Analysis and short-time extrapolation of stock market indexes through projection onto discrete wavelet subspaces

We consider the problem of short-time extrapolation of blue chips' stocks indexes in the context of wavelet subspaces following the theory proposed by X.-G. Xia and co-workers in a series of papers \cite{XLK,XKZ,LK,LXK}. The idea is first to approximate the oscillations of the corresponding stock index at some scale by means of the scaling function which is part of a given multi-resolution analysis of $L^2(\Re)$. Then, since oscillations at a finer scale are discarded, it becomes possible to extend such a signal up to a certain time in the future; the finer the approximation, the shorter this extrapolation interval. At the numerical level, a so--called Generalized Gerchberg-Papoulis (GGP) algorithm is set up which is shown to converge toward the minimum $L^2$ norm solution of the extrapolation problem. When it comes to implementation, an acceleration by means of a Conjugate Gradient (CG) routine is necessary in order to obtain quickly a satisfying accuracy. Several examples are investigated with different international stock market indexes.