人寿保险中的聚合 Markov 模型：性质与估值
Aggregate Markov models in life insurance: properties and valuation
AI总结：
本文提出聚合 Markov 模型以解决传统 Markov 链无法捕捉持续时间效应的问题，基于基本鞅的显式刻画推导了预期累积现金流和前瞻性准备金的矩阵表示，并通过数值算例验证了方法的有效性。
摘要：
在多状态人寿保险中，解析可处理性、计算效率与统计灵活性之间的充分平衡至关重要。这或许解释了 Markov 链建模的普及，其矩阵解析方法能够进行全面的处理。遗憾的是，Markov 链建模无法捕捉持续时间效应，因此本文提出聚合 Markov 模型作为替代。聚合 Markov 模型保留了 Markov 链的大部分解析可处理性，但属于非 Markov 性质，因而更具灵活性。基于对基本鞅的显式刻画，本文推导了预期累积现金流及相应前瞻性准备金的矩阵表示，适用于带或不带附带投保人行为的持续时间依赖型支付。全文特别关注了半 Markov 情形。最后，通过数值算例对方法与结果进行了说明。
英文摘要：
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.
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