Imprecise set and fuzzy valued probability

Abstract

a b s t r a c tSet valued probability and fuzzy valued probability theory is used for analyzing and modeling highly uncertain probability systems. In this paper the set valued probability and fuzzy valued probability are defined over the measurable space. They are derived from a set and fuzzy valued measure using restricted arithmetics. The range of set valued probability is the set of subsets of the unit interval and the range of fuzzy valued probability is the set of fuzzy sets of the unit interval. The expectation with respect to set valued and fuzzy valued probability is defined and some properties are discussed. Also, the fuzzy model is applied to binomial model for the price of a risky security.

Intuitionistic Fuzzy Probability of an Event

“…In such cases, fuzzy set theory successfully provides suitable tools for modeling probability of a uncertain event. The topic of probability theory with fuzzy information has been studied by many authors during the last decades [3,4,11,8,9,16,17,18,19,20,21,23,24,25,27,28,30,31,33,34,35,36,37,38,39,40,41,43,44]. The other resent approaches to fuzzy probability have been recently proposed are based on the concepts of sets of probability measures, distribution envelops, interval probabilities, p-box approach, and Dempster-Shafer theory (for instance, refer to [5,15].…”

A Note on Fuzzy Probability of a Fuzzy Event

“…Baldwin et al 2 introduced the probability of a fuzzy event using mass assignment theory techniques for processing uncertainty together with the t-norm definition of conditional probabilities. 1,4,[17][18][19]22,26,27,31,32). Plasecki 21 defined the probability of fuzzy events as a denumerable additivity measure.…”

“…In this way, imprecise assessment of the probability of an (crisp) event is expressed by a fuzzy set (e.g., see Ref. 1,4,[17][18][19]22,26,27,31,32). The other recent approaches to fuzzy probability have been recently proposed and which are based on the following concepts: imprecise probabilities, fuzzy probability theory has a relationship to concepts known as upper and lower probabilities, sets of probability measures, distribution envelops, interval probabilities, p-box approach, and Dempster-Shafer theory (for more details, see Ref.…”

New fuzzy integral based on gradual numbers

“…Motivated by the applications in several areas of applied science, such as mathematical economics, fuzzy optimal, process control and decision theory, much effort has been devoted to the generalization of different measure concepts and classical results to the case when outcomes of a random experiment are represented by fuzzy sets, such as the concepts of fuzzy measures (see, e.g., [2,[12][13][14]) and fuzzy integrals (see, e.g., [7,8,11]).…”