 Dubins upcrossing inequality

For a nonnegative supermartingale and $0<a<b$, let $U$ count completed upcrossings of $[a,b]$, allowing the first entry at time zero. The displayed bound holds for every integer $k\ge 1$. Construct the multiplicative upcrossing supermartingale, stop at the $k$th completed crossing, and use its initial expected value and nonnegativity. The bound is exponential in the number of crossings, and complements the first-moment Doob upcrossing inequality.