Filtrations Indexed by Ordinals; Application to a Conjecture of S. Laurent

Abstract

The following fact has been conjectured by Stéphane Laurent (Conjecture 3.18, page 160 of Séminaire de Probabilités XLIII): Let $\,\mathcal{F} = (\mathcal{F}_{t})$ and $\,\mathcal{G} = (\mathcal{G}_{t})$ be two filtrations on some probability space, and suppose that every $\,\mathcal{F}$ -martingale is also a $\,\mathcal{G}$ -martingale. For s < t, if $\,\mathcal{G}_{t}$ is generated by $\,\mathcal{G}_{s}$ and by countably many events, then $\,\mathcal{F}_{t}$ is generated by $\,\mathcal{F}_{s}$ and by countably many events. In this statement, “and by countably many events” can equivalently be replaced with “and by some separable σ-algebra”, or with “and by some random variable valued in some Polish space”. We propose a rather intuitive proof of this conjecture, based on the following necessary and sufficient condition: Given a probability space, let $\,\mathcal{D}$ be a σ -algebra of measurable sets and $\,\mathcal{C}$ a sub-σ-algebra of $\,\mathcal{D}$. Then $\,\mathcal{D}$ is generated by $\,\mathcal{C}$ and by countably many events if and only if there exists no strictly increasing filtration $\mathcal{F} = (\mathcal{F}_{\alpha })_{\alpha <\boldsymbol\aleph _{1}}$, indexed by the set $\,\lfloor \lceil 0,\boldsymbol\aleph _{1}\lfloor \lceil$ of all countable ordinals, and satisfying $\,\mathcal{C}\subseteq \mathcal{F}_{\alpha }\subseteq \mathcal{D}\,$ for each α. Another question then arises: can the martingale hypothesis on $\mathcal{F}$ and $\mathcal{G}$ be replaced by a more general condition involving the null events but not the values of the probability? We propose such a weaker hypothesis, but we are no longer able to derive the conclusion from it; so the question is left open.

The support of the ANR programme ProbaGeo is gratefully acknowledged.

[…] attention, car, à partir de ce moment, nous voilà aux prises avec le cardinal.

A. Dumas, Les trois mousquetaires.