Stochastic Calculus and Financial Applications / Edition 1

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Shastic calculus has important applications to mathematical finance. This book will appeal to practitioners and students who want an elementary introduction to these areas.

From the reviews: "As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book." —ZENTRALBLATT MATH

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Editorial Reviews

From the Publisher
From the reviews:


"…on the whole, the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of shastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently."


"This is a world of 'lovely exercises' that are 'very good good for the soul', 'honest martingales', 'bedrock approximations', portfolios that are 'born to lose', 'intuitive but bogus arguments', and 'embarrassingly crude insights'. In short, this is a book on shastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition.The main results are reinforced with simple special cases, and only when the intuitive foundations are laid does the auhtor resort to the formalism of probability. The coverage is limited to the essentials but nevertheless includes topics that will catch the eye of experts (such as the wavelet construction of Brownian motion). This is one of the most interesting and easiest reads in the discipline; a gem of a book."


"The book is indeed well written, with many insightful comments. I certainly would recommend it to students wishing to learn shastic calculus and its applications to the Black-Sholes option-pricing theory…I thoroughly enjoyed reading this book. The author is to be complimented for his efforts in providing many useful insights behind the various theories. It is a superb introduction to shastic calculus and Brownian motion…An interesting feature in this book is its coverage of partial differential equations."

"It is clear that this is a fairly comprehensive introduction to the tools of (classical) mathematical finance. … the text has much to offer. … In addition, the writing style is refreshingly informal and makes a book about a rather technical subject surprisingly enjoyable to read. In short, despite the recent deluge of textbooks in this area, I know of no better book for self-study." (Christian Kleiber, Statistical Papers, Vol. 46 (2), 2005)

"Steele’s book is a sophisticated introduction to shastic calculus with applications from basic Black-Scholes theory. … I highly recommend the book. His style is wonderful, and concepts really build on one another. … it offers one of the most elegant treatments of the subject that I know of." (, May, 2006)

"As is clear from the title of this book, it is concerned with applications of shastic calculus to finance. … one naturally judges the book by three criteria: topic selection, organization, and exposition. In all three domains the book succeeds. The topics selected are rich enough … he or she will benefit from the book. … there are innovations as well … from the pedagogic standpoint." (Philip Protter, SIAM Review, Vol. 43 (4), 2001)

"This book offers rich information and a mathematically honest treatment of shastic calculus and of its use in the theory of finance … . The author gradually builds the reader’s ability to grasp shastic concepts and techniques … . the author’s presentation of shastic models in finance and economy is precise and extensive … . Each chapter is accompanied by a collection of rather challenging exercises … ." (EMS Newsletter, December, 2002)

"The present book ‘is designed for students who want to develop professional skill in shastic calculus and its application to problems in finance’. … the textbook … retains a lovely lecture style focusing basic ideas and not formalities and technical details of shastic processes needed for finance. I can strongly recommend this book to students of mathematics and physics as well as non-experts in probability theory who are interested in shastic finance." (H. –J. Girlich, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (4), 2002)

"The last few years have been a fertile period for books on shastic calculus and its financial implications, but this one differs from the many mainstream treatments … . The style of the book creates the atmosphere of a lively lecture … . Each chapter ends with a section of carefully chosen exercises, preceded by some motivating remarks. … I really liked the book." (R. Grübel, Statistics & Decisions, Vol. 20 (4), 2002)

"This book gives an introduction to shastic calculus … with applications in mathematical finance. … As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book. … on the whole, the results are presented carefully and thoroughly … ." (Martin Schweizer, Zentralblatt MATH, Vol. 962, 2001)

"This is a book on shastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition. The main results are reinforced with simple special cases … . This is one of the most interesting and easiest reads in the discipline; a gem of a book." (D. L. McLeish, Short Book Reviews, Vol. 21 (1), 2001)

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Product Details

  • ISBN-13: 9781441928627
  • Publisher: Springer New York
  • Publication date: 12/1/2010
  • Series: Stochastic Modelling and Applied Probability Series , #45
  • Edition description: Softcover reprint of hardcover 1st ed. 2001
  • Edition number: 1
  • Pages: 302
  • Product dimensions: 0.65 (w) x 9.21 (h) x 6.14 (d)

Table of Contents

Random Walk and First Step Analysis
• First Martingale Steps
• Brownian Motion
• Martingale—Next Steps
• Richness of Paths
• Itô Integration
• Localization and Itô's Integral
• Itô's Formula
• Shastic Differential Equations
• Arbitrage and SDE's
• The Diffusion Equation
• Representation Theorems
• Girsanov Theory
• Arbitrage and Martingales
• The Feynman-Kac Connection

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Customer Reviews

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Sort by: Showing 1 Customer Reviews
  • Anonymous

    Posted March 9, 2003

    Riskfree profit !

    The book is at the interface of three areas, math, statistics, and finance. While connections between the first two have a long history, it was the connection to finance that caught my attention. Coming from math myself, I needed first to take a closer look at the book to orient myself. The mathematical subjects, smooth sailing, include stochastic differential equations (SDE) as they relate to PDEs; and the ideas from probability and statistics include Brownian motion, martingales, stochastic processes, and the Feynman-Kac connection. Browsing the chapters I found them to be a lovely presentation of ideas with which I am familiar. For me, it was chapter 10 that turned out to have stuff that I wasn't familiar with. That is the finance part, and it is based on a model for Option Pricing developed in 1973 by Fischer Black and Myron Scholes. An arbitrage opportunity [simplified] amounts to the simultaneous purchase and sale of related securities which is guaranteed to produce a *riskless* profit. It was after reading more in this chapter I understood why the book is used in a course at the Wharton School at the University of Pennsylvania. I am impressed with the level of math in this course. Part of the motivation in the applications to finance is that arbitrage enforces the price of most derivative securities. And I learned from ch 10 that the SDE of the Black-Scholes model governs the processes which represent the two variables S, the price of a stock, and B the price of a bond, both S and B representing stochastic variables depending of time t, i.e., both stochastic processes. In the model, S is a geometric Brownian motion, and B is a deterministic process with exponential growth. The two are determined as solutions to the SDE of Black-Scholes.

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