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On the Equivalence of Multiparameter Gaussian Processes

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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.

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References

  1. Baudoin F., Nualart D. (2003). Equivalence of Volterra processes. Stochastic Process. Appl. 107(2):327–350

    Article  MathSciNet  Google Scholar 

  2. Cheridito, P. (2002). Regularizing fractional Brownian motion with a view towards stock price modelling. Ph. D. thesis. ETH Zurich.

  3. Dozzi M. (1989). Stochastic Processes with a Multidimensional Parameter. Longman Scientific and Technical, Harlow

    MATH  Google Scholar 

  4. Erraoui M., Nualart D., Ouknine Y. (2003). Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet. Stoch. Dyn. 3(2):121–139

    Article  MathSciNet  Google Scholar 

  5. Hida, T., and Hitsuda, M. (1993). Gaussian Processes. Translations of Mathematical Monographs, Vol. 120, AMS, Providence, RI.

  6. Hitsuda M. (1968). Representation of Gaussian processes equivalent to Wiener process. Osaka J. Math. 5:299–312

    MathSciNet  Google Scholar 

  7. Houdre, C., and Villa, J. (2002). An example of infinite dimensional quasi-helix. Contemp. Math. 336.

  8. Kallianpur G. (1980). Stochastic Filtering Theory. Springer, New York

    MATH  Google Scholar 

  9. Kallianpur G., Oodaira H. (1973). Non-anticipative representation of equivalent Gaussian processes. Ann. Probability 1(1):104–122

    MathSciNet  Google Scholar 

  10. Pérez-Abreu, V., and Tudor, C. (2002). A transfer principle for multiple stochastic fractional integrals. Bolletin Sociedad Matematica Mexicana, 29 pp.

  11. Pipiras V., Taqqu M. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118(2): 251–291

    Article  MathSciNet  Google Scholar 

  12. Pipiras V., Taqqu M. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete?. Bernoulli 7(6):873–897

    MathSciNet  Google Scholar 

  13. Samko S.G., Kilbas A.A., Marichev O.I. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, Yverdon

    MATH  Google Scholar 

  14. Shepp L.A. (1966). Radon–Nikodym derivatives of Gaussian measures. Ann. Math. Stat. 37:321–354

    MathSciNet  Google Scholar 

  15. Sottinen T. (2004). On Gaussian processes equivalent in law to fractional Brownian motion. J. Theoret. Probab. 17(2):309–325

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  17. Tudor, C., and Tudor, M. (2005). On the multiparameter fractional Brownian motion. Adv. Math. Res., 6.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Tommi Sottinen

  2. Laboratoire de Probabilités, Université de Paris 6, 4, place Jussieu, 75252, Paris, France

    Ciprian A. Tudor

Authors
  1. Tommi Sottinen
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  2. Ciprian A. Tudor
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Correspondence to Tommi Sottinen.

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Sottinen, T., Tudor, C.A. On the Equivalence of Multiparameter Gaussian Processes. J Theor Probab 19, 461–485 (2006). https://doi.org/10.1007/s10959-006-0022-5

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  • DOI: https://doi.org/10.1007/s10959-006-0022-5

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