 Malliavin calculus

The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables.
The original motivation for the development of the subject was the desirability to provide a stochastic proof that Hörmander's condition is sufficient to ensure that the solution of a stochastic differential equation has a density (which was earlier established by partial differential equation techniques). The calculus also allows important regularity bounds to be obtained for this density. It has recently been applied to stochastic partial differential equations.
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 on random variables. This may be used in financial mathematics to compute sensitivities of financial derivatives (also known as the Greeks).
Contents
Invariance principle
The usual invariance principle for Lebesgue integration 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 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 CameronMartinGirsanov direction. Indeed, let h_{s} be a squareintegrable predictable process and set
If X 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 F in the direction φ and the integral appearing on the right hand side should be interpreted as an Itô integral. This expression remains true (by definition) also if h is not adapted, provided that the right hand side is interpreted as a Skorokhod integral.^{[citation needed]}
ClarkOcone formula
Main article: Clark–Ocone theoremOne 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 φ 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 D_{t} in the above statement of the result.^{[citation needed]}
Skorokhod integral
Main article: Skorokhod integralThe 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.
References
 Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto 1982, pp 271306
 Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", J. Faculty Sci. Uni. Tokyo Sect. 1A Math., 32 pp 176
 Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", J. Faculty Sci. Univ. Tokyo Sect. 1A Math., 34 pp 391442
 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 edition 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. Thesis, 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 on 20070417. http://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf. Retrieved 20070723. Lecture Notes, 43 pages
Categories: Stochastic calculus
 Integral calculus
 Mathematical finance
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