This paper considers an extension of the classical discrete-time risk model for which an INAR(1) process is utilized to model a temporal dependence between the number of claims. We apply a recursive method for deriving the Laplace transform of the aggregate claims with or without discounting in this framework. This methodology is implemented for the class of INAR(1) processes with an arbitrary innovations' distribution. Three risk models via specific INAR(1) processes are studied when the distribution of the individual claim sizes belongs to the class of mixed Erlang distributions. These different models allow us to discuss the frequent manifestations of equidispersion, overdispersion and zero inflation, and to evaluate the distribution of the (discounted) aggregate claims. Numerical examples are performed in order to illustrate the results obtained in this paper.

本文考虑了经典离散时间风险模型的扩展，其中使用 INAR(1) 过程来模拟索赔数量之间的时间依赖性。我们应用递归方法来推导在此框架中使用或不使用折扣的聚合索赔的拉普拉斯变换。该方法适用于具有任意创新分布的 INAR(1) 过程类。当单个索赔大小的分布属于混合 Erlang 分布类时，研究了通过特定 INAR(1) 过程的三个风险模型。这些不同的模型使我们能够讨论等分散、过度分散和零通货膨胀的常见表现，并评估（贴现）总债权的分布。