Malliavin calculus
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.
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.
Contents
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 CameronMartinGirsanov direction. Indeed, let be a squareintegrable 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 lefthand 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.^{[citation needed]}
ClarkOcone formula
One of the most useful results from Malliavin calculus is the ClarkOcone theorem, which allows the process in the martingale representation theorem 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" denoted in the above statement of the result.^{[citation needed]}
Skorokhod integral
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 , 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 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.
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.
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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 3540434313
 Nualart, David (2006). The Malliavin calculus and related topics (Second ed.). SpringerVerlag. ISBN 9783540283287.
 Bell, Denis. (2007) The Malliavin Calculus, Dover. ISBN 0486449947
 Schiller, Alex (2009) Malliavin Calculus for Monte Carlo Simulation with Financial Applications. Thesis, Department of Mathematics, Princeton University
 Øksendal, Bernt K..(1997) An Introduction To Malliavin Calculus With Applications To Economics. Lecture Notes, Dept. of Mathematics, University of Oslo (Zip file containing Thesis and addendum)
 Di Nunno, Giulia, Øksendal, Bernt, Proske, Frank (2009) "Malliavin Calculus for Lévy Processes with Applications to Finance", Universitext, Springer. ISBN 9783540785712
External links
 Friz, Peter K. (20050410). "An Introduction to Malliavin Calculus" (PDF). Archived from the original (PDF) on 20070417. Retrieved 20070723. Lecture Notes, 43 pages
 Zhang, H. (20041111). "The Malliavin Calculus" (PDF). Retrieved 20041111. Thesis, 100 pages