Introduction to Stochastic Integration de Hui-Hsiung Kuo

Introduction to Stochastic Integration: Universitext

Autor Hui-Hsiung Kuo

en Limba Engleză Paperback – 15 noi 2005

Prezentat sub formă de manual de nivel avansat, Introduction to Stochastic Integration constituie o introducere sistematică în calculul stochastic, fiind o resursă esențială pentru cercetătorii care navighează între teoria probabilităților și analiza matematică. Notăm cu interes modul în care Hui-Hsiung Kuo construiește argumentația: plecând de la limitările calculului clasic Leibniz-Newton în fața funcțiilor aleatorii, autorul demonstrează necesitatea aparatului matematic introdus de Kiyosi Itô în 1944. Întrucât traiectoriile mișcării browniene nu sunt derivabile în sens clasic, volumul explică detaliat formula lui Itô ca echivalent al regulii lanțului, incluzând termenul de corecție specific.

Această lucrare extinde cadrul propus de Brownian Motion Calculus de Ubbo F. Wiersema, care se concentrează pe aplicații financiare, oferind aici o rigoare teoretică superioară și o perspectivă mai largă asupra integralelor pentru martingale. În contextul operei sale anterioare, precum White Noise, Hui-Hsiung Kuo păstrează predilecția pentru spațiile infinit-dimensionale, însă aici rafinează tratamentul didactic pentru a fi accesibil celor aflați la începutul cercetării. Structura este una progresivă, pornind de la construcția mișcării browniene și culminând cu aplicații complexe ale ecuațiilor diferențiale stocastice. Apreciem includerea seturilor de exerciții în fiecare capitol, fapt ce transformă monografia dintr-o expunere teoretică într-un instrument de lucru activ. Față de abordările din Brownian Motion, Martingales, and Stochastic Calculus de Jean-François Le Gall, textul de față rămâne mai strâns ancorat în fundamentarea istorică și mecanică a proceselor Itô, fiind ideal pentru cei care doresc să înțeleagă nu doar utilizarea, ci și geneza acestor operatori.

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Specificații

ISBN-13: 9780387287201
ISBN-10: 0387287205
Pagini: 296
Ilustrații: XIII, 279 p. 2 illus.
Dimensiuni: 155 x 235 x 17 mm
Greutate: 0.45 kg
Ediția:2006
Editura: Humana
Colecția Universitext
Seria Universitext

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Recomandăm acest manual cercetătorilor și studenților la masterat în matematică sau fizică teoretică. Cititorul câștigă o înțelegere profundă a calculului Itô, învățând să gestioneze funcții aleatorii care sfidează regulile derivării clasice. Este un punct de plecare solid pentru oricine dorește să stăpânească ecuațiile diferențiale stocastice, oferind un echilibru rar între demonstrații riguroase și exerciții aplicative, totul în cadrul prestigioasei serii Universitext.

Descriere scurtă

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.

Cuprins

Brownian Motion.- Constructions of Brownian Motion.- Stochastic Integrals.- An Extension of Stochastic Integrals.- Stochastic Integrals for Martingales.- The Itô Formula.- Applications of the Itô Formula.- Multiple Wiener-Itô Integrals.- Stochastic Differential Equations.- Some Applications and Additional Topics.

Recenzii

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)

Textul de pe ultima copertă

The theory of stochastic integration, also called the Ito calculus, has a large spectrum of applications in virtually every scientific area involving random functions, but it can be a very difficult subject for people without much mathematical background. The Ito calculus was originally motivated by the construction of Markov diffusion processes from infinitesimal generators. Previously, the construction of such processes required several steps, whereas Ito constructed these diffusion processes directly in a single step as the solutions of stochastic integral equations associated with the infinitesimal generators. Moreover, the properties of these diffusion processes can be derived from the stochastic integral equations and the Ito formula. This introductory textbook on stochastic integration provides a concise introduction to the Ito calculus, and covers the following topics:
* Constructions of Brownian motion;
* Stochastic integrals for Brownian motion and martingales;
* The Ito formula;
* Multiple Wiener-Ito integrals;
* Stochastic differential equations;
* Applications to finance, filtering theory, and electric circuits.
The reader should have a background in advanced calculus and elementary probability theory, as well as a basic knowledge of measure theory and Hilbert spaces. Each chapter ends with a variety of exercises designed to help the reader further understand the material.
Hui-Hsiung Kuo is the Nicholson Professor of Mathematics at Louisiana State University. He has delivered lectures on stochastic integration at Louisiana State University, Cheng Kung University, Meijo University, and University of Rome "Tor Vergata," among others. He is also the author of Gaussian Measures in Banach Spaces (Springer 1975), and White Noise Distribution Theory (CRC Press 1996), and a memoir of his childhood growing up in Taiwan, An Arrow Shot into the Sun (Abridge Books 2004).

Caracteristici

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