Annihilation and Creation Operators

Abstract

In this chapter we present a first example of a pair of gradient and diver- gence operators satisfying the duality Assumption 3.1.1, the Clark formula Assumption 3.2.1 and the stability Assumption 3.2.10 of Section 3.1. This construction is based on annihilation and creation operators acting on multi- ple stochastic integrals with respect to a normal martingale. In the following chapters we will implement several constructions of such operators, respec- tively when the normal martingale (Mt)t?R+ is a Brownian motion or a compensated Poisson process. Other examples of operators satisfying the above assumptions will be built in the sequel by addition of a process with vanishing adapted projection to the gradient D, such as in Section 7.7 on the Poisson space