Introduction to Stochastic Integration (Paperback)

Hui-Hsiung Kuo

ISBN 10: 0387287205 / ISBN 13: 9780387287201

Published by Springer-Verlag New York Inc., New York, NY, 2005

About this Item

Paperback. Also called Ito calculus, the theory of stochastic integration has applications in virtually every scientific area involving random functions. This introductory textbook provides a concise introduction to the Ito calculus. From the reviews: "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." --THE MATHEMATICAL SCIENCES DIGITAL LIBRARY It was the beginning of the It o calculus, the counterpart of the LeibnizNewton calculus for random functions. The It o formula is the chain rule for the Itocalculus.Butitcannotbe expressed as in the LeibnizNewton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability.

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Bibliographic Details

• Title: Introduction to Stochastic Integration (Paperback)
• Publisher: Springer-Verlag New York Inc., New York, NY
• Publication Date: 2005
• Language: English
• ISBN 10: 0387287205
• ISBN 13: 9780387287201
• Binding: Paperback
• Edition: 1st Edition
• Condition: new

About this title

Synopsis

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.

About the Author

Hui-Hsiung Kuo is the Nicholson Professor of Mathematics at Louisiana State University.