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, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations. 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. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations. Invariance principle 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 and hence This can be used to derive the integration by parts formula since, setting f = gh, it implies A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let be a square-integrable predictable process and set If is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle: Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula: Here, the left-hand side is the Malliavin derivative of the random variable in the direction and the integral appearing on the right hand side should be interpreted as an Itô integral. This expression also remains true (by definition) if is not adapted, provided that the right hand side is interpreted as a Skorokhod integral. Clark–Ocone