THE GENERALIZED BIFRACTIONAL BROWNIAN MOTION
Main Article Content
Abstract
To extend several known centered Gaussian processes, we introduce a new centered Gaussian process, named the generalized bifractional Brownian motion. This process depends on several parameters, namely α > 0 , β>0 , 0<H<1 and 0<K≤1 . When K=1, we investigate its convexity properties. Then, when 2HK≤ 1, we prove that this process is an element of the QHASI class, a class of centered Gaussian processes, which was introduced in 2015
Downloads
Article Details
References
Kahane J-P. Some random series of func-tions. Second Edition. Cambridge studies in advanced mathematics, Cambridge Univer-sity Press, 1985.
Houdré C., Villa J. An example of infinite dimensional quasi-helix. // Comtemp. Math., 2003, Vol. 336, pp. 195-201.
Bojdecki T., Gorostiza L.G., Talarczyk A. Sub-fractional Brownian motion and its relation to occupation times // Statist. Probab. Lett., 2004, Vol. 69, pp. 405-419.
El-Nouty C., Journé J-L., The sub-bifractional Brownian motion // Studia Sci. Math. Hungar., 2013, Vol. 50, pp. 67-121.
Zili M., Generalized fractional Brownian motion, Modern Stochastics: Theory and Applications, 2017, Vol. 4, pp. 15-24.
Berman S.M. Limit theorems for the max-imum term in stationary sequences // Ann. Math. Statist, 1964, Vol. 35, pp. 502-516.
Samorodnitsky G., Taqqu M.S., Stable non-Gaussian random processes. Chap-man & Hall, 1994.
El-Nouty C., Journé J-L. Upper classes of the bifractional Brownian motion // Studia Sci. Math. Hungar, 2011, Vol. 48, pp. 371-407.
El-Nouty C., On approximately stationary Gaussian processes // International Journal for Computational Civil and Structural En-gineering, 2015, Vol. 11, pp. 15-26.
Bojdecki T., Gorostiza L.G., Talarczyk A. Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems // Electron. Comm. Probab., 2007, Vol. 12, pp. 161-172.
El-Nouty C. The increments of a bifractional Brownian motion // Studia Sci. Math. Hungar., 2009, Vol. 46, pp. 449–478.
El-Nouty C., Journé J-L., The sub-bifractional Brownian motion // Studia Sci. Math. Hungar, 2013, Vol. 50, pp. 67-121.
Russo F., Tudor C.A. On bifractional Brownian motion // Stochastic Process. Appl., 2006, Vol. 116, pp. 830-856.