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  "body": "\n## Summary\n\nThe quadratic variation of a process is the limit of the sums of squared increments over finer and finer partitions of a time interval.\nFor smooth paths it is zero, but for Brownian motion it equals the elapsed time.\nIt is the correction term in [[Itô's lemma|2J1RVQB0CM]] and a basic tool for [[semimartingales|4EJB6E2AZK]].\n\n## Explanations\n\n### L1 human\n\nCut the time interval $[0, t]$ into many small pieces, square the change of the path on each piece, and add.\nFor a smooth curve each change is about the length of the piece, so the squares are tiny and the sum goes to zero.\nA Brownian path is so rough that its change over a piece of length $h$ is about $\\sqrt{h}$.\nThen each square is about $h$, and the sum tends to $t$.\n",
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