We introduce a continuous-time framework for the prediction of outstanding liabilities, in which chain-ladder development factors arise as a histogram estimator of a cost-weighted hazard function running in reversed development time. We use this formulation to show that under our assumptions on the individual data chain-ladder is consistent. Consistency is understood in the sense that both the number of observed claims grows to infinity and the level of aggregation tends to zero. We propose alternatives to chain-ladder development factors by replacing the histogram estimator with kernel smoothers and by estimating a cost-weighted density instead of a cost-weighted hazard. Finally, we provide a real-data example and a simulation study confirming the strengths of the proposed alternatives.

中文翻译：

---

带付费数据的连续链梯

我们引入了一个用于预测未偿债务的连续时间框架，其中链梯发展因素作为在反向发展时间内运行的成本加权风险函数的直方图估计器而出现。我们使用这个公式来表明在我们对单个数据链梯的假设下是一致的。一致性被理解为观察到的声明数量增长到无穷大和聚合水平趋于零。我们通过用核平滑器替换直方图估计器并通过估计成本加权密度而不是成本加权风险来提出链梯发展因素的替代方案。最后，我们提供了一个真实数据示例和一个模拟研究，证实了所提出的替代方案的优势。