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Some Properties of Pettis Integrable Multivalued Martingales

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Innovations in Smart Cities and Applications (SCAMS 2017)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 37))

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Abstract

We present a new result of Pettis integrable multivalued martingale. A result presented in this paper is a new version of uniformly integrable martingale in Pettis integration. A classical theorem of vector uniformly integrable martingale in Bochner integration is stated by Egghe [11]. A multivalued version of this result is proved by Hiai and Umegaki in [20].

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References

  1. Akhiat, F., Castaing, C., Ezzaki, F.: Some variational convergence results for multivalued martingales. Adv. Math. Econ. 13, 1–33 (2010)

    Article  MATH  Google Scholar 

  2. Amrani, A., Castaing, C.: Weak compactness in Pettis integration. Bull. Polish Acad. Sci. 44(2), 67–79 (1996)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Castaing, A., Valadier, M.: Convex Analysis and measurable multifunctions. University des sciences et techniques de languedoc U.E.R de, Mathématiques (1975)

    MATH  Google Scholar 

  5. Castaing, C., Saadoune, M.: Dunford Pettis types theorem and convergences in set valued integration. J. Nonlinear Convex Anal. 1(1), 37–71 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Castaing, C., Ezzaki, F., Lavie, M., Saadoune, M.: Weak star convergence of martingales in a dual space. Working paper, Département de Mathématiques, Université Montpellier II, September 2009

    Google Scholar 

  7. Castaing, C., Ezzaki, F.: SLLN for convex random sets and random lower semicontinuous integrands. Atti. Sem. Mat. Fis. Univ. Modena XLV, 527–553 (1997)

    Google Scholar 

  8. Castaing, C., Ezzaki, F.: Some convergence results for multivalued martingales in the limit. In: Séminaire d Analyse Convexe, Montpellier, exposé No 1 (1990)

    Google Scholar 

  9. Castaing, C., Ezzaki, F.: Mosco convergence for multivalued martingales in the limit. C. R. Acad. Sci. Paris Ser. I(312), 695–698 (1991)

    MATH  Google Scholar 

  10. Castaing, C., Ezzaki, F.: Convergence of conditional expectation for unbounded closed convex random sets. Stud. Math. 124(2), 133–148 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Egghe, L.: Stopping Time Techniques for Analysts and Probabilists. London Mathematical Society Lecture Note Series, no. 100. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  12. Egghe, L.: Convergence of adapted sequences of Pettis-integral of closed valued multifunctions. Pacific J. Math. 114(2), 345–366 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. El Amri, K.: intégral de Pettis et convergence, Thèse de Doctorat Univérsité Mohamed V Faculté des Sciences Rabat (2000)

    Google Scholar 

  14. El Harami, M.: Contribution aux espérences conditionnelles des ensembles aléatoires Pettis intégrables et Applications à la convergence des classes des martingales, Thèse de Doctorat Univérsité Sidi Mohamed Ben Abdellah Faculté des Sciences et Techniques Fès (2012)

    Google Scholar 

  15. Ezzaki, F.: A general dominated convergence theorem for unbounded random sets. Bull. Polish Acad. Sci. Math. 44(3), 353–361 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Ezzaki,F.: Contributions aux problèmes de convergence des suites adaptées et des ensembles aléatoires. Thèse de Doctorat, Université de Montpellier II (1993)

    Google Scholar 

  17. Ezzaki, F., El Harami, M.: General Pettis conditional expectation and convergence theorems. Int. J. Math. Statist. 11, 91–111 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Geitz, R.: Pettis integration. Proc. Am. Math. Soc. 8, 81–86 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Godet-Thobie, C., Satco, B.: Decomposability and uniform integrability in Pettis integration. Quaestiones Math. 29, 39–58 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7, 149–182 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marraffa, V.: Stochastic process of vector valued Pettis and MacShane integrable functions. Folia Math. 12(1), 25–37 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Musial, K.: Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces. Atti Sem. Mat. Fis. Univ. Modena XXXV, 159–166 (1987)

    Google Scholar 

  23. Musial, K.: The weak Radonn-Nikodym property in Banach space. Studia Math. 64(2), 151–173 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Musial, K.: Martingales of Pettis integrable functions. In: Proceedings of the Conference on Measure Theory. Oberwolfach Lecture Notes in Mathematics, vol. 794, pp. 324–339. Springer, Berlin (1980)

    Google Scholar 

  25. Musial, K.: Pettis integrability of multifunctions with values in arbitrary Banach space. J. Convex Anal. 18(3), 769–810 (2011)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Modélisation et Calcul Scientifique, Département de Mathématiques, Faculté des Sciences et Techniques, Fès, Morocco

    M’hamed El-Louh & Fatima Ezzaki

Authors
  1. M’hamed El-Louh
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  2. Fatima Ezzaki
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Correspondence to M’hamed El-Louh .

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Editors and Affiliations

  1. Computer Sciences Department, Faculty of Sciences and Techniques, Abdelmalek Essaadi University, Tangier, Morocco

    Mohamed Ben Ahmed

  2. Computer Sciences Department, Faculty of Sciences and Techniques, Abdelmalek Essaadi University, Tangier, Morocco

    Anouar Abdelhakim Boudhir

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El-Louh, M., Ezzaki, F. (2018). Some Properties of Pettis Integrable Multivalued Martingales. In: Ben Ahmed, M., Boudhir, A. (eds) Innovations in Smart Cities and Applications. SCAMS 2017. Lecture Notes in Networks and Systems, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-74500-8_92

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  • DOI: https://doi.org/10.1007/978-3-319-74500-8_92

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74499-5

  • Online ISBN: 978-3-319-74500-8

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