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  "body": "\n## Summary\n\nThe Stratonovich integral $\\int Y \\circ dX$ is a stochastic integral.\nIt evaluates the integrand at the midpoint of each time step, not at the left end.\nIt differs from the [[Itô integral|JA95NH206E]] by half the [[quadratic covariation|X4DHMZD5JA]] of $Y$ and $X$.\nWith this integral the ordinary chain rule of calculus holds, which makes it natural in physics and in stochastic differential geometry.\n\n## Explanations\n\n### L1 human\n\nTo integrate against a Brownian path, one sums values of the integrand times small increments of the path.\nThe Itô integral takes the value at the start of each step, while the Stratonovich integral takes the average of the start and end values.\nFor smooth paths both choices give the same limit, but for Brownian paths they do not.\nFor example, $\\int_0^t B_s \\circ dB_s = \\tfrac12 B_t^2$ when $B_0 = 0$, just as in ordinary calculus.\nThe Itô version instead gives $\\tfrac12 (B_t^2 - t)$.\n",
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