On the stochastic behaviour of optional processes up to random times

TL;DR

This paper explores the properties of optional processes up to random times, demonstrating that assuming randomised stopping times simplifies analysis and offers advantages in theory and applications, including finance and Brownian motion studies.

Contribution

It shows that distributional properties of optional processes up to random times can be studied assuming randomised stopping times, and provides a new proof of the Jeulin-Yor decomposition formula.

Findings

Random times can be assumed as randomised stopping times without loss of generality.

Applications to financial mathematics and Brownian motion are demonstrated.

A novel proof of the Jeulin-Yor decomposition formula via Girsanov's theorem is presented.

Abstract

In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin-Yor decomposition formula via Girsanov's theorem is provided.