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.
Applied Mathematics and Optimization – Springer Journals
Published: Oct 1, 2003
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