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Introduction to Stochastic Integration

  • Textbook
  • © 2006

Overview

Authors:
  1. Hui-Hsiung Kuo

    1. Department of Mathematics, Louisiana State University, Baton Rouge, USA

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  • Provides a concise introduction to the theory of stochastic integration, also called the Ito calculus
  • Closes the gap between more technically advanced books like Karatzas and Shreve (Springer) and less rigourous but more intuitive approaches such as that of Thomas Mikosch (World Scientific)
  • Each chapter includes a variety of exercises designed to help the reader further understand the material
  • Contains an unusually diverse selection of examples, and an attractive selection of topics
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    Pages 1-6

  3. Brownian Motion

    Pages 7-22

  4. Constructions of Brownian Motion

    Pages 23-36

  5. Stochastic Integrals

    Pages 37-60

  6. An Extension of Stochastic Integrals

    Pages 61-74

  7. Stochastic Integrals for Martingales

    Pages 75-92

  8. The Itô Formula

    Pages 93-114

  9. Applications of the Itô Formula

    Pages 115-146

  10. Multiple Wiener-Itô Integrals

    Pages 147-184

  11. Stochastic Differential Equations

    Pages 185-230

  12. Some Applications and Additional Topics

    Pages 231-266

  13. Back Matter

    Pages 267-278
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Keywords

About this book

In the Leibniz–Newton calculus, one learns the di?erentiation and integration of deterministic functions. A basic theorem in di?erentiation is the chain rule, which gives the derivative of a composite of two di?erentiable functions. The chain rule, when written in an inde?nite integral form, yields the method of substitution. In advanced calculus, the Riemann–Stieltjes integral is de?ned through the same procedure of “partition-evaluation-summation-limit” as in the Riemann integral. In dealing with random functions such as functions of a Brownian motion, the chain rule for the Leibniz–Newton calculus breaks down. A Brownian motionmovessorapidlyandirregularlythatalmostallofitssamplepathsare nowhere di?erentiable. Thus we cannot di?erentiate functions of a Brownian motion in the same way as in the Leibniz–Newton calculus. In 1944 Kiyosi Itˆ o published the celebrated paper “Stochastic Integral” in the Proceedings of the Imperial Academy (Tokyo). It was the beginning of the Itˆ o calculus, the counterpart of the Leibniz–Newton calculus for random functions. In this six-page paper, Itˆ o introduced the stochastic integral and a formula, known since then as Itˆ o’s formula. The Itˆ o formula is the chain rule for the Itˆocalculus.Butitcannotbe expressed as in the Leibniz–Newton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. The Itˆ o formula can be interpreted only in the integral form. Moreover, there is an additional term in the formula, called the Itˆ o correction term, resulting from the nonzero quadratic variation of a Brownian motion.

Reviews

From the reviews:

"This textbook is a self-contained and systematic introduction to Itô’s stochastic integration with respect to martingales. The author gives special emphasis to the Brownian motion case. … Exercises are given in each chapter." (Jorge A. León, Mathematical Reviews, Issue 2006 e)

"Introduction to Stochastic Integration is exactly what the title says. I would maybe just add a ‘friendly’ introduction because of the clear presentation and flow of the contents. … Given its clear structure and composition, the book could be useful for a short course on stochastic integration. The concepts are easy to grasp … . Problems are given in each chapter and naturally are proof-based." (Ita Cirovic Donev, The Mathematical Sciences Digital Library, June, 2006)

"This is a very good book on stochastic integration covering subjects from a construction of a Brownian motion to stochastic differential equations. It grew up from lecture notes the author elaborated during several years, and can be equally well used for teaching and self-education. The text is extremely clear and concise both in language and mathematical notation. Every topic is illustrated by simple and motivating examples. … is a timely, happily designed and well written book. It will be useful for unprepared and advanced readers." (Ilya Pavlyukevich, Zentralblatt MATH, Vol. 1101 (3), 2007)

"This book covers stochastic integration with respect to square-integrable martingales. … I am sure that this book will be very welcomed by students and lectures of this subject … who will find many illustrative exercises provided. Reader also should not miss out on the Preface, which includes some anecdotes about K. Itô." (Thorsten Rheinländer, Journal of the American Statistical Association, Vol. 103 (483), September, 2008)

Authors and Affiliations

  • Department of Mathematics, Louisiana State University, Baton Rouge, USA

    Hui-Hsiung Kuo

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