Identification of Nonstandard Multifractional Brownian Motions under White Noise by Multiscale Local Variations of Its Sample Paths

The Hurst exponent and variance are two quantities that often characterize real-life, high-frequency observations. Such real-life signals are generally measured under noise environments. We develop a multiscale statistical method for simultaneous estimation of a time-changing Hurst exponent and a variance parameter in a multifractional Brownian motion model in the presence of white noise. The method is based on the asymptotic behavior of the local variation of its sample paths which applies to coarse scales of the sample paths. This work provides stable and simultaneous estimators of both parameters when independent white noise is present. We also discuss the accuracy of the simultaneous estimators compared with a few selected methods and the stability of computations with regard to adapted wavelet filters.