On the bifractional Brownian motion

Francesco Russo (1) , Ciprian A. Tudor (2, 3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 CES - Centre d'économie de la Sorbonne
3 SAMOS - Statistique Appliquée et MOdélisation Stochastique

This paper is devoted to analyze several properties of the bifractional Brownian motion introduced by Houdré and Villa. This process is a self-similar Gaussian process depending on two parameters $H$ and $K$ and it constitutes natural generalization of the fractional Brownian motion (which is obtained for $K=1$). We adopt the strategy of the stochastic calculus via regularization. Particular interest has for us the case $HK=\frac{1}{2}$. In this case, the process is a finite quadratic variation process with bracket equal to a constant times $t$ and it has the same order of self-similarity as the standard Brownian motion. It is a short memory process even though it is neither a semimartingale nor a Dirichlet process.

Mots clés: Bifractional Brownian motion; Dirichlet processes; Self-similar processes; Calculus via regularization

Domaines: Probabilités [math.PR]

Francesco Russo, Ciprian A. Tudor. On the bifractional Brownian motion. Stochastic Processes and their Applications, 2006, 116 (6), pp.830-856. ⟨10.1016/j.spa.2005.11.013⟩. ⟨hal-00130627⟩