On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays

In this paper, we consider an insurance portfolio containing several types of policies which may simultaneously face claims arising from the same catastrophe. A renewal counting process for the number of events causing claims and multivariate claim severities which are dependent on the occurrence time and/or the delay in reporting or payment are assumed. A unified model is proposed to study the time-dependent loss quantities such as the discounted aggregate reported/unreported claims and the number of the incurred but not reported (IBNR) claims. We then derive the joint moments of (i) different types of discounted aggregate claims until time t; and (ii) different types of discounted aggregate reported/unreported claims (including the total numbers of IBNR as special case) until time t. Finally, some numerical examples involving covariances and correlations of the aforementioned quantities are provided.