= Single marked exponential jump process
{title2=$X_t=U\mathbf1_{\{T\leq t\}},\quad T\sim\operatorname{Exp}(1)$}
Let $T$ be a rate-one <exponential random variable> independent of a <uniform random variable> $U$ on $(0,1)$. In its natural <filtration>, this <Markov jump process> jumps once from zero to its mark and then stays there. Its destination rate kernel is $q(x,dy)=\mathbf1_{\{x=0\}}\mathbf1_{(0,1)}(y)dy$. For a Borel mark set $B$, the <martingale> $\mathbf1_{\{T\leq t,U\in B\}}-|B|(t\wedge T)$ follows directly from the memoryless residual lifetime. Information about the future mark or lifetime must not be inserted into the initial filtration.
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