Malliavin calculus

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 . In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the of variations.

Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a ; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to 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, .

Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a ; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to .

The usual invariance principle for Lebesgue integration over the whole real line is that, for any real number ε and integrable function f, the following holds

This can be used to derive the integration by parts formula since, setting f = gh and differentiating with respect to ε on both sides, it implies

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