Fuzzy Random Variables On Noncompact Fuzzy Number Space

The paper mainly deals with fuzzy random variables that take values on noncompact fuzzy number space. Firstly, a general metric is defined on noncompact fuzzy number space and its completeness and separability are proved. Secondly, it turns out that some different concepts of noncompact fuzzy random variables are unified as the measurability is concerned. Furthermore, the variance and covariance of noncompact fuzzy random variables are introduced, and their properties are discussed. Finally, the fuzzy stochastic processes are also touched upon.The paper is organized as follows:Chapter 1 presents a summary of fuzzy random variables. Topics include a brief introduction of the idea of applying fuzzy random variables, an introduction of literature on fuzzy random variables and the problems considered in this thesis.Noncompact fuzzy number space is introduced in Chapter 2. A generalized metric is defined on the noncompact fuzzy number set. Some properties and theorems about noncompact fuzzy numbers are discussed. Compared with the separability and incompleteness of compact fuzzy number space, the generalized metric defined here is proved to be complete and separable.Noncompact fuzzy random variables are introduced in Chapter 3. We have compared some different concepts of noncompact fuzzy random variables. As the result, it will be shown that these concepts are unified as the measurability is concerned. The study extends some results of "A unified approach to fuzzy random variables (Volker Kratschmer, Fuzzy Sets andSystems, 2001, 123, 1-9)". The concepts and properties of noncompact p-order fuzzy random variables are presented as extension of the p-order random vector. Finally, we defined the expectation of noncompact fuzzy random variables by a new method.Noncompact fuzzy stochastic processes and their applications are put in Chapter 4. Some properties of noncompact 2-order fuzzy random variables and their applications are discussed. The results extend the study of "The variance and covariance of fuzzy random variables and their applications(Feng Yuhu.et al, Fuzzy Sets and Systems, 2001, 120, 487-497.) ". In addition, we get a result about the right-closed-right-continuous fuzzy martingale.