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 Incoming links: Doob's martingale inequality

Doob L2 maximal inequality Created 2026-09-24 Updated 2026-10-14

For a square-integrable martingale starting at zero,

E sup_{s \leq t} ∣ M_{s} ∣^{2} \leq 4 E ∣ M_{t} ∣^{2} .

For a continuous local martingale stopped so that its quadratic variation is integrable, the Itô isometry makes the right-hand side

4 E [M]_{t}

.

This is the

p = 2

specialization of Doob's martingale inequality.

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Doob Lp maximal inequality Created 2026-09-28 Updated 2026-10-14

For

p > 1

and a martingale

(M_{k})_{k \leq n}

,

∥ max_{k \leq n} ∣ M_{k} ∣ ∥_{p} \leq \frac{p}{p - 1} ∥ M_{n} ∥_{p} .

This strong

L^{p}

bound is one form of Doob's martingale inequality.

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