Fuzzy set-valued Gaussian processes and Brownian motions

Authors:
Shoumei Li;Li Guan
Affiliations:
Department of Applied Mathematics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing 100022, PR China;Department of Applied Mathematics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing 100022, PR China
Venue:
Information Sciences: an International Journal
Year:
2007

Citing 5
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Abstract

Gaussian processes and Brownian motion are concepts and tools in modelling important uncertain systems in many areas. In view of uncertainty complexity in many real-world problems, we extend these tools to the case where stochastic processes can take on fuzzy sets as values. In this paper, we discuss fuzzy set-valued Gaussian processes based on the results of [S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers, Dordrecht, 2002; S. Li, Y. Ogura, H.T. Nguyen, Gaussian processes and martingales for fuzzy valued variables with continuous parameter, Inform. Sci. 133 (2001) 7-21; S. Li, Y. Ogura, F.N. Proske, M.L. Puri, Central limit theorems for generalized set-valued random variables, J. Math. Anal. Appl. 285 (2003) 250-263; N.N. Lyashenko, On limit theorems for sums of independent compact random subsets in the Euclidean space, J. Soviet Math. 20 (1982) 2187-2196] and [M.L. Puri, D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373-1379]. We also introduce the concept of fuzzy set-valued Brownian motion, and then prove several properties of such processes.