This first volume contains four chapters. The first one was written by Peter Bank
and Hans F¨ollmer. It grew out of a seminar course at given at Princeton in 2002. It
reviews a recent approach to optimal stopping theory which complements the traditional
Snell envelop view. This approach is applied to utility maximization of a
satisfaction index, American options, and multi-armed bandits.
The second chapter was written by Fabrice Baudoin. It grew out of a course
given at CREST in November 2001. It contains an interesting, and very promising,
extension of the theory of initial enlargement of filtration, which was the topic of his
Ph.D. thesis. Initial enlargement of filtrations has been widely used in the treatment of
asymetric information models in continuous-time finance. This classical viewassumes
the knowledge of some random variable in the almost sure sense, and it is well
known that it leads to arbitrage at the final resolution time of uncertainty. Baudoin’s
chapter offers a self-contained reviewof the classical approach, and it gives a complete
VI Preface
analysis of the case where the additional information is restricted to the distribution
of a random variable.
The chapter contributed by Chris Rogers is based on a short course given during
the Montreal Financial Mathematics and Econometrics Conference organized in June
2001 by CIRANO in Montreal. The aim of this event was to bring together leading
experts and some of the most promising young researchers in both fields in order
to enhance existing collaborations and set the stage for new ones. Roger’s contribution
gives an intuitive presentation of the duality approach to utility maximization
problems in different contexts of market imperfections.
The last chapter is due to Mete Soner and Nizar Touzi. It also came out of seminar
course taught at Princeton University in 2001. It provides an overview of the duality
approach to the problem of super-replication of contingent claims under portfolio
constraints.A particular emphasis is placed on the limitations of this approach, which
in turn motivated the introduction of an original geometric dynamic programming
principle on the initial formulation of the problem. This eventually allowed to avoid
the passage from the dual formulation.