Bridges with random length: Gaussian-Markovian case

TL;DR

This paper investigates Gaussian bridges with random lengths, demonstrating that the Markov property is preserved under certain conditions, and establishing the filtration properties of these bridges.

Contribution

It introduces Gaussian bridges with random lengths and proves the preservation of the Markov property and filtration properties, extending previous Brownian bridge results.

Findings

Markov property is preserved in Gaussian bridges with random length when starting process is Markov.

The completed natural filtration of the bridge satisfies usual conditions.

Extension of Brownian bridge results to more general Gaussian processes.

Abstract

Motivated by the Brownian bridge on random interval considered by Bedini et al \cite{BBE}, we introduce and study Gaussian bridges with random length with special emphasis to the Markov property. We prove that if the starting process is Markov then this property was kept by the bridge with respect to the usual augmentation of its natural filtration. This leads us to conclude that the completed natural filtration of the bridge satisfies the usual conditions of right-continuity and completeness.