The Annals of Applied Probability

Functional quantization rate and mean regularity of processes with an application to Lévy processes

Harald Luschgy and Gilles Pagès

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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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Ann. Appl. Probab., Volume 18, Number 2 (2008), 427-469.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 60E99: None of the above, but in this section 60G51: Processes with independent increments; Lévy processes 60G15: Gaussian processes 60G52: Stable processes 60J60: Diffusion processes [See also 58J65]

Functional quantization Gaussian process Haar basis Lévy process Poisson process


Luschgy, Harald; Pagès, Gilles. Functional quantization rate and mean regularity of processes with an application to Lévy processes. Ann. Appl. Probab. 18 (2008), no. 2, 427--469. doi:10.1214/07-AAP459.

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