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Aumann Stochastic Integrals

  • Michał Kisielewicz
Chapter
  • 36 Downloads
Part of the Springer Optimization and Its Applications book series (SOIA, volume 157)

Abstract

In this chapter we present the definition and properties of Aumann stochastic integrals of set-valued stochastic processes \(F:\mathbb {R}^+\times \Omega \rightarrow \mathrm {Cl}(\mathbb {R}^d)\) and subsets of the space \(\mathbb {L}^p(\mathbb {R}^+\times \Omega ,\beta \otimes \mathcal {F},\mathbb {R}^d)\). We begin with the definition and properties of the Aumann integrals of subsets of the space \({\mathbb {L}}^p(T,\mathcal {F},\mu ,X)\), where (X, |⋅|) is a separable Banach space.

References

  1. 4.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Basel (1990)zbMATHGoogle Scholar
  2. 5.
    Aumann, R.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)MathSciNetCrossRefGoogle Scholar
  3. 13.
    Distel, J., Uhl, J.: Vector Measures. American Mathematical Society, Providence (1977)CrossRefGoogle Scholar
  4. 17.
    Fryszkowski, A.: Fixed Point Theory for Decomposable Sets. Kluwer Academic Publishers, Dordrecht (2003)zbMATHGoogle Scholar
  5. 22.
    Hess, C.: On the parametrized integral of multifunction: unbounded case. Set Valued Anal. 15, 1–20 (2007)MathSciNetCrossRefGoogle Scholar
  6. 24.
    Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7, 149–182 (1977)MathSciNetCrossRefGoogle Scholar
  7. 25.
    Hildebrand, W.: Core and Equilibria of a Large Economy. Princeton University Press, Princeton (1974)Google Scholar
  8. 27.
    Hu, Sh., Papageorgiou, N.S.: Handbook of Multivalued Analysis I. Kluwer Academic Publishers, Dordecht (1997)CrossRefGoogle Scholar
  9. 33.
    Jung, E.J., Kim, J.H.: On set-valued stochastic integrals. Stoch. Anal. Appl. 21, 401–418 (2003)MathSciNetCrossRefGoogle Scholar
  10. 38.
    Kisielewicz, M.: Differential Inclusions and Optimal Control. Kluwer Academic Publishers, New York (1991)zbMATHGoogle Scholar
  11. 39.
    Kisielewicz, M.: Set-valued stochastic integrals and stochastic inclusions. Disc. Math. 13, 119–126 (1993)MathSciNetzbMATHGoogle Scholar
  12. 40.
    Kisielewicz, M.: Properties of solutions set of stochastic inclusions. J. Appl. Math. Stoch. Anal. 6, 217–236 (1993)MathSciNetCrossRefGoogle Scholar
  13. 48.
    Kisielewicz, M.: Stochastic Differential Inclusions and Applications. Springer, Berlin (2013)CrossRefGoogle Scholar
  14. 53.
    Kisielewicz, M.: Approximation theorems for set-valued stochastic integrals. Stoch. Anal. Appl. 36(3), 495–520 (2018)MathSciNetCrossRefGoogle Scholar
  15. 56.
    Kisielewicz, M., Michta, M.: Integraby bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449, 1892–1910 (2017)MathSciNetCrossRefGoogle Scholar
  16. 58.
    Kisielewicz, M., Motyl, J.: Selection theorems for set-valued stochastic integrals. Stoch. Anal. Appl. 37(2), 243–270 (2019)MathSciNetCrossRefGoogle Scholar
  17. 60.
    Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley-Interscience Publication, New York (1984)zbMATHGoogle Scholar
  18. 63.
    Li, J., Li, S.: Set–valued stochastic Lebesgue integral and representation theorems. Int. J. Comput. Intell. Syst. 1(2), 177–187 (2008)Google Scholar
  19. 64.
    Li, J., Li, S.: Aumann type set–valued Lebesgue integral and representation theorem. Int. J. Comput. Intell. Syst. 2(1), 83–90 (2009)MathSciNetCrossRefGoogle Scholar
  20. 65.
    Li, J., Mitocha, I., Okazaki, Y., Zang J.: On set-valued stochastic integrals in an M-type Banach space. J. Math. Anal. Appl. 350, 216–233 (2009)MathSciNetCrossRefGoogle Scholar
  21. 68.
    Malinowski, M.T., Michta, M., Sobolewska, J.: Set-valued and fuzzy stochastic differential equations driven by semimartingales. Nonlinear Anal. 79, 204–220 (2013)MathSciNetCrossRefGoogle Scholar
  22. 95.
    Wang, Z., Wang, R.: Stieltjes set-valued integrals. J. Appl. Prob. Stat. 13(3), 303–316 (1947)zbMATHGoogle Scholar
  23. 96.
    Wang, L., Xue, H.: Lebesgue-Stieltjes set-valued integrals. Basic Sci. J. Text. Univ. 16(4), 317–320 (2004)zbMATHGoogle Scholar
  24. 99.
    Zhang, J., Qi, J.: Set-valued stochastic integrals with respect to finite variation processes. Adv. Pure Math. 3, 15–19 (2013)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Michał Kisielewicz
    • 1
  1. 1.Faculty of MathematicsUniversity of Zielona GóraZielona GóraPoland

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