Let $\left(\Omega, \mathcal{F}, \left\{ \mathcal{F}_{t} \right \}_{t \geq 0}, \mathbb{P}\right)$ be a filtered probability space, i.e. a probability space equipped with a filtration of $\sigma$-algebras. Then a random variable $\tau:\Omega\to [0,\infty)$ is called a stopping time if $$\{\omega\in\Omega:\tau(\omega)\le t\}\in\mathcal{F}_t$$