Introduction to Stochastic Integration.

Kuo, Hui-Hsiung -

ISBN 10: 0387287205 / ISBN 13: 9780387287201

Published by Berlin ; New York: Springer (Universitext), 2006

About this Item

Formateinband: Broschierte Ausgabe XIII, 278 S. (23,5 cm) 1st Edition; Sehr guter Zustand. Sprache: Englisch Gewicht in Gramm: 600 [Stichwörter: Stochastik, Stochastische Integration, Brownian Motion, Stochastic Integrals for Martingales, The Ito-Formula, Multiple Wiener-Ito Integrals, Stochastic Differential Equations etc.].

Seller Inventory # 62021

Bibliographic Details

• Title: Introduction to Stochastic Integration.
• Publisher: Berlin ; New York: Springer (Universitext)
• Publication Date: 2006
• Language: English
• ISBN 10: 0387287205
• ISBN 13: 9780387287201
• Binding: Soft cover
• Edition: 1st Edition
• Condition: Sehr gut
• Seller catalogs:
  • Mathematik

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.