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Positivity

, Volume 15, Issue 4, pp 617–637 | Cite as

Doob’s optional sampling theorem in Riesz spaces

  • J. J. GroblerEmail author
Article

Abstract

The notions of stopping times and stopped processes for continuous stochastic processes are defined and studied in the framework of Riesz spaces. This leads to a formulation and proof of Doob’s optional sampling theorem.

Keywords

Stochastic process with continuous parameter Vector lattice Stopping time Stopped process Conditional expectation Martingale 

Mathematics Subject Classification (2010)

06F20 46A40 60G44 60G07 

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References

  1. 1.
    Aliprantis C.D., Burkinshaw O.: Locally solid Riesz spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  2. 2.
    Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press Inc., Orlando (1985)zbMATHGoogle Scholar
  3. 3.
    DeMarr R.: A martingale convergence theorem in vector lattices. Canad. J. Math. 18, 424–432 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fremlin D.H.: Topological Riesz spaces and measure theory. Cambridge University Press, Cambridge (1974)zbMATHCrossRefGoogle Scholar
  5. 5.
    Grobler J.J.: Continuous stochastic processes in Riesz spaces: the Doob–Meyer decomposition. Positivity 14, 731–751 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Grobler J.J., de Pagter B.: Operators representable as multiplication-conditional expectation operators. J. Oper. Theory 48, 15–40 (2002)zbMATHGoogle Scholar
  7. 7.
    Karatzas I., Shreve S.E.: Brownian motion and stochastic calculus, Graduate Texts in Mathematics. Springer, New York (1991)CrossRefGoogle Scholar
  8. 8.
    Kuo, W.-C.: Stochastic processes on vector lattices, Thesis, University of the Witwatersrand, Johannesburg (2006)Google Scholar
  9. 9.
    Kuo W.-C., Labuschagne C.C.A., Watson B.A.: Discrete time stochastic processes on Riesz spaces. Indag. Math. 15, 435–451 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: An upcrossing theorem for martingales on Riesz spaces, Soft methodology and random information systems, pp. 101–108. Springer, Berlin (2004)Google Scholar
  11. 11.
    Kuo W.-C., Labuschagne C.C.A., Watson B.A.: Conditional expectation on Riesz spaces. J. Math. Anal. Appl. 303, 509–521 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Zero-one laws for Riesz space and fuzzy random variables, Fuzzy logic, soft computing and computational intelligence. pp. 393–397. Springer-Verlag and Tsinghua University Press, Beijing (2005)Google Scholar
  13. 13.
    Kuo W.-C., Labuschagne C.C.A., Watson B.A.: Convergence of Riesz space martingales. Indag. Math. 17, 271–283 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kuo W.-C., Labuschagne C.C.A., Watson B.A.: Ergodic theory and the strong law of large numbers on Riesz spaces. J. Math. Anal. Appl. 325, 422–437 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Luxemburg W.A.J., Zaanen A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)zbMATHGoogle Scholar
  16. 16.
    Meyer-Nieberg P.: Banach Lattices. Springer, Berlin (1991)zbMATHCrossRefGoogle Scholar
  17. 17.
    Schaefer H.H.: Banach lattices and positive operators. Springer, Berlin (1974)zbMATHGoogle Scholar
  18. 18.
    Stoica Gh.: Martingales in vector lattices I. Bull. Math. Soc. Sci. Math. Roumanie 34(4), 357–362 (1990)MathSciNetGoogle Scholar
  19. 19.
    Stoica Gh.: Martingales in vector lattices II. Bull. Math. Soc. Sci. Math. Roumanie 35(1–2), 155–158 (1991)MathSciNetGoogle Scholar
  20. 20.
    Stoica, Gh.: Order convergence and decompositions of stochastic processes, An. Univ. Bucuresti Mat. Anii XLII–XLIII (1993–1994), 85–91Google Scholar
  21. 21.
    Stoica Gh.: The structure of stochastic processes in normed vector lattices. Stud. Cerc. Mat. 46(4), 477–486 (1994)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Troitsky V.: Martingales in Banach lattices. Positivity 9, 437–456 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Vulikh, B.Z.: Introduction to the theory of partially ordered spaces, Translated from the Russian by L.F. Boron, Wolters-Noordhoff Scientific Publications Ltd. Groningen (1967)Google Scholar
  24. 24.
    Watson B.A.: An Ândo-Douglas type theorem in Riesz spaces with a conditional expectation. Positivity 13(3), 543–558 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Zaanen A.C.: Riesz spaces II. North-Holland, Amsterdam (1983)zbMATHGoogle Scholar
  26. 26.
    Zaanen A.C.: Introduction to Operator theory in Riesz spaces. Springer, Berlin (1991)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.North-West UniversityPotchefstroomSouth Africa

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