= Closed-level hitting times of continuous adapted processes
{title2=$T=\inf\{s:X_s\geq\lambda\}$}
For a continuous <adapted process> $X$ and $T=\inf\{s\geq0:X_s\geq\lambda\}$,
$$
\{T\leq t\}=\bigcap_{m\geq1}\bigcup_{r\in(\mathbb Q\cap[0,t])\cup\{t\}}\{X_r>\lambda-1/m\}.
$$
Continuity makes the supremum on $[0,t]$ attained and equals its supremum on this countable dense set. Each displayed event belongs to $\mathcal F_t$, so $T$ is a <stopping time> without requiring right-continuity of the filtration.
// Target: martingale.bigb
Back to article page
