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  "body": "\n## Summary\n\nA submartingale is an adapted, integrable [[stochastic process|0QH5ENKZM1]] whose [[conditional expectation|0TTDGTEFMN]] of a future value, given the present information, is at least the present value.\nIt models the fortune in a game that is favourable or fair to the player.\nEvery [[martingale|0D0GBNDKDH]] is a submartingale, and convex functions of martingales give many more examples.\nDoob's inequalities and convergence theorems are proved for submartingales.\n\n## Explanations\n\n### L1 human\n\nA submartingale tends to go up, or at least not down, on average.\nGiven everything known today, the best forecast of tomorrow's value is at least today's value.\nThe value can still go down on some days; only the average trend is upward.\nIf $S_n$ is a fair coin-tossing walk, then $S_n^2$ is a submartingale: squaring adds $1$ on average at each step.\n",
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