Some Classes of Proper Integrals and Generalized Ornstein–Uhlenbeck Processes

  • Andreas Basse-O’Connor
  • Svend-Erik Graversen
  • Jan Pedersen
Part of the Lecture Notes in Mathematics book series (LNM, volume 2046)


We give necessary and sufficient conditions for existence of proper integrals from 0 to infinity or from minus infinity to 0 of one exponentiated Lévy process with respect to another Lévy process. The results are related to the existence of stationary generalized Ornstein–Uhlenbeck processes. Finally, in the square integrable case the Wold-Karhunen representation is given.


Stochastic integration Lévy processes Generalized Ornstein–Uhlenbeck processes 


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  1. 1.
    O.E. Barndorff-Nielsen, N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63(2), 167–241 (2001)Google Scholar
  2. 2.
    A. Basse-O’Connor, S.-E. Graversen, J. Pedersen, A unified approach to stochastic integration on the real line. Thiele Research report 2010-08, 2010
  3. 3.
    A. Behme, Distributional properties of solutions of \(d{V }_{t} = {V }_{t-}d{U}_{t} + d{L}_{t}\) with Lévy Noise. Adv. Appl. Prob. 43, 688–711 (2011)Google Scholar
  4. 4.
    A. Behme, A. Lindner, R.A. Maller, Stationary solutions of the stochastic differential equation \(d{V }_{t} = {V }_{t-}d{U}_{t} + d{L}_{t}\) with Lévy Noise. Stochast. Process. Appl. 121(1), 91–108 (2011)Google Scholar
  5. 5.
    J. Bertoin, A. Lindner, R. Maller, On continuity properties of the law of integrals of Lévy processes. In Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol. 1934 (Springer, Berlin, 2008), pp. 137–159Google Scholar
  6. 6.
    P. Carmona, F. Petit, M. Yor, Exponential functionals of Lévy processes. In Lévy Processes (Birkhäuser Boston, Boston, MA, 2001), pp. 41–55Google Scholar
  7. 7.
    A. Cherny A. Shiryaev, On stochastic integrals up to infinity and predictable criteria for integrability. In Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol. 1857 (Springer, Berlin, 2005), pp. 165–185Google Scholar
  8. 8.
    R.A. Doney, R.A. Maller, Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theor. Probab. 15(3), 751–792 (2002)Google Scholar
  9. 9.
    K.B. Erickson, R.A. Maller, Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol. 1857 (Springer, Berlin, 2005), pp. 70–94Google Scholar
  10. 10.
    J. Jacod, Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics, vol. 714 (Springer, Berlin, 1979)Google Scholar
  11. 11.
    J. Jacod, P. Protter, Time reversal on Lévy processes. Ann. Probab. 16(2), 620–641 (1988)Google Scholar
  12. 12.
    J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. (Springer, Berlin (2003)Google Scholar
  13. 13.
    H. Kondo, M. Maejima, K. Sato, Some properties of exponential integrals of Lévy processes and examples. Electron. Commun. Probab. 11, 291–303 (electronic) (2006)Google Scholar
  14. 14.
    A. Lindner, R. Maller, Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes. Stochast. Process. Appl. 115(10), 1701–1722 (2005)Google Scholar
  15. 15.
    A. Lindner, K. Sato, Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein-Uhlenbeck processes. Ann. Probab. 37(1), 250–274 (2009)Google Scholar
  16. 16.
    A. Rocha-Arteaga, K. Sato, Topics in Infinitely Divisible Distributions and Lévy Processes, vol. 17 ofAportaciones Matemáticas: Investigación [Mathematical Contributions: Research] (Sociedad Matemática Mexicana, México, 2003)Google Scholar
  17. 17.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 ofCambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1999). Translated from the 1990 Japanese original, Revised by the authorGoogle Scholar
  18. 18.
    M. Yor, Exponential Functionals of Brownian Motion and Related Processes (Springer Finance. Springer, Berlin, 2001). With an introductory chapter by Hélyette Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. WilsonGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Basse-O’Connor
    • 1
  • Svend-Erik Graversen
    • 2
  • Jan Pedersen
    • 2
  1. 1.Department of MathematicsThe University of TennesseeKnoxvilleUSA
  2. 2.Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark

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