Conditional essential suprema with applications
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 diffusion and \(\alpha(.)\) is a control.
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