In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables ; this operation is used in mathematical finance to compute the sensitivities of financial derivatives.

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In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. The calculus has been applied to stochastic partial differential equations as well.

The calculus allows integration by parts with random variables ; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.

His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations. A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. One of the most useful results from Malliavin calculus is the Clark-Ocone theorem , which allows the process in the martingale representation theorem to be identified explicitly.

A simplified version of this theorem is as follows:. The existence of this adjoint follows from the Riesz representation theorem for linear operators on Hilbert spaces. The calculus has applications for example in stochastic filtering. From Wikipedia, the free encyclopedia. Main article: ClarkâOcone theorem. Main article: Skorokhod integral. This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations.

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## Differentiable measures and the Malliavin calculus

Wahrscheinlichkeitstheorie verw. Gebiete 56 , MR Zbl Paris , Cairoli , J. Walsh : Stochastic integrals in the plane. Acta Math.

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## Malliavin calculus

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