Frameable Non-stationary Processes and Volatility Applications

A crucial goal in many experimental fields and applications is achieving sparse signal approximations for the unknown signals or functions under investigation. This fact allows to deal with few significant structures for reconstructing signals from noisy measurements or recovering functions from indirect observations. We describe and implement approximation and smoothing procedures for volatility processes that can be represented by frames, particularly wavelet frames, and pursue these goals by using dictionaries of functions with adaptive degree of approximation power. Volatility is unobservable and underlying the realizations of stochastic processes that are noni.i.d., covariance non-stationary, self-similar and non-Gaussian; thus, its features result successfully detected and its dynamics well approximated only in limited time ranges and for clusters of bounded variability. Both jumps and switching regimes are usually observed though, suggesting that either oversmoothing or de-volatilization may easily occur when using standard and non-adaptive volatility models. Our methodological proposal combines wavelet-based frame decompositions with blind source separation techniques, and uses greedy de-noisers and feature learners.