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Product code: 14261
ISBN: 0387950168,
ISBN13: 9780387950167,
316 pages, hardback, published by Springer-Verlag New York Inc., 2001. Corr. 3rd edition, 2001
Description of Stochastic Calculus and Financial Applications
The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It' integral and aims to provide a development that is honest and complete without being pedantic. With the It' integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
Contents of Stochastic Calculus and Financial Applications
Random Walk and First Step Analysis
First Martingale Steps
Brownian Motion
Martingale Next Steps
Richness of Paths
It=F4 Integration
Localization and It=F4's Integral
It=F4's Formula
Stochastic Differential Equations
Arbitrage and SDE's
The Diffusion Equation
Representation Theorem
Girsanov Theory
Arbitrage and Martingales
The Feynman-Kac Connection
Reviews of Stochastic Calculus and Financial Applications
"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 stochastic 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."
- D.L. McLeish in 'Short Book Reviews', Vol. 21/1, April 2001
