Conditional essential suprema with applications
type: Item
instance of: scholarly article
title: Conditional essential suprema with applications
zbMATH Open document ID: 1049.60003
author: R. Jensen
author: Emmanuel N. Barron
author: Pierre Cardaliaguet
published in: Applied Mathematics and Optimization
publication date: 2004-05-27T00:00:00Z
review text: 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 arises whether a Radon-Nikodym theorem is available in \(L^\infty\) [see \textit{E. N. Barron, P. Cardaliaguet} and \textit{R. Jensen}, Appl. Math. Optimization 42, No.~2, 103--126 (2000; Zbl 0984.28002)]. The conditional supremum of a random variable \(X\) with respect to a sub-\(\sigma\)-algebra \(D\subset {\mathcal F}\) is a \(D\)-measurable random variable, denoted by \({\mathcal M}(X\mid D)\) which satisfies \[ \underset{\omega\in A} {\text{ess\,sup}}\, X(\omega)=\underset{\omega\in A} {\text{ess\,sup}}\,{\mathcal M}(X\mid D)(\omega),\quad A\in D. \] The conditional max has many properties enjoyed by conditional expectation. Applications of the conditional maximum are given in the remaining sections. The authors prove an ergodic theorem stating the time maxima are space maxima and various ramifications for measure-preserving maps. Next they introduce the concept of a maxingale as the analogue of a martingale. The fundamental martingale convergence theorems are applications of optimal sampling basically saying that if a sequence satisfies optional sampling, then it converges. In maxingale theory any submaxingale which satisfies optional sampling must in fact be increasing and then, when it is bounded above, it converges. In the last section the authors modify some recent work of \textit{H. M. Soner} and \textit{N. Touzi} [Commun. Partial Differ. Equations 27, No. 9--10, 2031--2053 (2002; Zbl 1036.49010); SIAM J. Control Optimization 41, No.~2, 404--424 (2002; Zbl 1011.49019) and J. Eur. Math. Soc. (JEMS) 4, No.~3, 201--236 (2002; Zbl 1003.49003)] where they showed that the stochastic target problems they were studying are connected to the worst case value function \[ u(t,x)= \inf_\alpha \underset {\omega \in\Omega}{\text{ess\,sup}}\;g(\xi(T)) \] where \(\xi(.)\) is a controlled diffu…
Mathematics Subject Classification ID: 60A10
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 49K45
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 2067293
zbMATH Keywords: conditional maximum
zbMATH Keywords: ergodic theory
zbMATH Keywords: maxingales
zbMATH Keywords: optimal stopping
zbMATH Keywords: worst case
MaRDI profile type: Publication
full work available at URL: s00245-003-0776-4
OpenAlex ID: W1979547553
DOI: 10.1007/S00245-003-0776-4
Recommended article: Necessary and Sufficient Conditions of Optimalcontrol for Infinite Dimensional SDEs
Recommended article: Nonconcave robust optimization with discrete strategies under Knightian uncertainty
Recommended article: On a generalization of the Dvoretzky-Wald-Wolfowitz theorem with an application to a robust optimization problem
Recommended article: Q3728793
Recommended article: A second-order maximum principle for singular optimal stochastic controls
Recommended article: Q3200970
Recommended article: A General Stochastic Maximum Principle for Optimal Control Problems
Recommended article: Martingale conditions for the optimal control of continuous time stochastic systems
Recommended article: Optimal locally absolutely continuous change of measure. finite set of decisions. part ii:optimization problems
description: scientific article; zbMATH DE number 2067293