Anticipating integrals and martingales on the Poisson space

Abstract

Let $\tilde{N}\_{t}$ be a standard compensated Poisson process on $[0,1]$. We prove a new characterization of anticipating integrals of the Skorohod type with respect to $\tilde{N}$, and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorohod integral processes with respect to $\tilde{N}$ admit a pointwise representation as usual It\^{o} integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorohod integral processes in terms of products of backward and forward Poisson martingales, (ii) develop a new It\^{o}-type calculus for anticipating integrals on the Poisson space, and (iii) write Burkholder-type inequalities for Skorohod integrals.