Upcrossing count
= Upcrossing count
{title2=$U_N[a,b]$}
The <upcrossing count> $U_N[a,b]$ counts completed <upcrossings> by time $N$, allowing the first purchase at time zero. For a <martingale>, the <Doob upcrossing inequality> bounds its <expected value> by $\mathbb E(M_N-a)^-/(b-a)$.