**Note:** This question is heavily related to a series of posts ([[1](https://mathoverflow.net/questions/480432/explicit-expression-for-the-expected-number-of-up-crossings-of-brownian-motion)], [[2](https://mathoverflow.net/questions/479673/sharpness-of-doobs-upcrossing-inequality)]) by user GJC20.

Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states: 

If $U(a,b)$ denotes the number of up-crossings of $X$ through $(a,b)$ up to time $1$, then 

$$\mathbb E \big[U(a,b)\big] \leq \frac{1}{b-a}\mathbb E\big[(a-X_1)^+\big].$$

This inequality should essentially never be sharp for an almost surely continuous martingale, except for those almost surely continuous in time, or those that never hit $a$ to begin with.

Suppose now $X$ is a Brownian motion. How much can we sharpen this bound?