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A Vector Girsanov Result and its Applications to Conditional Measures via the Birkhoff Integrability

  • Domenico Candeloro
  • Anna  Rita SambuciniEmail author
  • Luca Trastulli
Article
  • 41 Downloads

Abstract

Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and vice versa) are investigated to establish abstract versions of classical theorems of probability and stochastic processes. In particular, the Girsanov Theorem is extended and used with the treated methods.

Keywords

Birkhoff integral vector measure Girsanov Theorem 

Mathematics Subject Classification

28B20 58C05 28B05 46B42 46G10 18B15 

Notes

Acknowledgements

The Fondo Ricerca di Base 2018 University of Perugia—and the GNAMPA—INDAM (Italy) Project “Metodi di Analisi Reale per l’Approssimazione attraverso operatori discreti e applicazioni” (2019) supported this research.

References

  1. 1.
    Birkhoff, G.: Integration of functions with values in a Banach space. Trans. Am. Math. Soc. 38(2), 357–378 (1935)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Boccuto, A., Candeloro, D.: Differential calculus in Riesz spaces and applications to \(g\)-calculus. Med. J. Math. 8(3), 315–329 (2011).  https://doi.org/10.1007/s00009-010-0072-x MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boccuto, A., Candeloro, D., Sambucini, A.R.: Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 363–383 (2015).  https://doi.org/10.4171/RLM/710 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boccuto, A., Sambucini, A.R.: A note on comparison between Birkhoff and Mc Shane integrals for multifunctions. Real Anal. Exchange 37(2), 3–15 (2012).  https://doi.org/10.14321/realanalexch.37.2.0315 CrossRefGoogle Scholar
  5. 5.
    Candeloro, D., Croitoru, A., Gavriluţ, A., Sambucini, A.R.: An extension of the Birkhoff integrability for multifunctions. Mediterr. J. Math. 13(5), 2551–2575 (2016).  https://doi.org/10.1007/s00009-015-0639-7 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Candeloro, D., Di Piazza, L., Musial, K., Sambucini, A.R.: Relations among gauge and Pettis integrals for multifunctions with weakly compact convex values. Annali di Matematica 197(1), 171–183 (2018).  https://doi.org/10.1007/s10231-017-0674-z MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candeloro, D., di Piazza, L., Musial, K., Sambucini, A.R.: Some new result on the integration for multifunctions. Ricerche di Matematica 67(2), 361–372 (2018).  https://doi.org/10.1007/s11587-018-0376-x MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candeloro, D., Sambucini, A. R.: Order-type Henstock and Mc Shane integrals in Banach lattice setting. In: Proceedings of the SISY 2014—IEEE 12th International Symposium on Intelligent Systems and Informatics, pp. 55–59. ISBN 978-1-4799-5995-2 (2014).  https://doi.org/10.1109/SISY.2014.6923557
  9. 9.
    Candeloro, D., Sambucini, A.R.: Comparison between some norm and order gauge integrals in Banach lattices. PanAm. Math. J. 25(3), 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Candeloro, D., Labuschagne, C.C.A., Marraffa, V., Sambucini, A.R.: Set-valued Brownian motion. Ricerche di Matematica 67(2), 347–360 (2018).  https://doi.org/10.1007/s11587-018-0372-1 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Candeloro, D., Sambucini, A. R.: A Girsanov result through birkhoff integral. In: Gervasi, O. et al. (eds.) Computational Science and its Applications ICCSA 2018, LNCS 10960, pp. 676–683 (2018).  https://doi.org/10.1007/978-3-319-95162-1_47 CrossRefGoogle Scholar
  12. 12.
    Caponetti, D., Marraffa, V., Naralenkov, K.: On the integration of Riemann-measurable vector-valued functions. Monatsh. Math. 182, 513–536 (2017).  https://doi.org/10.1007/s00605-016-0923-z MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cascales, B., Rodríguez, J.: Birkhoff integral for multi-valued functions. Special issue dedicated to John Horváth. J. Math. Anal. Appl. 297(2), 540–560 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cascales, B., Rodríguez, J.: The Birkhoff integral and the property of Bourgain. Math. Ann. 331(2), 259–279 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
  16. 16.
    Cichoń, K., Cichoń, M.: Some applications of nonabsolute integrals in the theory of differential inclusions in Banach spaces. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds.) Vector Measures, Integration and Related Topics, Operator Theory: Advances and Applications, vol. 201, pp. 115–124. Birkha̋user, ISBN: 978-3-0346-0210-5 (2010)Google Scholar
  17. 17.
    Croitoru, A., Mastorakis, N.: Estimations, convergences and comparisons on fuzzy integrals of Sugeno, Choquet and Gould type. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1205–1212 (2014).  https://doi.org/10.1109/FUZZ-IEEE.2014.6891590
  18. 18.
    Croitoru, A., Gavriluţ, A., Iosif, A.E.: Birkhoff weak integrability of multifunctions. Int. J. Pure Math. 2, 47–54 (2015)Google Scholar
  19. 19.
    Croitoru, A., Gavriluţ, A.: Comparison between Birkhoff integral and Gould integral. Mediterr. J. Math. 12(2), 329–347 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Croitoru, A., Iosif, A., Mastorakis, N., Gavriluţ, A.: Fuzzy multimeasures in Birkhoff weak set-valued integrability. In: 2016 Third International Conference on Mathematics and Computers in Sciences and in Industry (MCSI), Chania, pp. 128–135 (2016).  https://doi.org/10.1109/MCSI.2016.034
  21. 21.
    Di Piazza, L., Marraffa e B.Satco, V.: Set valued integrability in non separable Fréchet spaces and applications. Math. Slovaca 66(5), 1119–1138 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fernández, A., Mayoral, F., Naranjo, F., Rodríguez, J.: On Birkhoff integrability for scalar functions and vector measures. Monatsh. Math. 157(2), 131–142 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fremlin, D.H.: Integration of vector-valued functions. Atti Semin. Mat. Fis. Univ. Modena 42, 205–211 (1994)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fremlin, D.H., Mendoza, J.: On the integration of vector-valued functions Illinois. J. Math. 38(1), 127–147 (1994)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fremlin, D.H.: The McShane and Birkhoff integrals of vector-valued functions, University of Essex Mathematics Department Research Report 92-10, version of 13.10.04. http://www.essex.ac.uk/math8/staff/fremlin/preprints.htm
  26. 26.
    Grobler, J.J., Labuschagne, C.C.A.: Girsanov’s theorem in vector lattices. Positivity 23(5), 1065–1099 (2019)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kaliaj, S.B.: Some full characterizations of differentiable functions. Mediterr. J. Math. 12, 639–646 (2015).  https://doi.org/10.1007/s00009-014-0458-20378-620X/15/030639-8 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kaliaj, S.B.: Differentiability and weak differentiability. Mediterr. J. Math. 13, 2801–2811 (2016).  https://doi.org/10.1007/s00009-015-0656-61660-5446/16/052801-11 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Labuschagne, C.C.A., Marraffa, V.: On set-valued cone absolutely summing map. Cent. Eur. J. Math. 8(1), 148–157 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Marraffa, V.: A Birkhoff type integral and the Bourgain property in a locally convex space. Real Anal. Exchange 32(2), 409–428 (2006–2007).  https://doi.org/10.14321/realanalexch.32.2.0409 MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mikosch, T.: Elementary Stochastic Calculus (with finance in view). World Scientific Publishing Co., Inc., River Edge, NJ (1998). ISBN: 981-02-3543-7 (source MathScinet)Google Scholar
  32. 32.
    Mushambi, N., Watson, B.A., Zinsou, B.: Generalization of the theorems of Barndorff-Nielsen and Balakrishnan-Stepanov to Riesz spaces. Positivity (2019).  https://doi.org/10.1007/s11117-019-00705-0 CrossRefGoogle Scholar
  33. 33.
    Novikov, A.: A certain identity for stochastic integrals. Theory Probab. Appl. 17(4), 717–720 (1972)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pettis, B.J.: On the integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Potyrala, M.: Some remarks about Birkhoff and Riemann-Lebesgue integrability of vector valued function. Tatra Mt. Math Publ. 35, 97–106 (2007)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Potyrala, M.: The Birkhoff and variational McShane integrals of vector valued functions. Folia Mathematica 13(1), 31–39 (2006)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Rodríguez, J.: Absolutely summing operators and integration of vector-valued functions. J. Math. Anal. Appl. 316(2), 579–600 (2006)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rodríguez, J.: Pointwise limits of Birkhoff integrable functions. Proc. Am. Math. Soc. 137(1), 235–245 (2009)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rodríguez, J.: Universal Birkhoff integrability in dual Banach spaces. Quaest. Math. 28(4), 525–536 (2005)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Stoica, G.: Limit laws for martingales in vector lattices. J. Math. Anal. Appl. 476(2), 715–719 (2019)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Trastulli, L.: Methods of integration with respect to non additive measures and some applications. Tesi Magistrale (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesUniversity of PerugiaPerugiaItaly

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