Sample path deviations of the Wiener and the Ornstein-Uhlenbeck process from its bridges

TL;DR

This paper compares different mathematical representations of Wiener and Ornstein-Uhlenbeck process bridges, analyzing their sample path deviations to understand their distinct behaviors despite being statistically equivalent.

Contribution

It provides a detailed comparison of sample path deviations for various bridge representations of Wiener and Ornstein-Uhlenbeck processes, highlighting their differences.

Findings

Different representations exhibit distinct sample path deviations.

Quantitative measures of deviations are calculated for each representation.

The behavior observed extends beyond Wiener processes to Ornstein-Uhlenbeck processes.

Abstract

We study sample path deviations of the Wiener process from three different representations of its bridge: anticipative version, integral representation and space-time transform. Although these representations of the Wiener bridge are equal in law, their sample path behavior is quite different. Our results nicely demonstrate this fact. We calculate and compare the expected absolute, quadratic and conditional quadratic path deviations of the different representations of the Wiener bridge from the original Wiener process. It is further shown that the presented qualitative behavior of sample path deviations is not restricted only to the Wiener process and its bridges. Sample path deviations of the Ornstein-Uhlenbeck process from its bridge versions are also considered and we give some quantitative answers also in this case.