Karatzas and
Shreve is well known to financial engineers. It is a very advanced text on
stochastic calculus. The authors claim to, in some areas, take readers to the
cutting edge of current research. I am not a researcher in the field, so I cannot
speak to this claim. However, based on the sophistication of the discourse, I
don't doubt it.
The book follows a
standard outline, opening with martingales and Brownian motion and proceeding to
stochastic integration. A chapter relates stochastic calculus to
(non-stochastic) partial differential equations. Other books tend to break up
this material, perhaps discussing the Dirchlet problem, heat equation and the
Feynman and Kac formulas separately in different chapters. Karatzas and Shreve
unify their treatment in a single chapter. I really like the chapter on
stochastic differential equations. It is extensive and very well written. A
closing chapter discusses Brownian local time.
Contents
Martingales, Stopping Times, and
Filtrations
Brownian Motion
Stochastic Integration
Brownian Motion and Partial Differential
Equations
Stochastic Differential Equations
Levy's Theory of Brownian Local Time
Discussions are at
a very high level of generality. Most results are stated for vector-valued
processes. In Karatzas and Shreve's hands, such generality is wonderful. They
write in a clear informal manner. Sure, the material is highly technical,
and their presentation is rigorous, but they always orient you to what is going
on. In a lot of books, it is easy to get lost among the trees. Karatzas and
Shreve keep you oriented to the forest.
Another aspect
that I really like is the exercises that are mixed into the body of the text. A
concept is introduced or a theorem is proven, and then you immediately get a
chance to try your own hand on a relevant exercise.
There is little
mention of applications; this is a math book. However, a brief section considers
applications in economics, including Black-Scholes theory.
I wouldn't read
this as a first book on stochastic calculus. However, it is a wonderful book to
graduate too. Don't be intimidated by its technical sophistication. The authors
deliver an accessible entree to advanced concepts in stochastic calculus.
