  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.
