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

Pavel V. Gapeev (), Monique Jeanblanc () and Dongli Wu ()
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Pavel V. Gapeev: London School of Economics
Monique Jeanblanc: Univ Évry, Université Paris Saclay
Dongli Wu: CCB Fintech

Journal of Theoretical Probability, 2025, vol. 38, issue 4, 1-36

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

Keywords: Marked point process; Compensator random measure; Conditional probability density; Jacod’s absolute continuity hypothesis; Initial and progressive enlargements of filtrations; Weak representation property; Changes of probability measures; Primary 60G44; 60J65; 60G40; Secondary 60G35; 60H10; 91G40 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01445-6

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