Appl Math Optim 48:229–253 (2003)
2003 Springer-Verlag New York Inc.
Conditional Essential Suprema with Applications
E. N. Barron,
and R. Jensen
Department of Mathematics and Statistics, Loyola University Chicago,
Chicago, IL 60626, USA
Departement de Math´ematiques, Universit´e de Brest,
29285 Brest Cedex, France
Abstract. The conditional supremum of a random variable X on a probability
space given a sub-σ -algebra is deﬁned and proved to exist as an application of
the Radon–Nikodym theorem in L
. After developing some of its properties we
use it to prove a new ergodic theorem showing that a time maximum is a space
maximum. The concept of a maxingale is introduced and used to develop the new
theory of optimal stopping in L
and the concept of an absolutely optimal stopping
time. Finally, the conditional max is used to reformulate the optimal control of the
worst-case value function.
Key Words. Conditional maximum, Ergodic theory, Maxingales, Optimal
stopping, Worst case.
AMS Classiﬁcation. 60A02, 37A02, 35D02.
The classical Radon–Nikodym theorem is the basis for the existence of conditional
expectation in probability theory, a fundamental tenet of the theory and applications.
The question naturally arises that once a Radon–Nikodym theorem is available in L
as it is from , can we use this theorem to develop a useful concept of conditional
maximum? The answer to that question is the content of this paper.
E. N. Barron and R. Jensen were partially supported by NSF-9972043 and NSF-0200169.