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Convergence in graph for fuzzy valued martingales and smartingales

  • Shoumei Li
  • Yukio Ogura
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 87)

Abstract

In this paper, we introduce the concept of convergence in graph for fuzzy-valued random variables, give an equivalent definition and then obtain convergence theorems for fuzzy-valued martingales, submartingales and supermartingales based on the results of our previous papers (Li and Ogura, 1996, 1998, 1999).

Keywords

Banach Space Conditional Expectation Strong Topology Fuzzy Random Variable Hausdorff Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aumann, R. J. (1965). Integrals of set-valued functions, J. Math. Anal. Appl. 12, 1–12.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ban, J. (1991). Ergodic theorems for random compact sets and fuzzy variables in Banach spaces, Fuzzy Sets and Systems 44, 71–82.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beer, G. (1993). Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, Holland.Google Scholar
  4. 4.
    Chatterji, S. D. (1968). Martingale convergence and the RN-theorems, Math. Scand. 22, 21–41.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hess, C. (1991). On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence, J. Multivar. Anal. 39, 175–201.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hess, C. (1998). On the almost sure convergence of sequences of random sets: martingales and extensions, (to appear).Google Scholar
  7. 7.
    Hiai, F. and Umegaki, H. (1977). Integrals, conditional expectations and martingales of multivalued functions, J. Multivar. Anal. 7, 149–182.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hiai, F. (1985). Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291, 613–627.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kendall, D. G. (1974). Foundations of a Theory of Random Sets, in Stochastic Geometry ( Harding, E. F. and Kendall D. G., Eds.), J. Wiley & Sons, New York.Google Scholar
  10. 10.
    Klement, E. P., Puri, L. M. and Ralescu, D. A. (1986). Limit theorems for fuzzy random variables, Proc. R. Soc. Lond. 407, 171–182.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Klein, E. and Thompson, A. C. (1984). Theory of Correspondences Including Applications to Mathematical Economics. J. Wiley & Sons, New York.zbMATHGoogle Scholar
  12. 12.
    Kim, B. K. and Kim, J. H. (1998) Stochastic integrals of set valued processes and fuzzy processes (preprint).Google Scholar
  13. 13.
    Kuratowski, K. (1965). Topology. 1, New York-London-Warszawa (Tranl. from French).Google Scholar
  14. 14.
    Li, S. and Ogura, Y. (1996). Fuzzy random variables, conditional expectations and fuzzy martingales, J. Fuzzy Math. 4, 905–927.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, S. and Ogura, Y. (1997). An optional sampling theorem for fuzzy valued martingales, Proc. IFSA’97, Prague 4, 9–14.Google Scholar
  16. 16.
    Li, S. and Ogura, Y. (1999). Convergence of set valued and fuzzy valued martingales, Fuzzy sets and Systems 101, 139–147.MathSciNetGoogle Scholar
  17. 17.
    Li, S. and Ogura, Y. (1998). Convergence of set valued sub-and super-martingales in the Kuratowski-Mosco sense, Ann. Probab. 26, 1384–1402.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    López-Diaz, M and Gil, M. A. (1998). Approximating integrably bounded fuzzy random variables in terms of the generalized Hausdorff metric, Inform. Sci. 104, 279–291.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Luu, D. Q. (1981). Representations and regularity of multivalued martingales, Acta Math. Vietn. 6, 29–40.zbMATHGoogle Scholar
  20. 20.
    Matheron, G. (1975). Random Sets and Integral Geometry, J. Wiley & Sons.Google Scholar
  21. 21.
    Molchanov, I. S. (1993). Limit Theorems for Unions of Random Closed Sets, Lect. Notes in Math. 1561, Springer-Verlag, Heidelberg.zbMATHGoogle Scholar
  22. 22.
    Mosco, U. (1969). Convergence of convex set and of solutions of variational inequalities, Advances Math. 3, 510–585.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mosco, U. (1971). On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35, 518–535.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Puri, M. L. and Ralescu, D. A. (1986). Fuzzy random variables, J. Math. Anal. Appl. 114, 406–422.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Puri, M. L. and Ralescu, D. A. (1991). Convergence theorem for fuzzy martingales, J. Math. Anal. Appl. 160, 107–121.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Râdström, H. (1952). An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3, 165–169.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ralescu, D. A. (1986) Radon-Nikodym theorem for fuzzy set-valued measures. In Fuzzy Sets Theory and Applications ( A. Jones et al., Eds.), D. Reidel Pub., Dordrecht, 39–50.CrossRefGoogle Scholar
  28. 28.
    Salinetti, G. and Roger J. B. Wets (1981). On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 226, 275–289.MathSciNetGoogle Scholar
  29. 29.
    Salinetti, G. and Roger J. B. Wets (1977). On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60, 211–226.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Thobie, C. G. (1974). Selections de multimesures, application a un theoreme de Radon-Nikodym multivoque. C. R. Acad. Sci. Paris Ser. A 279, 603–606.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wijsman, R. (1966) Convergence of sequences of convex sets, cones and fuctions, part 2, Trans. Amer. Math. Soc. 123, 32–45MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Zadeh, L. A. (1968). Probability measure of fuzzy events, J. Math. Anal. Appl. 23, 421–427.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Shoumei Li
    • 1
  • Yukio Ogura
    • 2
  1. 1.Department of Applied MathematicsBeijing Polytechnic UniversityChao Yang District, BeijingP. R. China
  2. 2.Department of MathematicsSaga UniversitySagaJapan

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