On the Identification of the Pointwise Holder Exponent of the Generalized Multifractional Brownian Motion

Antoine Ayache (1) , Jacques Lévy Véhel (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 COMPLEX - Artificial Evolution and Fractals

The Generalized Multifractal Brownian Motion (GMBM) is a continuous Gaussian process that extend the classical Fractional Brownian Motion (FBM) and Multifractal Brownian Motion (MBM). This work deals with the problem of identifying the pointwise Hölder functions H of the GMBM : While it does not seem easy to do so when H is an arbitrary liminf of continuous functions, we obtain below the following a priori unexpected result: as soon as the pointwise Hölder function of the GMBM belong to the first class of Baire it may be estimated almost surely at any point t. We also derive a CLT for our estimator. Thus, even very irregular variations of the Hölder regularity of the GMBM may be detected and estimated in practice. We illustrate our results on both simulated and real data.

Domaines: Probabilités [math.PR]

Antoine Ayache, Jacques Lévy Véhel. On the Identification of the Pointwise Holder Exponent of the Generalized Multifractional Brownian Motion. Stochastic Processes and their Applications, 2004, 111 (1), pp.119-156. ⟨10.1016/j.spa.2003.11.002⟩. ⟨inria-00559108⟩