Celebratio Mathematica — Yor

Dix thèmes de recherche sur les processus stochastiques qui me tiennent à cœur et m’ont longtemps occupé, I

Works connected to M. Weil

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M. Yor: “Sous-espaces denses dans $L^{1}$ ou $H^{1}$ et représentation des martingales” [Dense subspaces in $L^{1}$ or $H^{1}$ and representation of martingales], pp. 265–309 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. With an appendix by the author and J. de Sam Lazaro. MR 520008 Zbl 0391.60046 incollection

This paper was a considerable step in the study of the general martingale problem, i.e., of the set $ \mathcal{P} $ of all laws on a filtered measurable space under which a given set $ \mathcal{N} $ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R. G. Douglas (Michigan Math. J. 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in $ H^1 $. The main result can be stated as follows: given a law $ \mathbb{P}\in\mathcal{P} $, the set $ \mathcal{N} $ has the previsible representation property, i.e., $ \mathcal{F}_0 $ is trivial and stochastic integrals with respect to elements of $ \mathcal{N} $ are dense in $ H^1 $, if and only if $ \mathbb{P} $ is an extreme point of $ \mathcal{P} $. Many examples and applications are given.

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related