Anticipative Calculus for Filtered Poisson Processes

We construct the basis of a stochastic calculus for a new class of processes: filtered Poisson processes. These processes are defined by an fBm-like stochastic integral but a Poisson process is subsided to the Brownian motion. We use Malliavin calculus to first construct a gradient then a divergence operator, which will play the role of an anticipative stochastic integral. We study into details the sample-paths regularity of this integral and give an Ito ̂ formula for Itô-like processes