# Malliavin calculus

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.

## Overview and history

Paul Malliavin's stochastic calculus of variations extends the calculus of variations from functions to stochastic processes. In particular, it allows the computation of derivatives of random variables.

Malliavin invented his 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

${\displaystyle \int _{-\infty }^{\infty }f(x)\,d\lambda (x)=\int _{-\infty }^{\infty }f(x+\varepsilon )\,d\lambda (x).}$

This can be used to derive the integration by parts formula since, setting f = gh and differentiating with respect to ε on both sides, it implies

${\displaystyle \int _{-\infty }^{\infty }f'\,d\lambda =\int _{-\infty }^{\infty }(gh)'\,d\lambda =\int _{-\infty }^{\infty }gh'\,d\lambda +\int _{-\infty }^{\infty }g'h\,d\lambda .}$

A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let ${\displaystyle h_{s}}$ be a square-integrable predictable process and set

${\displaystyle \varphi (t)=\int _{0}^{t}h_{s}\,ds.}$

If ${\displaystyle X}$ is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle:

${\displaystyle E(F(X+\varepsilon \varphi ))=E\left[F(X)\exp \left(\varepsilon \int _{0}^{1}h_{s}\,dX_{s}-{\frac {1}{2}}\varepsilon ^{2}\int _{0}^{1}h_{s}^{2}\,ds\right)\right].}$

Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:

${\displaystyle E(\langle DF(X),\varphi \rangle )=E{\Bigl [}F(X)\int _{0}^{1}h_{s}\,dX_{s}{\Bigr ]}.}$

## Skorokhod integral

{{#invoke:main|main}} The Skorokhod integral 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 ${\displaystyle L^{2}([0,\infty )\times \Omega )}$, for F in the domain of the Malliavin derivative, we require

${\displaystyle E(\langle DF,u\rangle )=E(F\delta (u)),}$

where the inner product is that on ${\displaystyle L^{2}[0,\infty )}$ viz

${\displaystyle \langle f,g\rangle =\int _{0}^{\infty }f(s)g(s)\,ds.}$

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

${\displaystyle \delta (u)=\int _{0}^{\infty }u_{t}\,dW_{t},}$

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.

## Applications

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.

## References

• Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto 1982, pp 271–306
• Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", J. Faculty Sci. Uni. Tokyo Sect. 1A Math., 32 pp 1–76
• Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", J. Faculty Sci. Univ. Tokyo Sect. 1A Math., 34 pp 391–442
• Malliavin, Paul and Thalmaier, Anton. Stochastic Calculus of Variations in Mathematical Finance, Springer 2005, ISBN 3-540-43431-3
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