Electronic Journal of Probability

Müntz linear transforms of Brownian motion

Larbi Alili and Ching-Tang Wu

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We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Müntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 36, 15 pp.

Accepted: 22 March 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45D05: Volterra integral equations [See also 34A12]
Secondary: 60G15: Gaussian processes

Enlargement of filtration Gaussian process M\"untz polynomials noncanonical representation self-reproducing kernel Volterra representation

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Alili, Larbi; Wu, Ching-Tang. Müntz linear transforms of Brownian motion. Electron. J. Probab. 19 (2014), paper no. 36, 15 pp. doi:10.1214/EJP.v19-2424. https://projecteuclid.org/euclid.ejp/1465065678

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