Müntz linear transforms of Brownian motion

Preprint, Article, Other literature type English OPEN
Alili, Larbi ; Wu, Ching-Tang (2014)
  • Publisher: University of Washington. Dept. of Mathematics
  • Journal: (issn: 1083-6489)
  • Related identifiers: doi: 10.1214/EJP.v19-2424
  • Subject: QA | 60G15 | Enlargement of filtration | 45D05, 60G15 (Primary) 26C05, 46E22 (Secondary) | Mathematics - Probability | M\"untz polynomials | self-reproducing kernel | noncanonical representation | Volterra representation | 45D05 | Gaussian process

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.
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