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).

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

$\int_{-\infty}^\infty f(x)\, d \lambda(x) = \int_{-\infty}^\infty f(x+h)\, d \lambda(x) .$

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

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

A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let h_s be a square-integrable predictable process and set

$\varphi(t) = \int_0^t h_s\, d s .$

If X is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle:

$E(F(X + \varepsilon\varphi))= E \left [F(X) \exp \left ( \varepsilon\int_0^1 h_s\, d X_s - \frac{1}{2}\varepsilon^2 \int_0^1 h_s^2\, ds \right ) \right ].$

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

$E(\langle DF(X), \varphi\rangle) = E\Bigl[ F(X) \int_0^1 h_s\, dX_s\Bigr].$

Here, the left-hand 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]}

Clark-Ocone formula

Main article: Clark–Ocone theorem

One of the most useful results from Malliavin calculus is the Clark-Ocone theorem, which allows the process in the martingale representation theorem to be identified explicitly. A simplified version of this theorem is as follows:

For $F: C[0,1] \to \R$ satisfying $E(F(X)^2) < \infty$ which is Lipschitz and such that F has a strong derivative kernel, in the sense that for φ in C[0,1]

$\lim_{\varepsilon \to 0} (1/\varepsilon)(F(X+\varepsilon \varphi) - F(X) ) = \int_0^1 F'(X,dt) \varphi(t)\ \mathrm{a.e.}\ X$

then

$F(X) = E(F(X)) + \int_0^1 H_t \,d X_t ,$

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

$F(X) = E(F(X))+\int_0^1 E (D_t F | \mathcal{F}_t ) \, d X_t .$

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 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 $L^2([0,\infty) \times \Omega)$, for F in the domain of the Malliavin derivative, we require

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

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

$\langle f, g \rangle = \int_0^\infty f(s) g(s) \, ds.$

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

$\delta(u) = \int_0^\infty u_t\, d W_t ,$

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.

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
• Nualart, David (2006). The Malliavin calculus and related topics (Second edition ed.). Springer-Verlag. ISBN 978-3-540-28328-7. 
• 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 978-3-540-78571-2

External links

• Friz, Peter K. (2005-04-10). "An Introduction to Malliavin Calculus" (PDF). Archived from the original on 2007-04-17. http://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf. Retrieved 2007-07-23.  Lecture Notes, 43 pages