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  "body": "\n## Statement\n\nLet $(B_t)_{t \\ge 0}$ be a [[Brownian motion|22735FGJV6]] for a filtration $(\\mathcal{F}_t)$.\nLet $X$ be an Itô process\n$$X_t = X_0 + \\int_0^t \\mu_s \\, ds + \\int_0^t \\sigma_s \\, dB_s,$$\nwhere the second integral is an [[Itô integral|JA95NH206E]].\nThe processes $\\mu$ and $\\sigma$ are progressively measurable with $\\int_0^t (\\abs{\\mu_s} + \\sigma_s^2) \\, ds \u003c \\infty$ almost surely for every $t$.\nLet $f \\colon [0, \\infty) \\times \\R \\to \\R$ be once continuously differentiable in $t$ and twice continuously differentiable in $x$.\nThen, almost surely, for all $t \\ge 0$,\n$$f(t, X_t) = f(0, X_0) + \\int_0^t \\left( \\partial_t f + \\mu_s \\, \\partial_x f + \\tfrac12 \\sigma_s^2 \\, \\partial_{xx} f \\right)(s, X_s) \\, ds + \\int_0^t \\sigma_s \\, \\partial_x f(s, X_s) \\, dB_s.$$\nHere $\\partial_t$, $\\partial_x$ and $\\partial_{xx}$ are [[partial derivatives|09EGC6F8PT]].\n\n## Remarks\n\nThe proof expands $f$ by Taylor's formula to second order over a fine partition of $[0, t]$.\nThe sums of $(\\Delta X)^2$ converge to $\\int \\sigma_s^2 \\, ds$, the [[quadratic variation|X4DHMZD5JA]] of $X$, while higher-order terms vanish in the limit.\nIn differential form, $df = \\partial_t f \\, dt + \\partial_x f \\, dX + \\tfrac12 \\partial_{xx} f \\, (dX)^2$ with the rules $(dB)^2 = dt$ and $dt \\, dB = (dt)^2 = 0$.\nExample: for $f(x) = x^2$ and $X = B$ with $B_0 = 0$, it gives $B_t^2 = 2 \\int_0^t B_s \\, dB_s + t$.\nExample: for [[geometric Brownian motion|D48DR46B97]] $dS = \\mu S \\, dt + \\sigma S \\, dB$ with constants $\\mu, \\sigma$ and $S_0 \u003e 0$, the function $\\log$ gives $\\log S_t = \\log S_0 + (\\mu - \\tfrac12 \\sigma^2) t + \\sigma B_t$.\nThe $C^2$ hypothesis matters: for $f(x) = \\abs{x}$ the formula needs an extra local time term, which is [[Tanaka's formula|GEM0ERJ5ZB]].\n\n## Explanations\n\n### L1 human\n\nItô's lemma is the chain rule for functions of a random process driven by Brownian motion.\nCompared with the ordinary chain rule, it has one more term, half the second derivative times the squared noise intensity.\nThis term appears because a Brownian path moves by about $\\sqrt{dt}$ in time $dt$, so squares of its moves do not vanish.\n\n### L2 human\n\nThe lemma is the main computational tool of stochastic calculus.\nIt solves linear stochastic differential equations, such as those for geometric Brownian motion and the Ornstein–Uhlenbeck process.\nIt links expectations of diffusions to partial differential equations, as in the Kolmogorov backward equation and the Feynman–Kac formula.\nThe multidimensional and semimartingale versions have the same shape, with the quadratic covariations $d[X^i, X^j]$ in the second-order term.\n",
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