Résumé
This book is designed for students who want to develop
professional skills in stochastic calculus and its
application to problems in finance. 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 Itô
integral and aims to provide a development that is honest
and complete without being pedantic.
With the Itô 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.
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
- Random Walk and First Step Analysis
- First Martingale Steps
- Brownian Motion
- Martingale: The Next Steps
- Richness of Paths
- Itô Integration
- Localization and Itô's Integral
- Itô's Formula
- Stochastic Differential Equations
- Arbitrage and SDEs
- The Diffusion Equation
- Representation Theorem
- Girsanov Theory
- Arbitrage and Martingales
- The Feynman-Kac Connection
- Appendix I. Mathematical Tools
- Appendix II. Comments and Credits
- Bibliography
- Index.
L'auteur - J.Michael Steele
J. Michael Steele is C. F. Koo Professor of Statistics at the Wharton School, University of Pennsylvania. He is the author of more than 100 mathematical publications, including the books Probability Theory and Combinatorial Optimization and Stochastic Calculus and Financial Applications. He is also the founding editor of the Annals of Applied Probability.
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Springer |
| Auteur(s) | J.Michael Steele |
| Parution | 15/08/2001 |
| Nb. de pages | 300 |
| Format | 15,8 x 24 |
| Couverture | Relié |
| Poids | 555g |
| Intérieur | Noir et Blanc |
| EAN13 | 9780387950167 |
| ISBN13 | 978-0-387-95016-7 |
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