# Conditional Essential Suprema with Applications

Conditional Essential Suprema with Applications 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 \infty . 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 \infty 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Conditional Essential Suprema with Applications

, Volume 48 (3) – Oct 1, 2003
25 pages

/lp/springer_journal/conditional-essential-suprema-with-applications-avPbpqgloT
Publisher
Springer Journals
Subject
Philosophy
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-003-0776-4
Publisher site
See Article on Publisher Site

### 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 \infty . 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 \infty 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2003

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