Skip to main content
SpringerLink
Log in

Some Large Deviations Principles for Time-Changed Gaussian Processes

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract. Let X = (X(t))t≥0 (X(0) = 0) be a continuous centered Gaussian process on a probability space (Ω,F,P), and let (Yt)t∈[0,1] (Y0 = 0) be a continuous process (on the same probability space) with nondecreasing paths, independent of X. Define the time-changed Gaussian process Zt = X(Yt), t ∈ [0, 1]. In this paper, we investigate a problem of finite-dimensional large deviations and a problem of pathwise large deviations for time-changed continuous Gaussian processes. As applications, we considered subordinated Gaussian processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Alain and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer, Boston, MA, 2004.

    MATH  Google Scholar 

  2. R. Azencott, Grandes déviations et applications, in Eighth Saint Flour Probability Summer School—1978 (Saint Flour, 1978), Lect. Notes Math., Vol. 774, Springer, Berlin, 1980, pp. 1–176.

    Google Scholar 

  3. P. Carr and L. Wu, Time-changed Lévy processes and option pricing, J. Financ. Econ., 71(1):113–141, 2004.

    Article  Google Scholar 

  4. N.R. Chaganty, Large deviations for joint distributions and statistical applications, Sankhyā, Ser. A, 59(2):147–166, 1997.

    MathSciNet  MATH  Google Scholar 

  5. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Stoch. Model. Appl. Probab., Vol. 38, Springer, Berlin, 2010.

    Google Scholar 

  6. J.D. Deuschel and D.W. Stroock, Large Deviations, Pure Appl. Math., Vol. 137, Academic Press, Boston, MA, 1989.

    Google Scholar 

  7. B. Djehiche and A. Schied, Large deviations for hierarchical systems of interacting jump processes, J. Theor. Probab., 11(1):1–24, 1998.

    MathSciNet  MATH  Google Scholar 

  8. J. Gajda and M. Magdziarz, Large deviations for subordinated Brownian motion and applications, Stat. Probab. Lett., 88:149–156, 2014.

    MathSciNet  MATH  Google Scholar 

  9. F. Giorgi and B. Pacchiarotti, Large deviations for conditionalVolterra processes, Stochastic Anal. Appl., 35(2):191–210, 2017.

    MathSciNet  MATH  Google Scholar 

  10. M.G. Hahn, K. Kobayashi, J. Ryvkina, and S. Umarov, On time-changed Gaussian processes and their associated Fokker–Planck–Kolmogorov equations, Electron. Commun. Probab., 16:150–164, 2011.

    MathSciNet  MATH  Google Scholar 

  11. T. Hida and M. Hitsuda, Gaussian Processes, Transl. Math. Monogr., Vol.120, AMS, Providence, RI, 1993.

    Google Scholar 

  12. C. Macci and B. Pacchiarotti, Exponential tightness for Gaussian processes, with applications to some sequences of weighted means, Stochastics, 89(2):469–484, 2017.

    MathSciNet  MATH  Google Scholar 

  13. D.B. Madan and M. Yor, Representing the CGMY and Meixner Lévy processes as time changed Brownian motions, J. Comput. Finance, 12(1):27–47, 2008.

    MathSciNet  Google Scholar 

  14. M.M. Meerschaert, E. Nane, and Y.Xiao, Large deviations for local time fractional Brownian motion and applications, J. Math. Anal. Appl., 346(2):432–445, 2008.

    MathSciNet  MATH  Google Scholar 

  15. M.M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model. Nat. Phenom., 8(2):1–16, 2013.

    MathSciNet  MATH  Google Scholar 

  16. B. Pacchiarotti and A. Pigliacelli, Large deviations for conditionally Gaussian processes: Estimates of level crossing probability, Mod. Stoch., Theory Appl., 5(4):483–499, 2018.

    MathSciNet  MATH  Google Scholar 

  17. K. Sato, Lévy Processes and Infinitely Divisible Distributions, Camb. Stud. Adv. Math., Vol. 68, Cambridge Univ. Press, Cambridge, 2013.

  18. A. Schied, Criteria for exponential tightness in path spaces, in Nichtlineare Partielle Differentialgleichungen, Vol. 774, Rheinische Friedrich-Wilhelms-Universität Bonn, Sonderforschungsbeeich 256, 1994, pp. 1–176.

    Google Scholar 

  19. A. Schied, Moderate deviations and functional LIL for super-Brownian motion, Stochastic Processes Appl., 72(1): 11–25, 1997.

    MathSciNet  MATH  Google Scholar 

  20. W. Wang and Z. Chen, Large deviations for subordinated fractional Brownian motion and applications, J. Math. Anal. Appl., 458(2):1678–1692, 2018.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133, Roma, Italy

    Barbara Pacchiarotti

Authors
  1. Barbara Pacchiarotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barbara Pacchiarotti.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pacchiarotti, B. Some Large Deviations Principles for Time-Changed Gaussian Processes. Lith Math J 60, 513–529 (2020). https://doi.org/10.1007/s10986-020-09494-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-020-09494-6

MSC

Keywords

Search

Navigation