In multi-state life insurance, an adequate balance between analytic tractability, computational efficiency, and statistical flexibility is of great importance. This might explain the popularity of Markov chain modelling, where matrix analytic methods allow for a comprehensive treatment. Unfortunately, Markov chain modelling is unable to capture duration effects, so this paper presents aggregate Markov models as an alternative. Aggregate Markov models retain most of the analytical tractability of Markov chains, yet are non-Markovian and thus more flexible. Based on an explicit characterization of the fundamental martingales, matrix representations of the expected accumulated cash flows and corresponding prospective reserves are derived for duration-dependent payments with and without incidental policyholder behaviour. Throughout, special attention is given to a semi-Markovian case. Finally, the methods and results are illustrated in a numerical example.

在多州人寿保险中，分析易处理性、计算效率和统计灵活性之间的充分平衡非常重要。这可能解释了马尔可夫链建模的流行，其中矩阵分析方法允许进行综合处理。不幸的是，马尔可夫链模型无法捕获持续时间效应，因此本文提出聚合马尔可夫模型作为替代方案。聚合马尔可夫模型保留了马尔可夫链的大部分分析易处理性，但它是非马尔可夫的，因此更加灵活。基于基本鞅的明确特征，针对有或没有附带投保人行为的依赖于久期的付款，得出预期累积现金流量和相应预期准备金的矩阵表示。自始至终，特别关注半马尔可夫情况。最后通过数值算例说明了方法和结果。