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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 It o published the celebrated paper "Stochastic Integral" in the Proceedings of the Imperial Academy (Tokyo). It was the beginning of the It o calculus, the counterpart of the Leibniz-Newton calculus for random functions. In this six-page paper, It o introduced the stochastic integral and a formula, known since then as It o's formula. The It o formula is the chain rule for the It ocalculus.Butitcannotbe expressed as in the Leibniz-Newton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. The It o formula can be interpreted only in the integral form. Moreover, there is an additional term in the formula, called the It o correction term, resulting from the nonzero quadratic variation of a Brownian motion.
Zusammenfassung
It was the beginning of the Itˆ o calculus, the counterpart of the Leibniz–Newton calculus for random functions. 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.
Bericht
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)
