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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\ge 0}$: σ-algebras with $\\mathcal{F}_0 \\subseteq \\mathcal{F}_1 \\subseteq \\dots \\subseteq \\mathcal{F}$.\nA martingale with respect to $(\\mathcal{F}_n)$ is a sequence $(M_n)_{n \\ge 0}$ of real random variables with these three properties:\n1. $M_n$ is $\\mathcal{F}_n$-measurable for each $n$.\n2. $E[|M_n|] \u003c \\infty$ for each $n$.\n3. $E[M_{n+1} \\mid \\mathcal{F}_n] = M_n$ almost surely for each $n$.\n\n## Remarks\n\nIf \"$=$\" in property 3 is replaced by \"$\\ge$\", the sequence is a submartingale. If it is replaced by \"$\\le$\", it is a supermartingale.\nBy the [[law of total expectation|04RRB6HM00]], $E[M_n] = E[M_0]$ for each $n$.\nExamples: the simple random walk $S_n$ and $S_n^2 - n$ are martingales for the filtration of the steps.\nDoob's optional stopping theorem and Doob's martingale convergence theorem are the main tools of the theory.\n\n## Explanations\n\n### L1 human\n\nA martingale is a fair game: given everything you know now, your expected fortune after the next round is your fortune now.\nNo betting strategy can change the expected value of a martingale at fixed times.\n",
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