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Lévy Subordinators in Cones of Fuzzy Sets

  • Jan Schneider
  • Roman Urban
Article
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Abstract

The general problem of how to construct stochastic processes which are confined to stay in a predefined cone (in the one-dimensional but also multi-dimensional case also referred to as subordinators) is of course known to be of great importance in the theory and a myriad of applications. In this paper we see how this may be dealt with on the metric space of fuzzy sets/vectors: By first relating with each proper convex cone C in \(\mathbb {R}^{n}\) a certain cone of fuzzy vectors \(C^*\) and subsequently using some very specific Banach space techniques we have been able to produce as many pairs \((L^*_t, C^*)\) of fuzzy Lévy processes \(L^*_t\) and cones \(C^*\) of fuzzy vectors such that \(L^*_t\) are \(C^*\)-subordinators.

Keywords

Lévy processes in Banach spaces K-positive fuzzy Lévy process Cone-subordinators Cones in Banach spaces Pettis integral Bochner integral Convex sets Fuzzy sets Fuzzy vectors Support function Hausdorff distance 

Mathematics Subject Classification (2010)

G0G51 60G20 46C05 

References

  1. 1.
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  2. 2.
    Bañuelos, R., DeBlassie, D.: The exit distribution of iterated Brownian motion in cones. Stoch. Process. Appl. 116(1), 36–69 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bañuelos, R., Smits, R.G.: Brownian motion in cones. Probab. Theory Relat. Fields 108(3), 299–319 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bochner, S.: Integration von Funktionen, deren Wert die Elemente eines Vektorraumes sind. Fund. Math. 20(1), 262–276 (1933)CrossRefMATHGoogle Scholar
  5. 5.
    Bongiorno, E.G.: A note on fuzzy set-valued Brownian motion. Stat. Probab. Lett. 82(4), 827–832 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Davydov, Y., Molchanov, I., Zuyev, S.: Strictly stable distributions on convex cones. Electron. J. Probab. 13(11), 259–321 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge (2002)Google Scholar
  8. 8.
    DeBlassie, D.: Brownian motion in a quasi-cone. Probab. Theory Relat. Fields 154(1–2), 127–148 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. Fuzzy Sets Syst. 35(2), 241–249 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diamond, P., Kloeden, P.: Metric Spaces of Fuzzy Sets. World Scientific, Singapore (1994)MATHGoogle Scholar
  11. 11.
    Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. Fuzzy Sets Syst. 100(Supplement), 63–71 (1999)CrossRefGoogle Scholar
  12. 12.
    Feng, Y., Hu, L., Shu, H.: The variance and covariance of fuzzy random variables and their applications. Fuzzy Sets Syst. 120(2), 487–497 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Garbit, R., Raschel, K.: On the exit time from a cone for random walks with drift. Revista Matematica Iberoamericana 32(2), 511–532 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gikhman, I.I., Skorohod, A.V.: The Theory of Stochastic Processes, vol. II. Springer, Berlin (1975)Google Scholar
  15. 15.
    Körner, R.: Linear models with random fuzzy variables. Dissertation, TU Bergakademie Freiburg (1997)Google Scholar
  16. 16.
    Krätschmer, V.: Some complete metrics on spaces of fuzzy subsets. Fuzzy Sets Syst. 130(3), 357–365 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pérez-Abreu, A., Rocha-Arteaga, A.: On the Lévy–Khintchine representation of Lévy processes in cones of Banach spaces. Publ. Mat. Urug. 11, 41–55 (2006)MathSciNetGoogle Scholar
  18. 18.
    Pérez-Abreu, A., Rosiński, J.: Representation of infinitely divisible distributions on cones. J. Theor. Probab. 20(3), 535–544 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sato, K.-I.: Łprocesses and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2004)Google Scholar
  20. 20.
    Schneider, J., Urban, R.: A proof of Donsker’s invariance principle based on support functions of fuzzy random vectors. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 26(1), 27–42 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Viertl, R.: Statistical Methods for Fuzzy Data. Wiley, New York (2011)CrossRefMATHGoogle Scholar
  22. 22.
    Yosida, K.: Functional Analysis, 3rd edn. Springer, New York (1971)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science and ManagementWroclaw University of Science and TechnologyWrocławPoland
  2. 2.Institute of MathematicsWroclaw UniversityWrocławPoland

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