M\"untz linear transforms of Brownian motion

TL;DR

This paper studies a class of linear transforms of Brownian motion linked to M"untz Gaussian spaces, providing explicit kernels, connections to M"untz-Legendre polynomials, and conditions for infinite-dimensional cases and semimartingale properties.

Contribution

It introduces explicit kernels for M"untz linear transforms of Brownian motion and characterizes their properties, including links to M"untz-Legendre polynomials and semimartingale conditions.

Findings

Explicit kernels for M"untz linear transforms derived

Conditions for infinite-dimensional M"untz Gaussian spaces established

Analysis of when transformed processes remain semimartingales

Abstract

We consider a class of linear Volterra transforms of Brownian motion associated to a sequence of M\"untz Gaussian spaces and determine explicitly their kernels; some interesting links with M\"untz-Legendre polynomials are provided. This gives new explicit examples of progressive Gaussian enlargement of the Brownian filtration. By exploiting a link to stationarity, we give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional M\"untz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process.