discrete and continuous time Markov chains, Poisson processes, Brownian motion, and some renewal theory, including material on queuing networks, spatial Poisson processes, and a fine chapter on martingales which treats optional sampling and forms a good basis for later study of Brownian motion and applications to option pricing and the Black-Scholes formula.

Durrett wants his readers to be able to use stochastic processes to solve problems. He presents numerous examples to motivate and develop skills. Examples are explained in detail, sometimes including more than one solution. After stating a result, he frequently asks "Why is this true?" and then sketches a proof or offers an intuitive answer in order to develop the reader's insight (and to allow application-oriented readers to skip the details of a formal proof). He sometimes postpones or skips rigorous proofs so he can develop and apply the theory more quickly. A reader with applications in mind, especially one already familiar with the theory, will appreciate these features.

--Mathematical Reviews

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Product Details

  • ISBN-13: 9781461436140
  • Publisher: Springer New York
  • Publication date: 5/23/2012
  • Series: Springer Texts in Statistics Series
  • Edition description: 2nd ed. 2012
  • Edition number: 2
  • Pages: 266
  • Product dimensions: 6.20 (w) x 9.20 (h) x 0.80 (d)

Meet 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.

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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
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