Decomposition theorems for fuzzy supermartingales and submartingales
The aim of this paper is to generalize the Doob decomposition and the Riesz decomposition theorems of standard martingale theory to the case of fuzzy martingales. The fact that the fuzzy number space is not a linear space represents one of the main difficulties to obtain the intended results. In order to overcome this problem. Section 2 gives the representation theorem of fuzzy numbers on the basis of their support functions, and uses the concept of H-difference of fuzzy numbers and its properties. Different measurability concepts for fuzzy random variables, relationships among them, and two counterexamples are discussed. Section 3 studies the Doob decomposition of fuzzy supermartingales and submartingales. Since not all fuzzy supermartingales and submartingales are proved to have Doob decomposition, the author obtains necessary and sufficient conditions for a fuzzy supermartingale (respective submartingale) to provide Doob decomposition. Section 4 is devoted to Riesz decomposition for fuzzy supermartingales. Two notions of Riesz decomposition are defined, and necessary and sufficient conditions for a fuzzy supermartingale to have Riesz decomposition are analyzed.
scientific article; zbMATH DE number 1547847