Malliavin calculus

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

 \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 hs 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

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 Dt 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 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Malliavin — Paul Malliavin (* 11. September 1925 in Neuilly sur Seine) ist ein französischer Mathematiker. Malliavin promovierte 1954 bei Szolem Mandelbrojt an der Universität Paris (Sur quelques procédés d’extrapolation). Er war seit 1973 Professor an der… …   Deutsch Wikipedia

  • Malliavin's absolute continuity lemma — In mathematics specifically, in measure theory Malliavin s absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus.… …   Wikipedia

  • 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… …   Wikipedia

  • Paul Malliavin — at the ICM 2006 in Madrid, with Marie Paule Malliavin Born …   Wikipedia

  • Paul Malliavin — am ICM 2006 in Madrid; mit Marie Paule Malliavin Paul Malliavin (* 11. September 1925 in Neuilly sur Seine; † 3. Juni 2010 in Paris) war ein französischer Mathematiker. Malliavin promovierte 1954 bei Szolem Mandelbro …   Deutsch Wikipedia

  • 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. It is used to model systems that behave… …   Wikipedia

  • Moshe Zakai — Born 26 December 1926(1926 12 26) Sokółka, Poland …   Wikipedia

  • Skorokhod integral — In mathematics, the Skorokhod integral, often denoted delta; , is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies… …   Wikipedia

  • Ornstein–Uhlenbeck operator — Not to be confused with Ornstein–Uhlenbeck process. In mathematics, the Ornstein–Uhlenbeck operator can be thought of as a generalization of the Laplace operator to an infinite dimensional setting. The Ornstein–Uhlenbeck operator plays a… …   Wikipedia

  • Clark-Ocone theorem — In mathematics, the Clark Ocone theorem (also known as the Clark Ocone Haussmann theorem or formula) is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”