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Let  Ñ t   be a standard compensated Poisson process on [0, 1]. We prove a new characterization of anticipating integrals of the Skorokhod type with respect to  Ñ , 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 Skorokhod integral processes with respect to  Ñ  admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorokhod integral processes in terms of products of backward and forward Poisson martingales, (ii) develop a new Itô-type calculus for anticipating integrals on the Poisson space, and (iii) write Burkholder-type inequalities for Skorokhod integrals.

Anticipating integrals and martingales on the Poisson space