Upcrossing

An upcrossing of an interval $[a,b]$ consists of an observation at or below $a$ followed by a later observation at or above $b$. Successive completed upcrossings use disjoint buy-sell pairs.

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 $\mathbb{E}(M_N-a{)}^{-}/(b-a)$.