The derivative operator
The Malliavin calculus is an infinite dimensional calculus on a Gaussian space.
In this chapter we introduce the basic setup and define the derivative operator.
Suppose that H is a real separable Hilbert space with scalar product denoted
by · , ·H. Consider a Gaussian family of random variables W = {W (h), h H}
defined in a complete probability space (Ω , F, P ), with zero mean and covariance
(1.1) E(W (h)W (g)) = h , g
The mapping h W (h) provides a linear isometry between H and a closed sub-
space of H1 of
We assume that the σ-field F is generated by W and the
P -null sets.
Our basic example will be the case where the Hilbert space H is
B, µ),
where µ is a σ-finite measure without atoms on a measurable space (T, B). In this
case, for any set A B with µ(A) we make use of the notation W (A) =
W (1A). Then, A W (A) is a Gaussian random measure with independent in-
crements. That is, if A1, . . . , An are disjoint sets with finite measure, the random
variables W (A1), . . . , W (An) are independent, and for any A B with µ(A) ∞,
W (A) has the distribution N(0 , µ(A)). We will call {W (A), A B, µ(A) ∞} a
Gausian white noise on the measurable space (T, B) with intensity µ. Notice that
the covariance of the Gaussian family {W (A)} is
E(W (A)W (B)) = µ(A B).
Example 1.1. Let W be a white noise on [0, ∞) with intensity the Lebesgue
measure. Then, the stochastic process Wt = W (1[0,t]), t 0 is a standard Brow-
nian motion. In fact,
E(WtWs) = µ([0, t] [0, s]) = s t.
Example 1.2. Suppose that X is an n-dimensional random vector with density
(1.2) φ(x) =
(2π)− n
Then, if we take H = Rn, and W (h) =
hiXi, we get a Gaussian family
satisfying (1.1).
1.1. Hermite polynomials and chaos expansions
The Hermite polynomials are the coefficients in the expansion in powers of t of
the function F (x, t) = exp(tx
), that is,
(1.3) F (x, t) =

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