M ay 2 00 6 Functional quantization rate and mean pathwise regularity of processes with an application to Lévy processes

We investigate the connections between the mean pathwise regularity of stochastic processes and their L r (P)-functional quantization rate as random variables taking values in some L p ([0, T ], dt)-spaces (0 < p ≤ r). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (like for the Brownian motion) or not (like for the Poisson process). As a first application we establish the O((log N) − 1 2)-upper bound for general Itô processes which include multi-dimensional diffusions. Then, we focus on the specific family of Lévy processes for which we derive a general quantization rate based on the regular variation properties of its Lévy measure at 0. The case of compound Poisson processes which appears as degenerate in the former approach, is studied specifically: one observes some rates which are in-between finite dimensional and infinite dimensional " usual " rates.