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Gaussian processes and martingales for fuzzy valued random variables with continuous parameter. (English) Zbl 0988.60025
The purpose of this paper is to investigate fuzzy-valued Gaussian processes and martingales with continuous parameters. M. L. Puri and D. A. Ralescu [Ann. Probab. 13, 1373-1379 (1985; Zbl 0583.60011)] gave the first representation theorem for the fuzzy-valued random variables whose level sets are non-empty compact convex sets in $$\mathbb{R}^n$$ by using the embedding method under the Lipschitz condition. The main result of the paper consists in rebuilding the embedding theorem, and proving the same representation theorem but for the fuzzy-valued random variables whose level sets are non-empty bounded closed convex sets in a separable Banach space, without the Lipschitz condition. Then this result is extended to the fuzzy-valued Gaussian processes and to the set-valued martingales.

##### MSC:
 60G15 Gaussian processes 60G44 Martingales with continuous parameter 26E50 Fuzzy real analysis
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##### References:
 [1] Li, S.; Ogura, Y., Fuzzy random variables conditional, expectations and fuzzy martingales, J. fuzzy math., 4, 905-927, (1996) · Zbl 0879.60001 [2] Li, S.; Ogura, Y., Convergence of set valued and fuzzy valued martingales, Fuzzy sets and systems, 101, 453-461, (1999) · Zbl 0933.60041 [3] Li, S.; Ogura, Y., Convergence of set valued sub- and super-martingales in the Kuratowski-mosco sense, Ann. probab., 26, 1384-1402, (1998) · Zbl 0938.60031 [4] S. Li, Y. Ogura, An optional sampling theorem for fuzzy valued martingales, in: Proceedings of the IFSA’97, vol. 4, 1997, pp. 9-13 [5] S. Li, Y. Ogura, Convergence in graph for fuzzy valued martingales and smartingales, in: C. Bertoluzza, M.A. Gil, D.A. Ralescu (Eds.), Statistical Modeling, Analysis and Management of Fuzzy Data, Physica (in press) · Zbl 0917.60046 [6] Y. Ogura, S. Li, Separability for graph convergence of sequences of fuzzy valued random variables, Fuzzy Sets and Systems (in press) · Zbl 0994.60035 [7] Puri, M.L.; Ralescu, D.A., Differentials of fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009 [8] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 13, 1373-1379, (1985) · Zbl 0583.60011 [9] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 406-422, (1986) · Zbl 0605.60038 [10] Puri, M.L.; Ralescu, D.A., Convergence theorem for fuzzy martingales, J. math. anal. appl., 160, 107-121, (1991) · Zbl 0737.60005 [11] Klement, E.P.; Puri, L.M.; Ralescu, D.A., Limit theorems for fuzzy random variables, Proc. roy. soc. London, 407, 171-182, (1986) · Zbl 0605.60038 [12] F.N. Proske, M.L. Puri, Limit theorems for fuzzy random variables (to appear) · Zbl 0990.60020 [13] Colubi, A.; López-Dı́az, M.; Domı́nguez-Menchero, J.S.; Gil, M.A., A generalized strong law of large numbers, Probab. theory related fields, 114, 401-417, (1999) · Zbl 0933.60023 [14] Kim, B.K.; Kim, J.H., Stochastic integrals of set-valued processes and fuzzy processes, J. math. anal. appl., 236, 480-502, (1999) · Zbl 0954.60043 [15] Hiai, F.; Umegaki, H., Integrals, conditional expectations and martingales of multivalued functions, J. multivar. anal., 7, 149-182, (1977) · Zbl 0368.60006 [16] Papageorgiou, N.S., On the theory of Banach space valued multifunctions, 1, integration and conditional expectation, J. multiv. anal., 17, 185-206, (1985) · Zbl 0579.28009 [17] Luu, D.Q., Representations and regularity of multivalued martingales, Acta math. Vietnam, 6, 29-40, (1981) · Zbl 0522.60045 [18] Alo, R.A.; de Korvin, A.; Roberts, C., The optional sampling theorem for convex set valued martingales, J. reine angew. math., 310, 1-6, (1979) · Zbl 0416.60012 [19] Hess, C., On multivalued martingales whose values may be unbounded: martingale selectors and mosco convergence, J multivar. anal., 39, 175-201, (1991) · Zbl 0746.60051 [20] Klein, E.; Thompson, A.C., Theory of correspondences including applications to mathematical economics, (1984), Wiley New York · Zbl 0556.28012 [21] Lyashenko, N.N., Statistics of random compacts in Euclidian space, J. soviet math., 20, 2187-2196, (1982) · Zbl 0489.60041 [22] Ma, M., On embedding problems of fuzzy number space: part 5, Fuzzy sets and systems, 55, 313-318, (1993) · Zbl 0798.46058 [23] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer New York · Zbl 0734.60060
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