Stopping Time

Stopping Time

A random variable $\tau :\Omega \to \mathbb{N}\cup \{\infty \}$ is called a stopping time for $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ if $\{\omega :\tau (\omega )\le n\}\in {\mathcal{F}}_n$ for each $n\in \mathbb{N}$.

Theorem

If $(M_n{)}_{n\in \mathbb{N}}$ is a martingale and $\tau$ is a stopping time for $({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$, then the stopped process $(M_{n\wedge \tau }{)}_{n\in \mathbb{N}}$ is a martingale.