Estimation of the pointwise Hölder exponent of hidden multifractional Brownian motion using wavelet coefficients

Abstract

We propose a wavelet-based approach to construct consistent estimators of the pointwise Hölder exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our estimator are discussed, and we introduce an application to the problem of estimating the functional parameter of a nonlinear model.

A general class of multifractional processes and stock price informativeness

“…the observed process is of the form Φ(θ(t)X(t)), with Φ and θ being unknown C 2 -functions. In both [5] and [26], estimators of PHE with fine convergence rates are constructed and strategies for selecting input parameters are discussed.…”

“…There are so far a number of estimation strategies existing in literature. We refer to [17,18,10,5,26,39] and the references therein.…”

“…Wendt et al [42] have developed the wavelet leader based multifractal analysis for estimating 2D functions (images). Inspired by the above works, Jin et al [26] have provided a wavelet-based estimator of the time-varying PHE of a class of multifractional processes with a fine convergence rate, when the observations are the wavelet coefficients of some unknown function of a multiple of mBm, i.e. the observed process is of the form Φ(θ(t)X(t)), with Φ and θ being unknown C 2 -functions.…”

“…Note that in our paper we also consider a model more general than the one in [26], in that it allows Φ to be a function of both t and x, i.e. we assume the observed signal is some unknown function of time t and mBm X: Φ(t, X(t)).…”

“…We apply the LGQV-based approach to estimate the PHE of Φ(t, X(t)), when one of its discrete paths is observed. Similar to Jin et al [26], an estimator with fine convergence rate is constructed and appropriate parameter selection is discussed. In [39,40] the oscillation estimation method, which could be applied to estimate the PHE of all processes with continuous paths, is discussed.…”