We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.
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References
Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lect. Notes, Monograph Series, vol 12, Hayward, California.
Adler R.J. (2004). Gaussian random fields on manifolds, Stoch. Analysis, Random Fields and Appl., 4, 3–20
Alos E. Mazet O., Nualart D. (2001). Stochastic calculus with respect to Gaussian processes. Annals of Prob. 29, 766–801
Benassi A., Jaffard S., Roux D. (1998). Elliptic Gaussian random processes. Rev. Mat. Ibe. 13, 19–89
Dudley R.M. (1973). Sample functions of the Gaussian process. Annals of Prob. 1(1): 66–103
Goldie C.M., Greenwood P.E. (1986). Characterizations of set-indexed Brownian motion and associated conditions forfinite-dimensional convergence. Annals of Prob. 14, 802–816
Herbin, E. (2006). From N-parameter fractional Brownian motions to N-parameter multifractional Brownian motions, to appear in Rocky Mountains. J. Math.
Herbin, E., and Lévy-Véhel, J. (2004). Fine analysis of the regularity of Gaussian processes: Stochastic 2-microlocal analysis, preprint.
Hida, T., Kuo, H. H., Potthoff, J., and Streit, L. (1993). White Noise: An Infinite Dimensional Calculus, Kluwer Academic Press.
Hu, Y., and Øksendal, B. (2003). Fractional white noise calculus and application to finance. Infinite Dimensional Analysis, Quantum Prob. and Related Topics. 6, 1–32.
Ivanoff, G. (2003). Set-indexed processes: distributions and weak convergence, in: Topics in Spatial Stochastic Processes, Lecture Notes in Mathematics, 1802, Springer, 85–126.
Ivanoff, G., and Merzbach, E. (2000). Set-Indexed Martingales, Chapman & Hall/CRC.
Ivanoff G., Merzbach E. (2002). Random censoring in set-indexed survival analysis. Ann. Appl. Prob., 12(3): 944–971
Kamont A. (1996). On the fractional anisotropic Wiener field. Prob. and Math. Stat. 16(1): 85–98
Kuo, H. H. (1996). White Noise Distribution Theory, CRC Press.
Khoshnevisan, D. (2002). Multiparameter Processes: an introduction to random fields, Springer.
Lévy-Véhel, J., and Riedi, R. (1997). Fractional Brownian motion and data traffic modeling: The other end of the spectrum. In Fractals in Engineering, Springer, 185–202.
Peltier, R., and Lévy-Véhel, J. (1995). Multifractional Brownian motion: definition and preliminary results, Rapport de recherche INRIA 2645.
Pesquet-Popescu, B. (1998). Modélisation bidimensionnelle de processus non stationnaires et application à l’étude du fond sous-marin, thèse de l’ENS Cachan.
Pyke R. (1983). A uniform central limit theorem for partial sum processes indexed by sets. London Math. Soc. Lect. Notes 79, 219–240
Sottinen, T. (2003). Fractional Brownian motion finance and queueing, PhD thesis of University of Helsinki.
Talagrand M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Annals of Prob., 23(2): 767–775
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable non-Gaussian random processes, Chapman & Hall/CRC.
Xiao Y. (1997). Hausdorff measure of the graph of fractional Brownian motion. Math. Proc. Camb. Phil. Soc. 122, 565–576
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Herbin, E., Merzbach, E. A Set-indexed Fractional Brownian Motion. J Theor Probab 19, 337–364 (2006). https://doi.org/10.1007/s10959-006-0019-0
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DOI: https://doi.org/10.1007/s10959-006-0019-0