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  "body": "\n## Statement\n\nLet $n \\ge 0$, let $(X_k)_{0 \\le k \\le n}$ be a [[submartingale|EQ055QMACW]] with respect to a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_k)_{0 \\le k \\le n}$, and let $\\lambda \u003e 0$.\nThen\n$$\\lambda\\, P\\Bigl(\\max_{0 \\le k \\le n} X_k \\ge \\lambda\\Bigr) \\le E\\Bigl[X_n \\mathbf{1}\\bigl\\{\\max_{0 \\le k \\le n} X_k \\ge \\lambda\\bigr\\}\\Bigr] \\le E[X_n^+],$$\nwhere $X_n^+ = \\max(X_n, 0)$.\n\n## Remarks\n\nLet $\\tau$ be the first $k \\le n$ with $X_k \\ge \\lambda$, and split the event $A = \\{\\max_k X_k \\ge \\lambda\\}$ into the disjoint events $\\{\\tau = k\\}$.\nSince $\\{\\tau = k\\} \\in \\mathcal{F}_k$, the submartingale property gives $E[X_n \\mathbf{1}\\{\\tau = k\\}] \\ge E[X_k \\mathbf{1}\\{\\tau = k\\}] \\ge \\lambda P(\\tau = k)$.\nSumming over $k$ gives the first inequality, and $X_n \\mathbf{1}_A \\le X_n^+$ gives the second.\nIf $(M_k)$ is a [[martingale|0CDVK9RQBH]], then $\\abs{M_k}$ is a submartingale, so $P(\\max_{k \\le n} \\abs{M_k} \\ge \\lambda) \\le E[\\abs{M_n}]/\\lambda$.\nFor $M_k$ a sum of independent centred square-integrable variables, applying this to the submartingale $M_k^2$ gives [[Kolmogorov's inequality|D9B4HE4B4Q]].\nThe inequality is stronger than [[Markov's inequality|05PWWHYBX5]] for $X_n$ alone: it controls the whole path up to time $n$ by the last value.\nFor a non-negative submartingale and $p \u003e 1$, it implies Doob's $L^p$ inequality $E[\\max_{k \\le n} X_k^p] \\le \\bigl(\\tfrac{p}{p-1}\\bigr)^p E[X_n^p]$.\n\n## Explanations\n\n### L1 human\n\nFor a process that tends to increase, the chance that it ever climbs above a level $\\lambda$ before time $n$ is controlled by its final value alone.\nThe bound is the same as the Markov bound for the single variable $X_n$.\nSo looking at the maximum over the whole path costs nothing extra.\n\n### L2 human\n\nThe inequality is the basic maximal inequality of martingale theory.\nIt gives Kolmogorov's inequality, which is a step in the strong law of large numbers and the three-series theorem.\nIts $L^p$ form is used to prove $L^p$ convergence of martingales and to define the norm on spaces of continuous martingales.\nIn continuous time it holds for right-continuous submartingales, with the maximum replaced by the supremum over $[0, t]$.\n",
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