Bifractional Brownian motion: existence and border cases

Lifshits, Mikhail; Volkova, Ksenia

doi:10.1051/ps/2015015

Lifshits, Mikhail ^1,² ; Volkova, Ksenia ¹

¹ St. Petersburg State University, 28 Stary, Peterhof, Bibliotechnaya pl.,2, 198504 St. Petersburg, Russia.
² MAI, Linköping University, 58183 Linköping, Sweden.

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 766-781.

Résumé

Bifractional Brownian motion (bfBm) is a centered Gaussian process with covariance $R^{(H, K)} (s, t) = 2^{- K} ({(| s |}^{2 H} + {| t |}^{2 H})^{K} - {| t - s |}^{2 HK}), s, t \in ℝ$

We study the existence of bfBm for a given pair of parameters $(H, K)$ and encounter some related limiting processes.

Zbl

DOI : 10.1051/ps/2015015

Classification : 60G15, 42A82
Mots clés : Bifractional Brownian motion, Gaussian process, fractional Brownian motion

Affiliations des auteurs :

Lifshits, Mikhail ^{1, 2} ; Volkova, Ksenia ¹

¹ St. Petersburg State University, 28 Stary, Peterhof, Bibliotechnaya pl.,2, 198504 St. Petersburg, Russia.
² MAI, Linköping University, 58183 Linköping, Sweden.

@article{PS_2015__19__766_0,
     author = {Lifshits, Mikhail and Volkova, Ksenia},
     title = {Bifractional {Brownian} motion: existence and border cases},
     journal = {ESAIM: Probability and Statistics},
     pages = {766--781},
     publisher = {EDP-Sciences},
     volume = {19},
     year = {2015},
     doi = {10.1051/ps/2015015},
     zbl = {1333.60075},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2015015/}
}

TY  - JOUR
AU  - Lifshits, Mikhail
AU  - Volkova, Ksenia
TI  - Bifractional Brownian motion: existence and border cases
JO  - ESAIM: Probability and Statistics
PY  - 2015
SP  - 766
EP  - 781
VL  - 19
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2015015/
DO  - 10.1051/ps/2015015
LA  - en
ID  - PS_2015__19__766_0
ER  -

%0 Journal Article
%A Lifshits, Mikhail
%A Volkova, Ksenia
%T Bifractional Brownian motion: existence and border cases
%J ESAIM: Probability and Statistics
%D 2015
%P 766-781
%V 19
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2015015/
%R 10.1051/ps/2015015
%G en
%F PS_2015__19__766_0

Lifshits, Mikhail; Volkova, Ksenia. Bifractional Brownian motion: existence and border cases. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 766-781. doi : 10.1051/ps/2015015. http://www.numdam.org/articles/10.1051/ps/2015015/

Bibliographie
Cité par

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover Publ., New York (1972).

X. Bardina and K. Es-Sebaiy, An extension of bifractional Brownian motion. Commun. Stoch. Anal. 5 (2011) 333–340. | Zbl

O.E. Barndorff-Nielsen and V. Perez-Abreu, Stationary and selfsimilar processes driven by Lévy processes. Stochastic Processes Appl. 84 (1999) 357–369. | Zbl

T. Bojdecki, L.G. Gorostiza and A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Electron. Commun. Probab. 12 (2007) 161–172. | Zbl

P. Cheredito, W. Kawaguchi and M. Maejima, Fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 (2003) 1–14. | Zbl

M. Lifshits and S. Cohen, Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres. ESAIM: PS 16 (2012) 165–221. | Zbl

D.S. Egorov, Annual student’s memoir. St. Petersburg State University (2014).

P. Embrechts and M. Maejima, Selfsimilar Processes. Princeton University Press (2002). | Zbl

C. Houdré and J. Villa, An example of infinite dimensional quasi-helix. Stochastic Models. Ser.: Contemp. Math. 336 (2003) 195–201. | Zbl

P. Lei and D. Nualart, A decomposition of the bifractional Brownian motion and some applications. Stat. Probab. Lett. 79 (2009) 619–624. | Zbl

M. Lifshits, Random Processes by Example. World Scientific, Singapore (2014). | Zbl

M. Lifshits and K. Volkova, Bifractional Brownian motion: existence and border cases. Preprint arXiv:1502.02217 (2015).

M.A. Lifshits, R. Schilling and I. Tyurin, A probabilistic inequality related to negative definite functions, in High Dimensional Probability VI, Vol. 66 of Ser. Progress in Probability, edited by C. Houdré et al. Birkhäuser, Basel. Preprint: arXiv:1205.1284 (2013). | Zbl

C. Ma, The Schoenberg–Lévy kernel and relationships among fractional Brownian motion, bifractional Brownian motion and others. Theory Probab. Appl. 57 (2013) 619–632. | Zbl

M. Marouby, Micropulses and different types of Brownian motion. J. Appl. Probab. 48, (2011) 792–810. | Zbl

R. Schilling, R. Song and Z. Vondraček, Bernstein Functions. De Gruyter, Berlin. Stud. Math. (2010) 37. | Zbl

G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994). | Zbl

C.A. Tudor and Y. Xiao, Sample path properties of bifractional Brownian motion. Bernoulli 13 (2007) 1023–1052. | Zbl

W. Wang, On $p$ -variation of bifractional Brownian motion. Appl. Math. J. Chinese Univ. 26 (2011) 127–141. | Zbl

Cité par Sources :