Bounded stopping time
= Bounded stopping time
{title2=$T\leq N$}
A <bounded stopping time> is a <stopping time> bounded <almost surely> by a deterministic finite constant. In discrete time, a bound $T\leq N$ makes $M_T$ integrable whenever $M_0,\ldots,M_N$ are <integrable random variables>, and the <optional stopping theorem> applies to a <martingale> without additional limiting hypotheses. Almost-sure finiteness alone is weaker than boundedness.