Thin times and random times’ decomposition

Abstract

The paper defines and studies thin times which are random times whose graph is contained in a countable union of 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 τ, we introduce ${\mathbb{F}}^{\mathrm{\tau }}$, the smallest right-continuous filtration containing $\mathbb{F}$ and making τ a stopping time, and we show that, for a thin time τ, each $\mathbb{F}$-martingale is an ${\mathbb{F}}^{\mathrm{\tau }}$-semimartingale, i.e., the hypothesis $({\mathcal{H}}^{\prime })$ for $(\mathbb{F},{\mathbb{F}}^{\mathrm{\tau }})$ 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.