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April 2008 Functional quantization rate and mean regularity of processes with an application to Lévy processes
Harald Luschgy, Gilles Pagès
Ann. Appl. Probab. 18(2): 427-469 (April 2008). DOI: 10.1214/07-AAP459


We investigate the connections between the mean pathwise regularity of stochastic processes and their Lr(ℙ)-functional quantization rates as random variables taking values in some Lp([0, T], dt)-spaces (0<pr). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (e.g., for Brownian motion or symmetric stable processes) so that the rate is optimal as a universal upper bound. As a first application, we establish the O((log N)−1/2) upper bound for general Itô processes which include multidimensional 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 appear as degenerate in the former approach, is studied specifically: we observe some rates which are between the finite-dimensional and infinite-dimensional “usual” rates.


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Harald Luschgy. Gilles Pagès. "Functional quantization rate and mean regularity of processes with an application to Lévy processes." Ann. Appl. Probab. 18 (2) 427 - 469, April 2008.


Published: April 2008
First available in Project Euclid: 20 March 2008

zbMATH: 1158.60005
MathSciNet: MR2398762
Digital Object Identifier: 10.1214/07-AAP459

Primary: 60E99 , 60G15 , 60G51 , 60G52 , 60J60

Keywords: Functional quantization , Gaussian process , Haar basis , Lévy process , Poisson process

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.18 • No. 2 • April 2008
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