Single marked exponential jump process ($X_t=U{1}_{\{T\le t\}},T\sim \mathrm{Exp}(1)$)

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 $q(x,dy)=1_{\{x=0\}}{1}_{(0,1)}(y)dy$. For a Borel mark set $B$, the martingale $1_{\{T\le t,U\in B\}}-|B|(t\wedge T)$ follows directly from the memoryless residual lifetime. Information about the future mark or lifetime must not be inserted into the initial filtration.