Let $T\ge 1$ be some fixed integer. Consider a discrete-time martingale $(X_t)_{t=0,1,\ldots, T}$ or a continous-time martingale $(X_t)_{0\le t\le T}$ (the latter can be continuous or cadlag if it helps) s.t.
$$X_0=1/2 \quad\mbox{ and } \quad \mathbb P(X_T\in \{0,1\})=1.\quad\quad\quad (\ast)$$
For every (fixed) $\epsilon\in (0,1/2)$, denote by $U_{\epsilon}$ (resp. $D_{\epsilon}$) the number of upcrossing (resp. downcorssing) of $X$ accross $[1/2-\epsilon,1/2+\epsilon]$ over $[0,T]$, see e.g. Upcrossings, Downcrossings, and Martingale Convergence . Does there exist a martingale satisfying $(\ast)$ that maximizes $\mathbb E[U_{\epsilon}+D_{\epsilon}]$ (or $\mathbb E[\max(U_{\epsilon},D_{\epsilon})]$)?
Any answer, comments and references are highly appreciated.
PS : In view of Doob’s upcrossing/downcrossing lemma, one has
$$2\epsilon\mathbb E[U_{\epsilon}]\le \mathbb E[\max(1/2-\epsilon-X_T,0)]=1/4-\epsilon/2 \quad\mbox{and} \quad 2\epsilon\mathbb E[D_{\epsilon}]\le \mathbb E[\max(X_T-1/2+\epsilon,0)]= 1/4-\epsilon/2.$$
Could we expect the maximum is equal to $1/4\epsilon-1/2$?
