The Malliavin calculus, named after Paul Malliavin, extends the calculus of variations from functions to stochastic processes. The Malliavin calculus is also called the stochastic calculus of variations. In particular, it allows the computation of derivatives of random variables.
Malliavin ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. 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 for example in stochastic filtering.
Read more about Malliavin Calculus: Overview and History, Invariance Principle, Clark-Ocone Formula, Skorokhod Integral, Applications
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