Let $\tau$ be a stopping time and $X_t^{\tau} (\omega) := X_{\min(\tau (\omega), t)}$, $t \geq 0, \omega \in \Omega$ be the stopped process. Then $\min( \tau, t ) \in \mathcal{F}_t$ and if $X$ is also progressive measurable, we obtain also that $X_{\min( \tau, t)}$ with values in $(E, \mathcal{E})$ is $\mathcal{F}_t$-measurable. Then for $\Gamma \in \mathcal{E}$ we have that
$\{ X_{\tau} \in \Gamma \} \cap \{ \tau \leq t \} = \{ X_{\min(\tau, t)} \in \Gamma \} \cap \{ \tau \leq t \} \in \mathcal{F}_t$
and it follows $\{ X_{\tau} \in \Gamma \} \in \mathcal{F}_{\tau}$.
My question now is, shouldn't it be $\{ X_{\tau} \in \Gamma \} \in \mathcal{F}_{t}$ instead? But why is it $F_{\tau}$ ?
Thank you.
By definition $A \in \mathcal F_{\tau}$ iff $A \cap \{\tau \leq t\} \in \mathcal F_t$ for every $t$. Hence $\{X_{\tau} \in \Gamma\} \in \mathcal F_{\tau}$.