# 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 $$\mathbb E[\max(U_{\epsilon},D_{\epsilon})]$$)?

$$2\epsilon\mathbb E[U_{\epsilon}]\le \mathbb E[\max(1/2-\epsilon-X_T,0)]=1/4-\epsilon/2 \quad\mbox{and} \quad 2\epsilon\mathbb E[D_{\epsilon}]\le \mathbb E[\max(X_T-1/2+\epsilon,0)]= 1/4-\epsilon/2.$$
Could we expect the maximum is equal to $$1/4\epsilon-1/2$$?