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2nd Edition

Introduction to Stochastic Processes




ISBN 9781584886518
Published May 16, 2006 by Chapman and Hall/CRC
248 Pages 13 B/W Illustrations

 
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Book 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

    ...
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    Author(s)

    Biography

    Greogory F. Lawler