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The fractional mixed fractional brownian motionand fractional brownian sheet

Published online by Cambridge University Press:  17 August 2007

Charles El-Nouty
Charles El-Nouty*
Affiliation:
U.F.R. de mathématiques, Université Paris VI, 175 rue du Chevaleret, 75013 Paris, France; elnouty@ccr.jussieu.fr
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Abstract


We introduce the fractional mixed fractional Brownian motion and fractionalBrownian sheet, and investigate the small ball behavior of its sup-norm statistic.Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

Keywords

Fractional Brownian motionfractional Brownian sheetlowerclassessmall ball probabilities.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 448 - 465
Copyright
© EDP Sciences, SMAI, 2007

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References

Ayache, A., Leger, S. and Pontier, M., Drap Brownien fractionnaire. Potential Anal. 178 (2002) 31–43. CrossRef
Ayache, A. and Xiao, Y., Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407–439. CrossRef
Belinsky, E. and Linde, W., Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators. J. Theoret. Probab. 15 (2002) 589–612. CrossRef
Borell, C., Convex measures on locally convex space. Math. Ark. Math. 12 (1974) 239–252. CrossRef
Cheridito, P., Mixed fractional Brownian motion. Bernoulli 7 (2001) 913–934. CrossRef
N.J. Cutland, P.E. Kopp and W. Willinger, Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. in Seminar on Stochastic Analysis, Random Fields and Applications, Progr. Probab. E. Bolthausen, M. Dozziand F. Russo Eds., Basel: Birkhauser 36 (1995) 327–351.
El-Nouty, C., On the lower classes of fractional Brownian motion. Studia Sci. Math. Hungar. 37 (2001) 363–390.
C. El-Nouty, Lower classes of fractional Brownian motion under Hölder norms, Limit Theorems in Probability and Statistics, Balatonlelle, 1999, I. Berkes, E. Csáki, M. Csörgő Eds., János Bolyai Mathematical Society, Budapest (2002) 7–34.
El-Nouty, C., The fractional mixed fractional Brownian motion. Statist. Probab. Lett. 65 (2003) 111–120. CrossRef
El-Nouty, C., Lower classes of integrated fractional Brownian motion. Studia Sci. Math. Hungar. 41 (2004) 17–38.
El-Nouty, C., The influence of a log-type small ball factor in the study of the lower classes. Bull. Sci. math. 129 (2005) 318–338. CrossRef
F. Gassmann and D. Bürki, Experimental investigation of atmospheric dispersion over the Swiss Plain – Experiment SIESTA, Boundary-Layer Meteorology, Springer Netherlands 41 (1987) 295–307.
F. Gassmann, P. Gaglione, S.E. Gryning, H. Hasenjäger, E. Lyck, H. Richner, B. Neiniger, S. Vogt and P. Thomas, Experimental Investigation of Atmospheric Dispersion over Complex Terrain in a Prealpine Region (experiment SIESTA) Swiss Federal Institute for Reactor Research EIR 604 (1986).
Kühn, T. and Linde, W., Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002) 669–696.
M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer Verlag, Berlin (1994).
Li, W.V., Gaussian, A correlation inequality and its applications to small ball probabilities. Elect. Comm. in Probab. 4 (1999) 111–118. CrossRef
Li, W.V. and Linde, W., Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris 326 (1998) 1329–1334. CrossRef
W.V. Li and Q.M. Shao, Gaussian Processes: Inequalities, Small Ball Probabilities and Applications, Stochastic Processes: Theory and Methods, Handbook of Statistics 19 (2001).
M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers, Dordrecht (1995).
Mason, D.M. and Shi, Z., Small Deviations for Some Multi-Parameter Gaussian Processes. J. Theoret. Probab. 14 (2001) 213–239. CrossRef
Monrad, D. and Rootzen, H., Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields 101 (1995) 173–192. CrossRef
P. Révész, Random walk in random and non-random environments, World Scientific Publishing Co., Teaneck, NJ (1990).
Samuelson, P.A., Rational theory of warrant pricing. Indust. Management Rev. 6 (1965) 13–31.
Talagrand, M., Lower classes of fractional Brownian motion. J. Theoret. Probab. 9 (1996) 191–213. CrossRef
Xiao, Y. and Zhang, T., Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 (2002) 204–226. CrossRef