Provided a martingale $X$, discrete-time $X=(X_n, n\in\mathbb N)$ or continuous-time $X=(X_t, t\ge 0)$, Doob's upcrosssing inequality states that :
If $U_N(a,b)$ denotes the number of up-crossings of $X$ through $(a,b)$ up to time $N$, then
$$\mathbb E \big[U_N(a,b)\big] \leq \frac{1}{b-a}\mathbb E\big[(a-X_N)^+\big].$$
Is the upper bound above sharp? Or equivalently, does there exist a martingale such that this upper bound is achieved? If not, is there any result on the sharp upper bound? I'm mostly interested in the case $a<X_0<b$.
PS : Iosif's computation implies that the upper bound can be achieved by non constant martingale, see Explicit expression for the expected number of up-crossings of Brownian motion and I look for more examples.
