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  "body": "\n## Summary\n\nItô calculus is the calculus of integrals with respect to [[Brownian motion|22735FGJV6]] and, more generally, [[semimartingales|4EJB6E2AZK]].\nIts central object is the Itô integral $\\int_0^t H_s \\, dB_s$ of an adapted process $H$.\nIts chain rule is [[Itô's lemma|2J1RVQB0CM]], which has an extra second-order term coming from the [[quadratic variation|X4DHMZD5JA]] of the path.\nIt is the language of stochastic differential equations and of continuous-time models in finance and physics.\n\n## Explanations\n\n### L1 human\n\nA Brownian path is too rough to have a derivative, so the usual integral $\\int H \\, dB$ cannot be defined path by path.\nItô's idea is to approximate by sums $\\sum H_{t_i} (B_{t_{i+1}} - B_{t_i})$ that always use the value of $H$ at the left end of each interval.\nSuch a sum is a fair bet: the stake is fixed before the random increment is seen.\nThe limit of these sums is the Itô integral.\nFor example, $\\int_0^t B_s \\, dB_s = \\tfrac12 (B_t^2 - t)$, and the term $-t/2$ shows that the rules of ordinary calculus change.\n",
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