We consider the initial enlargement \({{\mathbb {F}}}^{(\zeta )}\) of a filtration \({{\mathbb {F}}}\) (called the reference filtration) generated by a marked point process with a random variable \(\zeta \) . We assume Jacod’s absolute continuity hypothesis, that is, the existence of a nonnegative conditional density for this random variable with respect to \({{\mathbb {F}}}\) . Then, we derive explicit expressions for the coefficients that appear in the integral representation for the optional projection of an \({{\mathbb {F}}}^{(\zeta )}\) -(square integrable) martingale on \({{\mathbb {F}}}\) . In the case in which \(\zeta \) is strictly positive (called a random time in that case), we also derive explicit expressions for the coefficients, that appear in the related representation for the optional projection of an \({{\mathbb {F}}}^{(\zeta )}\) -martingale on \({{\mathbb {G}}}\) , the reference filtration progressively enlarged by \(\zeta \) . We also provide similar results for the \({{\mathbb {F}}}\) -optional projection of any martingale in \({{\mathbb {G}}}\) . The arguments of the proof are built on the methodology that was developed in our paper (Gapeev et al. in Electron J Probab 26:1–24 2021) in the Brownian motion setting under the more restrictive Jacod’s equivalence hypothesis.
