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  "body": "\n## Definition\n\nLet $T \u003e 0$, let $(B_t)_{0 \\le t \\le T}$ be a [[Brownian motion|22735FGJV6]] on $(\\Omega, \\mathcal{F}, P)$, and let $\\mathcal{F}$ be generated by $B$.\nLet $H = L^2([0, T])$, and for $h \\in H$ let $W(h) = \\int_0^T h(t) \\, dB_t$ be the [[Itô integral|JA95NH206E]].\nA *smooth random variable* is $F = f(W(h_1), \\ldots, W(h_n))$ with $h_1, \\ldots, h_n \\in H$ and $f \\in C^\\infty(\\R^n)$, where $f$ and all its partial derivatives have polynomial growth.\nIts *Malliavin derivative* is the $H$-valued random variable\n$$D_t F = \\sum_{i=1}^n \\partial_i f(W(h_1), \\ldots, W(h_n)) \\, h_i(t), \\quad 0 \\le t \\le T.$$\nThe operator $D$ is closable from $L^2(\\Omega)$ to $L^2(\\Omega; H)$.\nIts closure is again written $D$, and its domain $\\mathbb{D}^{1,2}$ is the completion of the smooth random variables for the norm $\\norm{F}_{1,2}^2 = E[F^2] + E\\left[\\norm{DF}_H^2\\right]$.\n*Malliavin calculus* is the study of $D$, of its adjoint $\\delta$, called the Skorokhod integral, and of their applications.\n\n## Remarks\n\nExample: $F = B_T^2 = W(\\mathbf{1}_{[0,T]})^2$ is smooth with $f(x) = x^2$, so $D_t F = 2 B_T$ for $0 \\le t \\le T$.\nNon-example: $F = \\mathbf{1}_{\\set{B_T \u003e 0}}$ is not in $\\mathbb{D}^{1,2}$; an indicator of an event $A$ lies in $\\mathbb{D}^{1,2}$ only if $P(A)$ is $0$ or $1$.\nThe basic integration by parts formula is $E[\\langle DF, h \\rangle_H] = E[F \\, W(h)]$ for smooth $F$ and $h \\in H$.\nThe Clark–Ocone formula $F = E[F] + \\int_0^T E[D_t F \\mid \\mathcal{F}_t] \\, dB_t$ for $F \\in \\mathbb{D}^{1,2}$ makes the [[martingale representation theorem|2E8MQXDYNV]] explicit.\nThe setting is often the [[classical Wiener space|5Z00K7K1HY]] of continuous paths.\n",
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