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

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.