Introduction to Stochastic Processes: 2nd Edition (Hardback) book cover

Introduction to Stochastic Processes

2nd Edition

By Gregory F. Lawler

Chapman and Hall/CRC

248 pages | 13 B/W Illus.

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Hardback: 9781584886518
pub: 2006-05-16
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Description

Emphasizing fundamental mathematical ideas rather than proofs, Introduction to Stochastic Processes, Second Edition provides quick access to important foundations of probability theory applicable to problems in many fields. Assuming that you have a reasonable level of computer literacy, the ability to write simple programs, and the access to software for linear algebra computations, the author approaches the problems and theorems with a focus on stochastic processes evolving with time, rather than a particular emphasis on measure theory.

For those lacking in exposure to linear differential and difference equations, the author begins with a brief introduction to these concepts. He proceeds to discuss Markov chains, optimal stopping, martingales, and Brownian motion. The book concludes with a chapter on stochastic integration. The author supplies many basic, general examples and provides exercises at the end of each chapter.

New to the Second Edition:

  • Expanded chapter on stochastic integration that introduces modern mathematical finance

  • Introduction of Girsanov transformation and the Feynman-Kac formula

  • Expanded discussion of Itô's formula and the Black-Scholes formula for pricing options

  • New topics such as Doob's maximal inequality and a discussion on self similarity in the chapter on Brownian motion

    Applicable to the fields of mathematics, statistics, and engineering as well as computer science, economics, business, biological science, psychology, and engineering, this concise introduction is an excellent resource both for students and professionals.

  • Table of Contents

    Preface to Second Edition

    Preface to First Edition

    PRELIMINARIES

    Introduction

    Linear Differential Equations

    Linear Difference Equations

    Exercises

    FINITE MARKOV CHAINS

    Definitions and Examples

    Large-Time Behavior and Invariant Probability

    Classification of States

    Return Times

    Transient States

    Examples

    Exercises

    COUNTABLE MARKOV CHAINS

    Introduction

    Recurrence and Transience

    Positive Recurrence and Null Recurrence

    Branching Process

    Exercises

    CONTINUOUS-TIME MARKOV CHAINS

    Poisson Process

    Finite State Space

    Birth-and-Death Processes

    General Case

    Exercises

    OPTIMAL STOPPING

    Optimal Stopping of Markov Chains

    Optimal Stopping with Cost

    Optimal Stopping with Discounting

    Exercises

    MARTINGALES

    Conditional Expectation

    Definition and Examples

    Optional Sampling Theorem

    Uniform Integrability

    Martingale Convergence Theorem

    Maximal Inequalities

    Exercises

    RENEWAL PROCESSES

    Introduction

    Renewal Equation

    Discrete Renewal Processes

    M/G/1 and G/M/1 Queues

    Exercises

    REVERSIBLE MARKOV CHAINS

    Reversible Processes

    Convergence to Equilibrium

    Markov Chain Algorithms

    A Criterion for Recurrence

    Exercises

    BROWNIAN MOTION

    Introduction

    Markov Property

    Zero Set of Brownian Motion

    Brownian Motion in Several Dimensions

    Recurrence and Transience

    Fractal Nature of Brownian Motion

    Scaling Rules

    Brownian Motion with Drift

    Exercises

    STOCHASTIC INTEGRATION

    Integration with Respect to Random Walk

    Integration with Respect to Brownian Motion

    Itô's Formula

    Extensions if Itô's Formula

    Continuous Martingales

    Girsanov Transformation

    Feynman-Kac Formula

    Black-Scholes Formula

    Simulation

    Exercises

    Suggestions for Further Reading

    Index

    About the Series

    Chapman & Hall/CRC Probability Series

    Learn more…

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General
    MAT029010
    MATHEMATICS / Probability & Statistics / Bayesian Analysis