The Malliavin calculus, named after Paul Malliavin

Paul Malliavin

Paul Malliavin was a French mathematician. He was Professor Emeritus at the Pierre and Marie Curie University. He was a member of the Academy of Sciences since 1979.-Domains of research:...

, is a theory of variational stochastic calculus

Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...

. In other words it provides the mechanics to compute derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s of random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s.

The original motivation for the development of the subject was the desirability to provide a stochastic proof that Hörmander's condition

Hörmander's condition

In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations...

 is sufficient to ensure that the solution of a stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

 has a density

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 (which was earlier established by partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

 techniques). The calculus also allows important regularity bounds to be obtained for this density. It has recently been applied to stochastic partial differential equation

Stochastic partial differential equation

Stochastic partial differential equations are similar to ordinary stochastic differential equations. They are essentially partial differential equations that have additional random terms. They can be exceedingly difficult to solve...

s.

While this original motivation is still very important the calculus has found numerous other applications; for example in stochastic filtering. A useful feature is the ability to perform integration by parts

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

 on random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s. This may be used in financial mathematics to compute sensitivities of financial derivative

Derivative (finance)

A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...

s (also known as the Greeks).

Invariance principle

The usual invariance principle for Lebesgue integration

Lebesgue integration

In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

 over the whole real line is that, for any real number h and integrable function f, the
following holds


This can be used to derive the integration by parts

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

 formula since, setting f = gh and differentiating with respect to h on both sides, 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

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

 and set


If is a Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

, the Girsanov theorem

Girsanov theorem

In probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure...

 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

Malliavin derivative

In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense...

 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 remains true (by definition) also if is not adapted, provided that the right hand side is interpreted as a Skorokhod integral

Skorokhod integral

In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod...

.

Clark-Ocone formula

One of the most useful results from Malliavin calculus is the Clark-Ocone theorem

Clark-Ocone theorem

In mathematics, the Clark–Ocone theorem is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itō integral with respect to that path...

, which allows the process in the martingale representation theorem

Martingale representation theorem

In probability theory, the martingale representation theorem states that a random variable which is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion....

 to be identified explicitly. A simplified version of this theorem is as follows:

For satisfying which is Lipschitz and such that F has a strong derivative kernel, in the sense that
for 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

Malliavin derivative

In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense...

" denoted in the above statement of the result.

Skorokhod integral

The Skorokhod integral

Skorokhod integral

In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod...

 operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative thus for u in the domain of the operator which is a subset of ,
for F in the domain of the Malliavin derivative, we require

where the inner product is that on viz

The existence of this adjoint follows from the Riesz representation theorem

Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space 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.