 Upward skip-free random walk

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.