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

Pavel V. Gapeev, Monique Jeanblanc and Dongli Wu

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: 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.

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 (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 36 pages
Date: 2025-12-31
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Published in Journal of Theoretical Probability, 31, December, 2025, 38(4). ISSN: 0894-9840

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