Dubins upcrossing inequality ($\mathbb{P}(U\ge k)\le (a/b{)}^k\mathbb{E}\min (X_0/a,1)$)

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.

Start with cash one, switch to $X/a$ when a nonnegative supermartingale first reaches at most $a$, and switch to cash $q=b/a$ when it next reaches at least $b$. Repeat with coefficients multiplied by $q$ after every completed upcrossing. At each switch the new value is at most the old one, so pasting supermartingales with downward jumps gives a supermartingale when stopped after any fixed number of switches. Its initial value is $\min (X_0/a,1)$, and its value at the $k$th completed crossing is exactly $q^k$. Applying its expected value bound proves the Dubins upcrossing inequality.