Thin times and random times' decomposition

TL;DR

This paper investigates the structure of random times, decomposing them into thin and thick components, and explores their properties and implications within different filtrations, with applications to honest times and their classifications.

Contribution

It introduces a decomposition of generic random times into thin and thick parts and analyzes the semimartingale property of martingales under enlarged filtrations for thin times.

Findings

Decomposition of random times into thin and thick parts.

Martingale property preservation under filtration enlargement for thin times.

Classification of honest times based on their support in filtrations.

Abstract

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration $\mathbb F$. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all $\mathbb F$-stopping times. Then, for a given random time $\tau$, we introduce ${\mathbb F}^\tau$, the smallest right-continuous filtration containing $\mathbb F$ and making $\tau$ a stopping time, and we show that, for a thin time $\tau$, each $\mathbb F$-martingale is an ${\mathbb F}^\tau$-semimartingale, i.e., the hypothesis $({\mathcal H}^\prime)$ for $(\mathbb F, {\mathbb F}^\tau)$ holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.