• Hardcover: 344 pages
• Publisher: Springer-Verlag New York Inc. (Jan 2001)
• Language English
• ISBN-10: 0387950168
• ISBN-13: 978-0387950167

Synopsis
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 Ito integral and aims to provide a development that is honest and
complete without being pedantic.With the Ito 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.

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