Malliavin calculus

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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 PDE techniques). The calculus also allows important regularity bounds to be obtained for this density.

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 financial mathematics to compute sensitivities of financial derivatives (also known as the Greeks).

1 Invariance principle
2 Clark-Ocone formula
3 Skorohod integral
4 References
5 External links

[edit] Invariance principle

The usual invariance principle for Lebesgue integration is for any real h the following holds

$\int f(x+h)\, d \lambda(x) = \int f(x)\, d \lambda(x).$

This can be used to derive the integration by parts formula since setting f = gh it implies

$0= \int f' \,d \lambda = \int (gh)' \,d \lambda = \int g h'\, d \lambda + \int g' h\, d \lambda.$

In the stochastic sense, for a Cameron-Martin-Girsanov direction

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

an analogue of the invariance principle can be derived and hence an integration by parts formula

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

[edit] 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 $\varphi$ in C[0,1]

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

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.

[edit] Skorohod integral

The Skorohod 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.

[edit] References

Kusuoka, S. and Stroock, D., Applications of Malliavin Calculus I, Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto 1982, pp 271-306 (1981)
Kusuoka, S. and Stroock, D. Applications of Malliavin Calculus II, J. Faculty Sci. Uni. Tokyo Sect. 1A Math., 32 pp 1-76 (1985)
Kusuoka, S. and Stroock, D. Applications of Malliavin Calculus III, J. Faculty Sci. Univ. Tokyo Sect. 1A Math., 34 pp 391-442 (1987)
Malliavin, Paul and Thalmaier, Anton. Stochastic Calculus of Variations in Mathematical Finance, Springer 2005, ISBN 3-540-43431-3
Nualart, David. The Malliavin Calculus and related topics, Springer 1995.