If $0 < S < T$ are stopping times, and $A \in \mathcal{F}_S$ show that $T = S1_A + T1_{A^c}$ is a stopping time.

Question

Let $0 < S < T$ two stopping times with respect to the discrete-time filtration $(\mathcal{F}_n)_{n \ge 0}$. Let $A \in \mathcal{F}_S$. Show that the random time $T'$ defined by:

\begin{align*} T' &= S1_A + T1_{A^c} \\ \end{align*}

is also a stopping time.

To show that $T'$ is a stopping time we need to show that for any $n$, $\{T' = n \} \in \mathcal{F}_n$.

\begin{align*} \{T' = n \} &= (\{S = n \} \cap A) \cup (\{T = n \} \cap A^c) \\ \end{align*}

Is this piece right? $A \in \mathcal{F}_S$ but not $A \in \mathcal{F}_n$, so I'm somewhat stuck on what to try next. thanks :)

Answer 1

$A \in \mathcal F_S$ implies $(S=n)\cap A \in \mathcal F_n$. Also, $\mathcal F_S \subseteq \mathcal F_T$ so $A \in \mathcal F_T$ and this gives $(T=n)\cap A \in \mathcal F_n$.