Operator scaled Wiener bridges

TL;DR

This paper introduces operator scaled Wiener bridges by integrating matrix scaling into the drift of multidimensional Wiener bridges, analyzing their properties and asymptotic behavior, and exploring the uniqueness of their law based on the scaling matrix.

Contribution

It presents a novel class of Wiener bridges with matrix-scaled drifts and provides conditions for their bridge property and law determination.

Findings

Derived a sufficient condition for the bridge property based on eigenvalues.

Analyzed the asymptotic behavior of the operator scaled Wiener bridges.

Discussed the potential non-uniqueness of the law determined by the scaling matrix.

Abstract

We introduce operator scaled Wiener bridges by incorporating a matrix scaling in the drift part of the SDE of a multidimensional Wiener bridge. A sufficient condition for the bridge property of the SDE solution is derived in terms of the eigenvalues of the scaling matrix. We analyze the asymptotic behavior of the bridges and briefly discuss the question whether the scaling matrix determines uniquely the law of the corresponding bridge.