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Optional Sampling Theorem and Representation of Set-Valued Amart

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Soft Methods for Integrated Uncertainty Modelling

Part of the book series: Advances in Soft Computing ((AINSC,volume 37))

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Abstract

In this paper, we shall prove some properties of set-valued asymptotic martingale (amart for short) and provide an optional sampling theorem. We also prove a quasi Risez decomposition theorem for set-valued amarts. Then we shall discuss the existence of selections of set-valued amarts and give a representation theorem.

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References

  1. R. Aumann (1965). Integrals of set valued functions, J. Math. Anal. Appl. 12, 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Bagchi (1985). On a.s. convergence of classes of multivalued asymptotic martingales, Ann. Inst. H. Poincaré, Probabilités et Statistiques, 21, 313–321.

    MATH  MathSciNet  Google Scholar 

  3. G. Beer (1993). Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers.

    Google Scholar 

  4. A. Bellow (1978). Some aspects of the theory of the vector-valued amarts, “Vector Space Measure and Applications I” (ed. R. Aron and S. Dineen), Lect. Notes Math., 644, 57–67.

    Google Scholar 

  5. A. Bellow (1978). Uniform amarts: A class of asymptotic martingales for which strong almost sure convergence obtains. Z. Wahr. verw. Gebiete, 41, 177–191.

    Article  MATH  MathSciNet  Google Scholar 

  6. R.V. Chacon and L. Sucheston (1975). On convergence of vector-valued asymptotic martingales. Z. Wahr. verw. Gebiete. 33, 55–59.

    Article  MATH  MathSciNet  Google Scholar 

  7. G.A. Edgar and L. Suchistion (1976). Amarts: A class of asympotic martingales B. Discrete parameter. J. Multiv. Anal., 6, 193–221.

    Article  Google Scholar 

  8. G.A. Edgar and L. Suchistion (1976). Amarts: A class of asympotic martingales A. continuous parameter. J. Multiv. Anal., 6, 572–591.

    Article  Google Scholar 

  9. G.A. Edgar and L. Suchistion (1976). The Riesz decomposition for vectorvalued amarts, Z. Wahr. verw. Gebiete. 36, 85–92.

    Article  MATH  Google Scholar 

  10. C. Hess (1983). Measurability and integrability of the weak upper limit of a sequence of multifunctions, J. Math. Anal. Appl., 153, 226–249.

    Article  MathSciNet  Google Scholar 

  11. F. Hiai and H. Umegaki (1977). Integrals, conditional expectations and martingales of multivalued functions, Jour. Multiv. Anal., 7 149–182.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. de Korvin and R. Kleyle (1985). A convergence theorem for convex set valued supermartingales. Stoch. Anal. Appl., 3, 433–445.

    Article  MATH  Google Scholar 

  13. S. Li and Y. Ogura (1998). Convergence of set valued sub-supermartingales in the Kuratowski-Mosco sense, Ann. Probab., 26, 1384–1402.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Li and Y. Ogura (1999). Convergence of set valued and fuzzy valued martingals. Fuzzy Sets and Syst., 101, 453–461.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Li, Y. Ogura and V. Kreinovich (2002). Limit Theorems and Applications of Set-Valued and Fuzzy Sets-Valued Random Variables, Kluwer Academic Publishers.

    Google Scholar 

  16. D.Q. Luu (1981). Representations and regularity of multivalued martingales, Acta Math. Vietn., 6, 29–40.

    MATH  Google Scholar 

  17. D.Q. Luu (1984). Applications of set-valued Randon-Nikodym theorms to convergence of multivalued L 1-amarts, Math. Scand., 54, 101–114.

    MATH  MathSciNet  Google Scholar 

  18. D.Q. Luu (1985). Quelques resultats des amarts uniform nultivoques dans les espaces de Banach, CRAS, Paris, 300, 63–65.

    MATH  Google Scholar 

  19. D.Q. Luu (1986). Representation theorems for multi-valued (regular) L 1- amarts, Math. Scand., 58, 5–22.

    MATH  MathSciNet  Google Scholar 

  20. N.S. Papageorgiou (1985). On the theorey of Banach space valued multifunctions. 1. integration and conditional wxpectation, J. Multiv. Anal., 17, 185–206.

    Article  MATH  Google Scholar 

  21. N.S. Papageorgiou (1987). A convergence theorem for set valued multifunctions. 2. set valued martingales and set valued measures, J. Multiv. Anal., 17, 207–227.

    Article  Google Scholar 

  22. N.S. Papageorgiou (1995). On the conditional expectation and convergence properties of random sets. Trans. Amer. Math. Soc., 347, 2495–2515.

    Article  MATH  MathSciNet  Google Scholar 

  23. W.X. Zhang, Z.P. Wang and Y. Gao (1996). Set-valued random processes, Science Publisher. (in Chinese).

    Google Scholar 

  24. Z.P. Wang and X. Xue (1994). On convergence and closedness of multivalued martingales, Trans. Ameri. Math. Soc., 341, 807–827.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Beijing University of Technology, Pingleyuan, Chaoyang District, 100022, Beijing, P.R.China

    Shoumei Li

Authors
  1. Shoumei Li
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  2. Li Guan
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Editor information

Editors and Affiliations

  1. AI Group Department of Engineering Mathematics, University of Bristol, BS8 1TR, Bristol, UK

    Jonathan Lawry

  2. Statistics and Operations Research, Rey Juan Carlos University, C-Tulip˙n s/n MÛstoles28933, Spain

    Enrique Miranda

  3. Intelligent Systems Group Department of Electronics & Computer Science, University of Santiago de Compostela Santiago de Compostela, Spain

    Alberto Bugarin

  4. Department of Applied Mathematics, Beijing University of Technology, 100022, Beijing, P.R. China

    Shoumei Li

  5. Dpto. Estadistica e I.O y D.M. Calle Calvo Sotelo s/n, Universiad de Oviedo Fac. Ciencias, 33071, Oviedo, Spain

    Maria Angeles Gil

  6. Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland

    Przemys aw Grzegorzewski

  7. Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland

    Olgierd Hyrniewicz

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© 2006 Springer

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Li, S., Guan, L. (2006). Optional Sampling Theorem and Representation of Set-Valued Amart. In: Lawry, J., et al. Soft Methods for Integrated Uncertainty Modelling. Advances in Soft Computing, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-34777-1_18

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  • DOI: https://doi.org/10.1007/3-540-34777-1_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34776-7

  • Online ISBN: 978-3-540-34777-4

  • eBook Packages: EngineeringEngineering (R0)

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