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.
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Barron, E., Cardaliaguet, P. & Jensen, R. Conditional Essential Suprema with Applications. Appl Math Optim 48, 229–253 (2003). https://doi.org/10.1007/s00245-003-0776-4
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DOI: https://doi.org/10.1007/s00245-003-0776-4