Single jump filtrations and local martingales

TL;DR

This paper characterizes local martingales in a filtration generated by a single jump time, providing a complete description and classification based on their global behavior, which is a novel result even in well-studied special cases.

Contribution

It introduces a new representation for local martingales in single jump filtrations and offers a comprehensive classification, extending existing literature.

Findings

Local martingales have a specific representation involving deterministic functions and random variables.

Complete classification of local martingales based on their global behavior.

The results are new even for the case where the filtration is generated by the jump time.

Abstract

A single jump filtration $({\mathscr{F}}_t)_{t\in \mathbb{R}_+}$ generated by a random variable $\gamma$ with values in $\overline{\mathbb{R}}_+$ on a probability space $(\Omega ,{\mathscr{F}},\mathsf{P})$ is defined as follows: a set $A\in {\mathscr{F}}$ belongs to ${\mathscr{F}}_t$ if $A\cap \{\gamma >t\}$ is either $\varnothing$ or $\{\gamma >t\}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){\mathbb{1}}_{\{t<\gamma \}}+L{\mathbb{1}}_{\{t\geqslant \gamma \}}$, where $F$ is a deterministic function and $L$ is a random variable such that $\mathsf{E}|M_t|<\infty$ and $\mathsf{E}(M_t)=\mathsf{E}(M_0)$ for every $t\in \{t\in \mathbb{R}_+:{\mathsf{P}}(\gamma \geqslant t)>0\}$. This result seems to be new even in a special case that has been studied in the literature, namely, where ${\mathscr{F}}$ is the smallest $\sigma$-field with respect to which $\gamma$ is measurable (and then the filtration is the smallest one with respect to which $\gamma$ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.