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  "body": "\n## Summary\n\nMalliavin calculus is a differential calculus for random variables that are functions of a [[Brownian motion|22735FGJV6]] path.\nIts basic operator is the Malliavin derivative $D$, which measures how a random variable changes when the path is perturbed in directions of finite energy.\nIts adjoint is the Skorokhod integral, which extends the [[Itô integral|JA95NH206E]] to integrands that are not adapted.\nIt is used to prove that laws of solutions of stochastic differential equations have smooth densities, and to compute sensitivities in finance.\n\n## Explanations\n\n### L1 human\n\nA random variable built from a Brownian path is a function on the space of all paths, an infinite-dimensional space.\nMalliavin calculus defines derivatives of such functions, much like partial derivatives in many variables.\nFor example, the value $B_T$ of the path at time $T$ has derivative $1$ at each time $t \\le T$.\nIndeed, a small push of the path at time $t$ moves $B_T$ by the same amount.\nAn integration by parts formula then turns questions about densities into questions about expectations.\n",
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