This book is intended for readers who are quite familiar with
probability and stochastic processes but know little or nothing about
?nance. It is written in the de?nition/theorem/proof style of modern
mathematics and attempts to explain as much of the ?nance motivation and
terminology as possible. A mathematical monograph on ?nance can be
written today only - cause of two revolutions that have taken place on
Wall Street in the latter half of the twentieth century. Both these
revolutions began at universities, albeit in economics departments and
business schools, not in departments of mathematicsor statistics.
Theyhaveledinexorably, however, to anes- lation in the level of
mathematics (including probability, statistics, partial di?erential
equations and their numerical analysis) used in ?nance, to a point where
genuine research problems in the former ?elds are now deeply intertwined
with the theory and practice of the latter. The ?rst revolution in
?nance began with the 1952 publication of "Po- folio Selection," an
early version of the doctoral dissertation of Harry Markowitz. This
publication began a shift away from the concept of t- ing to identify
the "best" stock for an investor, and towards the concept of trying to
understand and quantify the trade-o's between risk and - turn inherent
in an entire portfolio of stocks. The vehicle for this so-called
mean-variance analysis of portfolios is linear regression; once this
analysis is complete, one can then address the optimization problem of
choosing the portfolio with the largest mean return, subject to keeping
the risk (i. e.
