Malliavin calculus
mathematical techniques used in probability theory and related fields
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. P. Malliavin first initiated the calculus on infinite dimensional space. Then, significant contributors such as Seiichiro Kusuoka, Daniel W. Stroock, Jean-Michel Bismut, Shinzo Watanabe, Ichirō Shigekawa, and so on completed the foundations for the field.
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 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 in, for example, stochastic filtering.
Overview and history
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations.
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