Although the techniques in this chapter can be used to establish the general Malliavin calculus later on, it is possible to jump to the next chapter. Only the techniques in this chapter, not the results, are used later.

Following, we deal with calculus for discrete Lévy processes. In an application we obtain Malliavin calculus for Poisson processes and for Brownian motion with values in abstract Wiener spaces over ‘little’ l2. To obtain similar results for Lévy processes defined on the continuous timeline [0, ∞[, and for Brownian motion with values in abstract Wiener spaces over any separable Hilbert space, the space ℝℕ is replaced by an extension *(ℝℕ) of ℝℕ and ℕ is replaced by [0, ∞[. We will identify two separable Hilbert spaces only if there exists a canonical, i.e., basis independent, isomorphic isometry between them.

The seminal paper of Malliavin was designed to study smoothness of solutions to stochastic differential equations. Here the Itô integral and Malliavin derivative are used to obtain the Clark–Ocone formula. This formula plays an important role in mathematics of finance (cf. Aase et al. and Di Nunno et al.).

Smolyanov and von Weizsäcker use differentiability to study measures on ℝℕ. They admit products of different measures. In contrast to their work, our approach is based on chaos decomposition, and measures are included which are not necessarily smooth. However, each measure has to be the product of a single fixed Borel measure on ℝ.