Electronic Journal of Probability

Müntz linear transforms of Brownian motion

Larbi Alili and Ching-Tang Wu

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065678

Digital Object Identifier
doi:10.1214/EJP.v19-2424

Mathematical Reviews number (MathSciNet)
MR3183580

Zentralblatt MATH identifier
1292.60079

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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