Chapter 15 - Malliavin calculus

Summary

The Malliavin calculus (also known as stochastic calculus of variation) is a differential calculus for functions (i.e. random variables) defined on a space with a Gaussian measure. (In applications, the space is usually some version of the Wiener space.) In accordance with our general principle, we present here a version concentrating on the random variables without explicit mention of the underlying space.

We define in Sections 1–3 the basic derivative operators ∂_ξ and ▽ for an arbitrary Gaussian Hilbert space, and in Section 9 the dual divergence operator. We also give a detailed treatment of the Sobolev spaces D^{k, p} in Sections 5–8; this includes a proof of the important Meyer inequalities in Section 8. Results on existence and smoothness of densities are given in Sections 4 and 10; these results are central in many applications. Finally, a connection with the Skorohod integral is established in Section 11.

The first application of Malliavin calculus (Malliavin 1978) was to study smoothness of solutions to partial differential operators. Many other applications have been developed later, for example to stochastic differential equations and stochastic integrals. We will not treat any of these applications here; for applications, other versions of the theory and further results on analysis on Wiener space we refer to for example Bell (1987), Bouleau and Hirsch (1991), Ikeda and Watanabe (1984), Malliavin (1993, 1997), Nualart (1995, 1997+), Nualart and Zakai (1986), Ocone (1987), Peters (1997+), Stroock (1981), Üstünel (1995), Watanabe (1984), Zakai (1985).