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Incurred but not reported (IBNR) loss reserving is an important issue forProperty & Casualty (P&C) insurers. The modeling of the claim arrival process,especially its temporal dependence, has not been closely examined in many ofthe current loss reserving models.  In this paper, we propose modeling the claim arrival process together withits reporting delays as a marked Cox process. Our model is versatile inmodeling temporal dependence, allowing also for natural interpretations. Thispaper focuses mainly on the theoretical aspects of the proposed model. We showthat the associated reported claim process and  IBNR claim process are both marked Cox processes with easily convertibleintensity functions and marking distributions. The proposed model can alsoaccount for fluctuations in the exposure. By an order statistics property, weshow that the corresponding discretely observed process preserves all theinformation about the claim arrival epochs. Finally, we derive closed-formexpressions for both the autocorrelation function (ACF) and the distributionsof the numbers of reported claims and IBNR claims. Model estimation and itsapplications are considered in a subsequent paper, Badescu et al.(2015b)

A Marked Cox Model for IBNR Claims: Model and Theory