Projections in Enlargements of Filtrations under Jacod’s Absolute Continuity Hypothesis for Marked Point Processes

Abstract We consider the initial enlargement $${{\mathbb {F}}}^{(\zeta )}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> of a filtration $${{\mathbb {F}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> (called the reference filtration) generated by a marked point process with a random variable $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> . We assume Jacod’s absolute continuity hypothesis, that is, the existence of a nonnegative conditional density for this random variable with respect to $${{\mathbb {F}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . Then, we derive explicit expressions for the coefficients that appear in the integral representation for the optional projection of an $${{\mathbb {F}}}^{(\zeta )}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> -(square integrable) martingale on $${{\mathbb {F}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . In the case in which $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> 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 $${{\mathbb {F}}}^{(\zeta )}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> -martingale on $${{\mathbb {G}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , the reference filtration progressively enlarged by $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> . We also provide similar results for the $${{\mathbb {F}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -optional projection of any martingale in $${{\mathbb {G}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . 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.