Cones and decomposition of sub- and supermartingales

We study sub- and supermartingales with values in the class Fc of all convex fuzzy subsets of a Banach space having compact support. The terms "sub" and "super" are understood with respect to a partial order (not necessarily inclusion). In this general framework, orders with respect to which a Doob-type decomposition can hold are naturally associated to convex subcones with a special property termed Fc-properness. Decomposition theorems into the sum or difference of a martingale and a compensator are proven. In the "sub" case the martingale need exist only in a superspace of Fc.