Projections in enlargements of filtrations under Jacod's absolute continuity hypothesis for marked point processes

We consider the initial enlargement F(ζ) of a filtration F (called the reference filtration) generated by a marked point process with a random variable ζ. We assume Jacod’s absolute continuity hypothesis, that is, the existence of a nonnegative conditional density for this random variable with respect to F. Then, we derive explicit expressions for the coefficients that appear in the integral representation for the optional projection of an F(ζ)-(square integrable) martingale on F. In the case in which ζ 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 F(ζ)-martingale on G, the reference filtration progressively enlarged by ζ. We also provide similar results for the F-optional projection of any martingale in 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.