Stopping Time

Let $(\mathcal{F}_t)_{t \geq 0}$ be a filtration (Definition~Filtration and Adapted Process). A random variable $\tau : \Omega \to [0, \infty]$ is a stopping time with respect to $(\mathcal{F}_t)$ if \[ \{\tau \leq t\} \in \mathcal{F}_t \quad for all t \geq 0. \] The stopped $\sigma$-algebra is $\mathcal{F}_\tau := \{ A \in \mathcal{F} : A \cap \{\tau \leq t\} \in \mathcal{F}_t for all t \geq 0 \}$.