Stochastic fuzzy analysis and its applications

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Former study on limit theorems for random fuzzy sets was mainly for the case of compact level sets. This is due to its method which is based on the embedding theorem of convex compact sets into the space of bounded continuous functions on the unit closed ball in the dual space. The object of this project is to obtain limit theorems for wider class of random fuzzy sets, extending the former study.There are two cases where the compactness condition fails. One is the case of bounded but non compact sets, which occurs in infinite dimension cases. The other is that of unbounded sets, which occurs even in finite dimensional cases. In the former case, the first problem one may encounter is the closedness of the Aumann integrals. This integral is fundamental in the study of random sets or random fuzzy sets, and although a counter example for the closeness was known, a good sufficient condition was not known. In this project, we obtained that the Aumann integral is closed if the underlying spac … More e is a reflexive Banach space. We next dealt with the concept of convergence, which is significant in both former and latter cases. In most of former works, the fuzzy sets were treated as simple collections of level sets and the topologies sometimes differs from intuition. In this project, we introduced a new concept called graph convergence which fits intuition more, and improved various convergence theorems. Finally, we also obtained separability theorem for graph convergence, which ensures that one have only to check the convergence of level sets for countably many levels for a proof of convergence of random fuzzy sets.Based on the former theoretical advance, we obtained or improved various limit theorems on random sets and/or random fuzzy sets, which covers law of large numbers, limit theorems for martingale and s-martingales, regular representation theorem for martingales, optional sampling theorems, approximation of random fuzzy sets and representation theorem of Gaussian random fuzzy sets. Less