Single jump filtrations and local martingales

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.