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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $T \\subseteq [0, \\infty)$ be an index set, and let $(\\mathcal{F}_t)_{t \\in T}$ be a [[filtration|HNK1Y62G7V]].\nA *submartingale* with respect to $(\\mathcal{F}_t)$ is a family $(X_t)_{t \\in T}$ of real random variables such that:\n\n1. $(X_t)$ is [[adapted|YRNFE016FF]]: $X_t$ is $\\mathcal{F}_t$-measurable for each $t \\in T$.\n2. $E[\\abs{X_t}] \u003c \\infty$ for each $t \\in T$.\n3. $E[X_t \\mid \\mathcal{F}_s] \\ge X_s$ almost surely for all $s \\le t$ in $T$, where $E[\\cdot \\mid \\mathcal{F}_s]$ is the [[conditional expectation|0YAMJRK04E]].\n\n## Remarks\n\nLet $S_n = \\xi_1 + \\dots + \\xi_n$, where the $\\xi_i$ are independent with $P(\\xi_i = 1) = P(\\xi_i = -1) = 1/2$, and let $\\mathcal{F}_n = \\sigma(\\xi_1, \\dots, \\xi_n)$.\nThen $X_n = S_n^2$ is a submartingale: it is adapted, $\\abs{X_n} \\le n^2$, and $E[S_{n+1}^2 \\mid \\mathcal{F}_n] = S_n^2 + 2 S_n E[\\xi_{n+1}] + E[\\xi_{n+1}^2] = S_n^2 + 1 \\ge S_n^2$.\nThe process $-S_n^2$ is adapted and integrable but fails clause 3, since $E[-S_{n+1}^2 \\mid \\mathcal{F}_n] = -S_n^2 - 1$.\nIn discrete time $T = \\{0, 1, 2, \\dots\\}$, clause 3 is equivalent to $E[X_{n+1} \\mid \\mathcal{F}_n] \\ge X_n$ for each $n$, by the tower property.\nA process is a [[martingale|0CDVK9RQBH]] if and only if it is both a submartingale and a [[supermartingale|KYW1PV0SX5]].\nIf $(M_t)$ is a martingale and $\\varphi$ is convex with $E[\\abs{\\varphi(M_t)}] \u003c \\infty$ for all $t$, then $(\\varphi(M_t))$ is a submartingale by the conditional form of [[Jensen's inequality|0RWMRTPC87]].\nTaking expectations in clause 3 shows that $t \\mapsto E[X_t]$ is non-decreasing.\n",
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