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2012 | Book

Markov Chains Read chapter Read first chapter

Essentials of Stochastic Processes

Author: Richard Durrett

Publisher: Springer New York

Book Series : Springer Texts in Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Markov Chains
Abstract
The importance of Markov chains comes from two facts: (i) there are a large number of physical, biological, economic, and social phenomena that can be modeled in this way, and (ii) there is a well-developed theory that allows us to do computations. We begin with a famous example, then describe the property that is the defining feature of Markov chains
Richard Durrett
Chapter 2. Poisson Processes
Abstract
To prepare for our discussion of the Poisson process, we need to recall the definition and some of the basic properties of the exponential distribution. A random variable T is said to have an exponential distribution with rate λ, or T = exponential(λ), if
$$\begin{array}{rcl} P(T \leq t) = 1 - {e}^{-\lambda t}\quad \mbox{ for all }t \geq 0& &\end{array}$$
(2.1)
Here we have described the distribution by giving the distribution function F(t) = P(T ≤ t). We can also write the definition in terms of the density function f T (t) which is the derivative of the distribution function.
$$\begin{array}{rcl}{ f}_{T}(t) = \left \{\begin{array}{@{}l@{\quad }l@{}} \lambda {e}^{-\lambda t}\quad &\mbox{ for }t \geq 0 \\ 0 \quad &\mbox{ for }t < 0 \end{array} \right.& &\end{array}$$
(2.2)
Richard Durrett
Chapter 3. Renewal Processes
Abstract
In the Poisson process the times between successive arrivals are independent and exponentially distributed. The lack of memory property of the exponential distribution is crucial for many of the special properties of the Poisson process derived in this chapter. However, in many situations the assumption of exponential interarrival times is not justified. In this section we will consider a generalization of Poisson processes called renewal processes in which the times t 1, t 2, … between events are independent and have distribution F.
Richard Durrett
Chapter 4. Continuous Time Markov Chains
Abstract
In Chap. 1 we considered Markov chains X n with a discrete time index n = 0, 1, 2, … In this chapter we will extend the notion to a continuous time parameter t ≥ 0, a setting that is more convenient for some applications.
Richard Durrett
Chapter 5. Martingales
Abstract
In this chapter we will introduce a class of process that can be thought of as the fortune of a gambler betting on a fair game. These results will be important when we consider applications to finance in the next chapter. In addition, they will allow us to give more transparent proofs of some facts from Chap. 1 concerning exit distributions and exit times for Markov chains.
Richard Durrett
Chapter 6. Mathematical Finance
Abstract
To warm up for the developments in the next section we will look at two simple concrete examples under the unrealistic assumption that the interest rate is 0.
Richard Durrett
Backmatter
Metadata
Title
Essentials of Stochastic Processes
Author
Richard Durrett
Copyright Year
2012
Publisher
Springer New York
Electronic ISBN
978-1-4614-3615-7
Print ISBN
978-1-4614-3614-0
DOI
https://doi.org/10.1007/978-1-4614-3615-7

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