Journal of Theoretical Probability

, Volume 19, Issue 2, pp 461–485 | Cite as

On the Equivalence of Multiparameter Gaussian Processes

  • Tommi SottinenEmail author
  • Ciprian A. Tudor

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.


Brownian sheet fractional Brownian sheet equivalence of Gaussian processes Hitsuda representation Shepp representation canonical representation of Gaussian processes Girsanov theorem stochastic differential equations 

Mathematics Subject Classification (2000)

Primary 60G15 Secondary 60G30 Secondary 60H05 


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  1. 1.
    Baudoin F., Nualart D. (2003). Equivalence of Volterra processes. Stochastic Process. Appl. 107(2):327–350CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheridito, P. (2002). Regularizing fractional Brownian motion with a view towards stock price modelling. Ph. D. thesis. ETH Zurich.Google Scholar
  3. 3.
    Dozzi M. (1989). Stochastic Processes with a Multidimensional Parameter. Longman Scientific and Technical, HarlowzbMATHGoogle Scholar
  4. 4.
    Erraoui M., Nualart D., Ouknine Y. (2003). Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet. Stoch. Dyn. 3(2):121–139CrossRefMathSciNetGoogle Scholar
  5. 5.
    Hida, T., and Hitsuda, M. (1993). Gaussian Processes. Translations of Mathematical Monographs, Vol. 120, AMS, Providence, RI.Google Scholar
  6. 6.
    Hitsuda M. (1968). Representation of Gaussian processes equivalent to Wiener process. Osaka J. Math. 5:299–312MathSciNetGoogle Scholar
  7. 7.
    Houdre, C., and Villa, J. (2002). An example of infinite dimensional quasi-helix. Contemp. Math. 336.Google Scholar
  8. 8.
    Kallianpur G. (1980). Stochastic Filtering Theory. Springer, New YorkzbMATHGoogle Scholar
  9. 9.
    Kallianpur G., Oodaira H. (1973). Non-anticipative representation of equivalent Gaussian processes. Ann. Probability 1(1):104–122MathSciNetGoogle Scholar
  10. 10.
    Pérez-Abreu, V., and Tudor, C. (2002). A transfer principle for multiple stochastic fractional integrals. Bolletin Sociedad Matematica Mexicana, 29 pp.Google Scholar
  11. 11.
    Pipiras V., Taqqu M. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118(2): 251–291CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pipiras V., Taqqu M. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete?. Bernoulli 7(6):873–897MathSciNetGoogle Scholar
  13. 13.
    Samko S.G., Kilbas A.A., Marichev O.I. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, YverdonzbMATHGoogle Scholar
  14. 14.
    Shepp L.A. (1966). Radon–Nikodym derivatives of Gaussian measures. Ann. Math. Stat. 37:321–354MathSciNetGoogle Scholar
  15. 15.
    Sottinen T. (2004). On Gaussian processes equivalent in law to fractional Brownian motion. J. Theoret. Probab. 17(2):309–325CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tudor C.A., Viens F. (2003) Itô formula and local time for the fractional Brownian sheet. Electron. J. Prob. 8:1–31 (paper 14)MathSciNetGoogle Scholar
  17. 17.
    Tudor, C., and Tudor, M. (2005). On the multiparameter fractional Brownian motion. Adv. Math. Res., 6.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Laboratoire de ProbabilitésUniversité de Paris 6ParisFrance

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