Title: Positive parts of convergent fuzzy set-valued martingales

Authors: Coenraad C.A. Labuschagne

Addresses: School of Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, WITS, 2050, South Africa

Abstract: For any given filtration (Σ_i), we give a description of the space of D_∞-convergent set-valued fuzzy martingales with respect to (Σ_i). 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.

Keywords: hyperspace; fuzzy conditional expectations; fuzzy martingales; fuzzy sets; fuzzy logic.

DOI: 10.1504/IJAAC.2008.022186

International Journal of Automation and Control, 2008 Vol.2 No.2/3, pp.376 - 395

Published online: 22 Dec 2008 *

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