Malliavin Calculus — Mathematics Knowledge System

Overview

Malliavin calculus is a differential calculus on Wiener space -- the space of continuous paths of a Brownian motion. It provides derivatives and integration by parts formulas for functionals of Brownian motion. The main objects are the Malliavin derivative D (differentiating a functional F of Brownian motion with respect to a perturbation of the path), the Skorohod integral delta (the adjoint of D, extending the Ito integral to non-adapted processes), and the Ornstein-Uhlenbeck operator L = -delta*D. Malliavin calculus proves the absolute continuity of laws of SDE solutions and is used in finance for computing Greeks (sensitivities) of option prices.

Intuition

Ordinary calculus differentiates functions of real numbers. Ito calculus handles functions of Brownian paths via the chain rule (Ito's lemma). Malliavin calculus goes one step further: it differentiates RANDOM VARIABLES (functionals of the Brownian path) with respect to a perturbation of the entire path. If F = F(W) is a functional of Brownian motion W, the Malliavin derivative D_t F captures how F changes if you 'push' the Brownian path at time t. The integration by parts formula E[F * G'] = E[F' * G] for smooth densities becomes E[F * delta(u)] = E[<DF, u>_H] in the Malliavin framework.

Formal Definition

Definition

On the Wiener space (Omega = C([0,1], R), mu = Wiener measure), smooth cylindrical functionals F = f(W(t_1), ..., W(t_n)) have Malliavin derivative: D_s F = sum_{i: t_i >= s} (partial_i f)(W(t_1), ..., W(t_n)). This extends to a closed operator D: D^{1,2} -> L^2(Omega x [0,1]) where D^{1,2} is the Sobolev space. The Skorohod integral delta is the L^2 adjoint of D: E[<DF, u>_H] = E[F * delta(u)] for u in Dom(delta). For adapted processes u, delta(u) = integral u_t dW_t (Ito integral).

D_s F = \sum_{i:\, t_i \ge s} \partial_i f(W(t_1),\ldots,W(t_n))

Malliavin derivative of cylindrical functional

\mathbb{E}[F\, \delta(u)] = \mathbb{E}[\langle DF, u\rangle_H]

Integration by parts (Malliavin)

\delta(u) = \int_0^1 u_t\, dW_t \text{ for adapted } u

Skorohod integral extends Ito integral

\frac{\partial}{\partial x} \mathbb{E}[g(X_T)] = \mathbb{E}\!\left[g(X_T) \int_0^T H_t\, dW_t\right]

Clark-Ocone formula / Greek via Malliavin

Notation

Notation	Meaning
$D_t F$	Malliavin derivative of F at time t
$\delta(u)$	Skorohod integral of u (adjoint of D)
$\mathbb{D}^{1,2}$	Malliavin-Sobolev space: F in L^2 with DF in L^2(H)
$L = -\delta D$	Ornstein-Uhlenbeck operator

Theorems

Theorem 1: Malliavin's Criterion for Absolute Continuity

Let F = (F_1,...,F_m): Omega -> R^m with F_i in D^{1,2}. If the Malliavin matrix sigma_{ij} = <DF_i, DF_j>_H is non-degenerate (det sigma > 0 a.s. and (det sigma)^{-1} in L^p for all p >= 1), then the law of F is absolutely continuous with respect to Lebesgue measure on R^m and has a smooth density.

Theorem 2: Clark-Ocone Formula

For F in D^{1,2}, F admits the representation F = E[F] + integral_0^1 E[D_t F | F_t] dW_t. The integrand is E[D_t F | F_t] -- the conditional expectation of the Malliavin derivative. This gives an explicit Ito integral representation of any square-integrable functional.

Theorem 3: Integration by Parts for Greeks

For an SDE dX_t = b(X_t)dt + sigma(X_t)dW_t and payoff g(X_T), the sensitivity d/dx E[g(X_T)] can be written as E[g(X_T) * H] where H = delta(u) for an appropriate weight process u. This avoids computing the density of X_T and enables Monte Carlo simulation of Greeks.

Worked Examples

1
F = W_t^2 = f(W_t) with f(x) = x^2.
2
D_s W_t = 1_{s <= t} (the Malliavin derivative of W_t is the indicator function that s <= t).
$D_s W_t = \mathbf{1}_{s \le t}$
3
By the chain rule for D: D_s F = D_s (W_t^2) = 2 W_t * D_s W_t = 2 W_t * 1_{s <= t}.
$D_s W_t^2 = 2W_t \cdot \mathbf{1}_{s \le t}$

✓ Answer

D_s(W_t^2) = 2*W_t * 1_{s <= t}. For s > t, the derivative is zero (you can't affect the past by perturbing at time s > t).

Practice Problems

Hardproof writing

State and prove the integration by parts formula E[D_t F * G] = E[F * D^*_t G] for the Malliavin derivative, where D^* = delta is the Skorohod integral.

Common Mistakes

Common Mistake

Thinking the Skorohod integral always equals the Ito integral.

For adapted integrands u, delta(u) = integral u_t dW_t (Ito integral). For non-adapted u, delta(u) is the Skorohod integral -- a generalisation. The Ito integral requires adaptedness; the Skorohod integral does not, but it also does not have the same martingale property.

Quiz

The Malliavin derivative D_t F measures:

The Ito integral of F up to time tHow the functional F changes when the Brownian path is perturbed at time tThe conditional expectation of F given F_tThe variance of F at time t

Historical Background

Malliavin calculus was introduced by Paul Malliavin in 1976 to give a probabilistic proof of Hormander's theorem (hypoellipticity of second-order PDEs). Bismut (1981) developed a simplified version via integration by parts. Nualart's 1995 monograph systematised the theory. The application to mathematical finance (computing Greeks) was developed by Fournié, Lasry, Lebuchoux, Lions, and Touzi in the 1990s.

1976
Malliavin introduces the calculus on Wiener space
Paul Malliavin
1978
Bismut develops an alternative approach via Girsanov and integration by parts
Jean-Michel Bismut
1995
Nualart publishes the definitive textbook on Malliavin calculus
David Nualart
1999
Fournié et al. apply Malliavin calculus to compute financial Greeks
Eric Fournie

Summary

Malliavin calculus differentiates functionals of Brownian motion: D_t F is the 'derivative' w.r.t. path perturbation at t.
The Skorohod integral delta is the adjoint of D; for adapted integrands it equals the Ito integral.
The Clark-Ocone formula gives an explicit Ito representation: F = E[F] + integral E[D_t F | F_t] dW_t.
Malliavin's criterion: non-degenerate Malliavin matrix implies absolutely continuous law with smooth density.

References

BookNualart, D. The Malliavin Calculus and Related Topics. Springer, 2006.
BookDi Nunno, G., Oksendal, B., and Proske, F. Malliavin Calculus for Levy Processes with Applications to Finance. Springer, 2009.
WebsiteWikipedia -- Malliavin calculus