Definition

Let $(\mathcal F_n)_{n\ge 0}$ be a filtration. A random variable

$$ T:\Omega\to \{0,1,2,\dots\}\cup\{\infty\} $$

is a stopping time if for every $n\ge 0$,

$$ \{T=n\}\in \mathcal F_n. $$

Equivalently, for every $n\ge 0$,

$$ \{T\le n\}\in \mathcal F_n. $$

Interpretation

At time $n$, you can decide whether the process stops at time $n$ using only the information available up to time $n$.

Standard examples

  • Deterministic time, such as $T=5$.
  • Hitting time:
$$ T=\inf\{t\ge 0:X_t=x\}. $$
  • First exit time:
$$ T=\inf\{t\ge 0:X_t\notin A\}. $$

Standard non-example

The time of the maximum over a fixed horizon,

$$ T=\operatorname{argmax}_{0\le t\le N} X_t, $$

is not a stopping time, because deciding whether $T=n$ requires future values $X_{n+1},\dots,X_N$.