Let $S_n$ be a random walk with $S_0 = 0$ and $0 < p < 0.5$. How to use de Moivre’s martingale $Y_n = (q/p)^{S_n}$ to show $E(\sup_{0\leq m\leq n}S_m)≤p/(1-2p)$?
2026-03-27 23:14:03.1774653243
de Moivre’s martingale stopping time problem
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For positive integers $w,n$, we have by Doob's inequality that $$\mathbb{P}\left(\sup_{0\leq m\leq n}S_m\geq w\right)= \mathbb{P}\left(\sup_{0\leq m\leq n}Y_m\geq (q/p)^w\right)\leq (q/p)^{-w}\,\mathbb{E}(Y_n)=(q/p)^{-w}.$$ Adding over $w$ gives $$\mathbb{E}\left(\sup_{0\leq m\leq n}S_m\right)\leq \sum_{w=1}^\infty (q/p)^{-w}={p\over q-p}.$$