Stochastic decomposition and approximation of stock index return volatility

We show that decomposing a class of signals with overcomplete dictionaries of functions and combining multiresolution and independent component analysis allow for feature detection in complex non-stationary high frequency time series. Computational learning techniques are then designed through the Matching Pursuit algorithm, whose performance is monitored so to extract relevant information about the structure of the volatility function. We refer to wavelet and cosine packet dictionaries due to the fact that with intra-daily time series some features of the underlying stochastic processes may remain undetected when standard volatility models are applied to the observed data. Independent component analysis results are particularly encouraging and suggest a better compromise between time and frequency resolutions, and thus a more efficient and accurate Matching Pursuit performance.