Brownian Motion and Stochastic Calculus

  • Ioannis Karatzas
  • Steven E. Shreve

Part of the Graduate Texts in Mathematics book series (GTM, volume 113)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Ioannis Karatzas, Steven E. Shreve
    Pages 1-46
  3. Ioannis Karatzas, Steven E. Shreve
    Pages 47-127
  4. Ioannis Karatzas, Steven E. Shreve
    Pages 128-238
  5. Ioannis Karatzas, Steven E. Shreve
    Pages 239-280
  6. Ioannis Karatzas, Steven E. Shreve
    Pages 281-398
  7. Ioannis Karatzas, Steven E. Shreve
    Pages 399-446
  8. Back Matter
    Pages 447-470

About this book

Introduction

This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization).

This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.

Keywords

YellowSale2006 adopted-textbook Brownian motion differential equation integration local time Markov process Martingal Martingale measure probability Semimartingal Semimartingale stochastic calculus stochastic differential equation stochastic process stochastic processes

Authors and affiliations

  • Ioannis Karatzas
    • 1
  • Steven E. Shreve
    • 2
  1. 1.Departments of Mathematics and StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0949-2
  • Copyright Information Springer Science+Business Media, Inc. 1998
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-97655-6
  • Online ISBN 978-1-4612-0949-2
  • Series Print ISSN 0072-5285
  • About this book