Müntz linear transforms of Brownian motion
Preprint, Article, Other literature type
- Publisher: University of Washington. Dept. of Mathematics
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.