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). 1
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