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