= Upward skip-free random walk
{title2=$X_n\in\mathbb Z,\quad X_n\leq1$}
An upward skip-free <random walk> has integer-valued increments bounded above by one. Arbitrarily large downward jumps are permitted, but a first crossing of a nonnegative integer level $k$ from below must land exactly at $k$. This removes the overshoot from the <stopped likelihood ratio under exponential tilting> and permits the exact <geometric maximum of an upward skip-free random walk>. For a real threshold $k$, the first crossing is the crossing of $\lceil k\rceil$, so the overshoot-free formula must use that integer level.
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