Pasting ordered stopping times
= Pasting ordered stopping times
{title2=$U=S\mathbf1_A+T\mathbf1_{A^c}$}
If $S\leq T$ are <stopping times> and $A\in\mathcal F_S$, then $U=S\mathbf1_A+T\mathbf1_{A^c}$ is a <stopping time>. The key identity is
$$
A^c\cap\{T\leq t\}=\{T\leq t\}\setminus(A\cap\{S\leq t\}).
$$
This uses the ordering $S\leq T$: without it, $A$ need not be known when $T$ occurs. Such pastings test the <martingale> identity through <expectations> at <bounded stopping times>.