stochastic processes – Martigale that maximizes its expected number of upcrossings/downcrossings Answer

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stochastic processes – Martigale that maximizes its expected number of upcrossings/downcrossings

let $days 1$ be some fixed integer. Consider a discrete-time martingale $(X_t)_{t=0,1,\ldots, T}$ or a continuous-time martingale $(X_t)_{0 ≤ t ≤ T}$ (the latter can be continuous or cadlag if it helps) st

$$X_0=1/2 \quad\mbox{ and } \quad \mathbb P(X_T\in \{0,1\})=1.\quad\quad\quad (\ast)$$

For every (fixed) $\epsilon\in (0.1/2)$denote by $U_{\epsilon}$ (or $D_{\epsilon}$) the number of upcrossing (resp. downcrossing) of $X$ across $[1/2-\epsilon,1/2+\epsilon]$ over $[0,T]$see eg
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 $1/4\epsilon-1/2$?

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