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  "body": "\n## Definition\n\nLet $X$ and $Y$ be continuous real [[semimartingales|4EJB6E2AZK]] on a filtered probability space, and let $t \\ge 0$.\nThe *Stratonovich integral* of $Y$ with respect to $X$ is\n$$\\int_0^t Y_s \\circ dX_s = \\int_0^t Y_s \\, dX_s + \\frac12 [Y, X]_t.$$\nHere $\\int_0^t Y_s \\, dX_s$ is the [[Itô integral|JA95NH206E]].\nThe process $[Y, X]$ is the [[quadratic covariation|X4DHMZD5JA]] of $Y$ and $X$.\n\n## Remarks\n\nEquivalently, it is the limit in probability of the sums $\\sum_i \\tfrac12 (Y_{t_i} + Y_{t_{i+1}})(X_{t_{i+1}} - X_{t_i})$ over partitions $0 = t_0 \u003c \\cdots \u003c t_n = t$ whose mesh tends to $0$.\nExample: let $B$ be a [[Brownian motion|22735FGJV6]] with $B_0 = 0$.\nThen $\\int_0^t B_s \\, dB_s = \\tfrac12 (B_t^2 - t)$ and $[B, B]_t = t$, so $\\int_0^t B_s \\circ dB_s = \\tfrac12 B_t^2$.\nNon-example: the left-point sums $\\sum_i B_{t_i} (B_{t_{i+1}} - B_{t_i})$ converge to $\\tfrac12 (B_t^2 - t)$, not to the Stratonovich integral, because they omit the covariation term.\nFor $f$ three times continuously differentiable, the chain rule takes the classical form $f(X_t) = f(X_0) + \\int_0^t f'(X_s) \\circ dX_s$.\nThe notation $\\circ \\, dX$, with a small circle, marks the Stratonovich integral.\n",
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