Discretizing the fractional Levy area

Andreas Neuenkirch (1) , Samy Tindel (2, 3) , Jérémie Unterberger (2)

1 Institut für Mathematik
2 IECN - Institut Élie Cartan de Nancy
3 BIGS - Biology, genetics and statistics

In this article, we give sharp bounds for the Euler- and trapezoidal discretization of the Levy area associated to a d-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean-square convergence rate of the Euler scheme. For H<3/4 the exact convergence rate is n^{-2H+1/2}, where n denotes the number of the discretization subintervals, while for H=3/4 it is n^{-1} (log(n))^{1/2} and for H>3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error distribution of the Euler scheme. For H lesser than 3/4 one obtains a Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.

Mots clés: fractional Brownian motion; Levy area; approximation schemes

Domaines: Probabilités [math.PR]

Andreas Neuenkirch, Samy Tindel, Jérémie Unterberger. Discretizing the fractional Levy area. Stochastic Processes and their Applications, 2010, 120, pp.223-254. ⟨10.1016/j.spa.2009.10.007⟩. ⟨hal-00358325⟩