CHAPTER 1

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

H

.

The mapping h → W (h) provides a linear isometry between H and a closed sub-

space of H1 of

L2(Ω).

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

L2(T,

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

2

exp −

|x|2

2

.

Then, if we take H = Rn, and W (h) =

∑n

i=1

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 −

t2

2

), that is,

(1.3) F (x, t) =

∞

n=0

tnHn(x).

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http://dx.doi.org/10.1090/cbms/110/01