Chaotic and variational calculus in discrete and continuous time for the Poisson process

We study a new interpretation of the Poisson space as a triplet (H,B, P ) where H is a Hilbert space, B is the completion of H and P is the extension to the Borel σ−algebra of B of a cylindrical measure on B. A discrete chaotic decomposition of L2(B,P ) is defined, along with multiple stochastic integrals of elements of Hon. It turns out that the directional derivative of functionals in L2(B,P ) in the direction of an element of H is an annihilation operator on the discrete chaotic decomposition. By composition with the Poisson process, we deduce continuous-time operators of derivation and divergence that form the number operator on the discrete chaotic decomposition. Those results are applied to the representation of random variables in the Wiener-Poisson chaotic decomposition.