Positive parts of convergent fuzzy set-valued martingales

Abstract

For any given filtration (Σi), we give a description of the space of D∞-convergent set-valued fuzzy martingales with respect to (Σi). We show that this space of martingales can be endowed with a canonical metric in such a way that it is a complete metric space. We also show that this space of martingales is a join-semilattice. In particular, we derive a formula to calculate the join of any two such martingales. This enables us to also calculate positive parts of such martingales. Our order theoretic approach yields, as a bonus, a necessary and sufficient condition for a regular fuzzy set-valued martingale to be D∞-convergent. A description, in terms of constant sequences, is also given of the D∞-convergent fuzzy set-valued reversed martingales.