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.

Here we will give the full details on the construction of poly-saturated models of mathematics and prove the existence of such models. Compare these appendices with Chapter 7.

In this chapter a detailed introduction to martingale theory is presented. In particular, we study important Banach spaces of martingales with regard to the supremum norm and the quadratic variation norm. The main results show that the martingales in the associated dual spaces are of bounded mean oscillation. The Burkholder–Davis–Gandy (B–D–G) inequalities for Lp-bounded martingales are very useful applications. All results in this chapter are well known; I learned the proofs from Imkeller's lecture notes. We also need the B–D–G inequalities for special Orlicz spaces of martingales.

In this chapter we study martingales, defined on standard finite timelines. Later on the notion ‘finite’ is extended and the results in this chapter are transferred to a finite timeline, finite in the extended sense. We obtain all established results also for the new finite timeline. Then we shall outline some techniques to convert processes defined on this new finite timeline to processes defined on the continuous timeline [0,∞[ and vice versa. The reader is referred to the fundamental articles of Keisler, Hoover and Perkins and Lindstrøm.

From what we have now said it follows that we only need to study martingales defined on a discrete, even finite, timeline.