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Probability Theory and Related Fields

, Volume 164, Issue 1–2, pp 133–163 | Cite as

On infinitely divisible semimartingales

  • Andreas Basse-O’Connor
  • Jan RosińskiEmail author
Article

Abstract

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.

Keywords

Semimartingales Infinitely divisible processes Stationary processes Fractional processes 

Mathematics Subject Classification

60G48 60H05 60G51 60G17 

Notes

Acknowledgments

Jan Rosiński’s research was partially supported by a Grant #281440 from the Simons Foundation.

References

  1. 1.
    Barndorff-Nielsen, O.E.: Superposition of Ornstein–Uhlenbeck type processes. Teor. Veroyatnost. i Primenen. 45(2), 289–311 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basse, A.: Gaussian moving averages and semimartingales. Electron. J. Probab. 13(39), 1140–1165 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Basse, A.: Spectral representation of Gaussian semimartingales. J. Theor. Probab. 22(4), 811–826 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basse, A., Pedersen, J.: Lévy driven moving averages and semimartingales. Stoch. Process. Appl. 119(9), 2970–2991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Basse-O’Connor, A.: Representation of Gaussian semimartingales with application to the covariance function. Stochastics 82(4), 381–401 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Basse-O’Connor, A., Graversen, S.-E.: Path and semimartingale properties of chaos processes. Stoch. Process. Appl. 120(4), 522–540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Basse-O’Connor, A., Rosiński, J.: Characterization of the finite variation property for a class of stationary increment infinitely divisible processes. Stoch. Process. Appl. 123(6), 1871–1890 (2013)Google Scholar
  8. 8.
    Basse-O’Connor, A., Rosiński, J.: On the uniform convergence of random series in Skorohod space and representations of càdlàg infinitely divisible processes. Ann. Probab. 41(6), 4317–4341 (2013)Google Scholar
  9. 9.
    Beiglböck, M., Schachermayer, W., Veliyev, B.: A direct proof of the Bichteler–Dellacherie theorem and connections to arbitrage. Ann. Probab. 39(6), 2424–2440 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bender, C., Lindner, A., Schicks, M.: Finite variation of fractional Lévy processes. J. Theor. Probab. 25(2), 594–612 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bichteler, K.: Stochastic integration and \(L^p\)-theory of semimartingales. Ann. Probab. 9(1), 49–89 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)Google Scholar
  13. 13.
    Cheridito, P.: Gaussian moving averages, semimartingales and option pricing. Stoch. Process. Appl. 109(1), 47–68 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cherny, A. : When is a moving average a semimartingale? MaPhySto Research Report 2001-28 (2001)Google Scholar
  15. 15.
    Çinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54(2), 161–219 (1980)Google Scholar
  16. 16.
    Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49(3), 335–347 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Emery, M.: Covariance des semimartingales gaussiennes. C. R. Acad. Sci. Paris Sér. I Math. 295(12), 703–705 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fasen, V., Klüppelberg, C.: Extremes of supOU processes. In: Stochastic Analysis and Applications, Abel Symp., vol. 2, pp. 339–359. Springer, Berlin (2007)Google Scholar
  19. 19.
    Gal’chuk, L.I.: Gaussian semimartingales. In: Statistics and Control of Stochastic Processes (Moscow, 1984), Transl. Ser. Math. Eng., pp. 102–121. Optimization Software, New York (1985)Google Scholar
  20. 20.
    Hida, T., Hitsuda, M.: Gaussian processes. Translations of Mathematical Monographs, vol. 120. American Mathematical Society, Providence (1993). (Translated from the 1976 Japanese original by the authors)Google Scholar
  21. 21.
    Hudson, W.N., Tucker, H.G.: Asymptotic independence in the multivariate central limit theorem. Ann. Probab. 7(4), 662–671 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 2nd edn, vol. 288. Springer, Berlin (2003)Google Scholar
  23. 23.
    Jain, N.C., Monrad, D.: Gaussian quasimartingales. Z. Wahrsch. Verw. Gebiete 59(2), 139–159 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jeulin, T., Yor, M.: Moyennes mobiles et semimartingales. In: Séminaire de Probabilités, XXVII. Lecture Notes in Math., vol. 1557, pp. 53–77. Springer, Berlin (1993)Google Scholar
  25. 25.
    Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York). Springer, New York (1997)Google Scholar
  26. 26.
    Kardaras, C., Platen, E.: On the semimartingale property of discounted asset-price processes. Stoch. Process. Appl. 121(11), 2678–2691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Knight, F.B.: Foundations of the Prediction Process, Oxford Studies in Probability, vol. 1. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1992)Google Scholar
  28. 28.
    Ledoux, M., Talagrand, M.: Probability in Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23. Springer, Berlin (1991). (Isoperimetry and processes)Google Scholar
  29. 29.
    Liptser, R.S., Shiryayev, A.: Theory of Martingales, Mathematics and its Applications (Soviet Series), vol. 49. Kluwer Academic Publishers Group, Dordrecht (1989)Google Scholar
  30. 30.
    Meyer, P.-A.: Un résultat d’approximation. In: Séminaire de Probabilités, XVIII. Lecture Notes in Math., vol. 1059, pp. 268–270. Springer, Berlin (1984)Google Scholar
  31. 31.
    Mijatović, A., Urusov, M.:On the loss of the semimartingale property at the hitting time of a level. J. Theor. Probab. (2014). doi: 10.1007/s10959-013-0527-7
  32. 32.
    Protter, P.E.: Stochastic integration and differential equations. Stochastic Modelling and Applied Probability, 2nd edn, vol. 21 of Applications of Mathematics (New York). Springer, Berlin (2004)Google Scholar
  33. 33.
    Rajput, B.S., Rosiński, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82(3), 451–487 (1989)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rosiński, J.: On path properties of certain infinitely divisible processes. Stoch. Process. Appl. 33(1), 73–87 (1989)CrossRefzbMATHGoogle Scholar
  35. 35.
    Rosiński, J.: On series representations of infinitely divisible random vectors. Ann. Probab. 18(1), 405–430 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Lévy Processes, pp. 401–415. Birkhäuser, Boston (2001)Google Scholar
  37. 37.
    Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999). (Translated from the 1990 Japanese original, revised by the author)Google Scholar
  38. 38.
    Schnurr, A.: On the semimartingale nature of Feller processes with killing. Stoch. Process. Appl. 122(7), 2758–2780 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stricker, C.: Semimartingales gaussiennes—application au problème de l’innovation. Z. Wahrsch. Verw. Gebiete 64(3), 303–312 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Surgailis, D., Rosiński, J., Mandrekar, V., Cambanis, S.: Stable mixed moving averages. Probab. Theory Relat. Fields 97(4), 543–558 (1993)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Aarhus UniversityAarhusDenmark
  2. 2.University of TennesseeKnoxvilleUSA

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