Wavelet Methods for Continuous-time Prediction Using Representations of
Autoregressive processes in Hilbert spaces
   Anestis Antoniadis (University Joseph Fourier)


We consider the prediction problem of a continuous-time stochastic
process in terms of its recent past. We first discuss some known,
as well as recent, methods to predict such a stochastic process on
an entire time-interval. They are based on
the notion of autoregressive Hilbert processes that represent a
generalization of the classical autoregressive processes to random
variables with values in a Hilbert space. A careful analysis
reveals, in particular, that these methods are related to the
theory of function estimation in linear ill-posed inverse
problems. In the deterministic literature, such problems are
usually solved by Tikhonov-Phillips regularization. We describe
some recent approaches from the deterministic literature that can
be adapted to obtain fast and feasible solutions to our prediction
problem. For large sample sizes, however, these approaches are not
computationally efficient.


Correspondingly, we propose three wavelet methods to efficiently
address the aforementioned prediction problem. We present wavelet
regularization techniques for the sample paths of the stochastic
process and obtain consistency results of the resulting prediction
estimators. We illustrate the performance of the proposed wavelet
methods in finite sample applications by means of two real data
examples.


The work is joint with Theofanis Sapatinas.