Müntz linear transforms of Brownian motion. (English) Zbl 1292.60079

Alili, Larbi; Wu, Ching-Tang

Electron. J. Probab. 19, Paper No. 36, 15 p. (2014).

Summary: 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.

MSC:

60J65	Brownian motion
45D05	Volterra integral equations
60G15	Gaussian processes
26C05	Real polynomials: analytic properties, etc.
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Keywords:

enlargement of filtration; Gaussian process; Müntz polynomials; noncanonical representation; self-reproducing kernel; Volterra representation