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.
P. Malliavin first initiated the calculus on infinite dimensional space.
Then, the significant contributors such as S. Kusuoka, D. Stroock, J-M. Bismut, Shinzo 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.
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.
(a functional from the classical Wiener space) and consider the task of finding a derivative for it.
The natural idea would be to use the Gateaux derivative however this does not always exist.
Therefore it does make sense to find a new differential calculus for such spaces by limiting the directions.
The toy model of Malliavin calculus is an irreducible Gaussian probability space
exists a canonical irreducible Gaussian probability space
Properties of a Gaussian probability space that do not depend on the particular choice of basis are called intrinsic and such that do depend on the choice extrensic.
[1] We denote the countably infinite product of real spaces as
be the canonical Gaussian measure, by transferring the Cameron-Martin theorem from
, denote we get a canonical representation of the additive group
acting on the endomorphisms by defining One can show that the action of
is extrinsic meaning it does not depend on the choice of basis for
the integration by parts formula 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, it implies A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction.
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
A simplified version of this theorem is as follows: Consider the standard Wiener measure on the canonical space
in C[0,1] then where H is the previsible projection of F'(x, (t,1]) which may be viewed as the derivative of the function F with respect to a suitable parallel shift of the process X over the portion (t,1] of its domain.
This may be more concisely expressed by Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals F by replacing the derivative kernel used above by the "Malliavin derivative" denoted
[citation needed] The Skorokhod integral operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative in the white noise case when the Hilbert space is an
viz The existence of this adjoint follows from the Riesz representation theorem for linear operators on Hilbert spaces.
It can be shown that if u is adapted then where the integral is to be understood in the Itô sense.
Thus this provides a method of extending the Itô integral to non adapted integrands.
The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives.