ABSTRACT

A discrete-time risk process is considered where the full distribution of the claim size X is not completely known to the insurance company. Rather, it assumes that the distribution of X given $Z=\zeta$ is $F_{\zeta }$ where Z is some structural random variable for which a prior is available. The main emphasis of the paper is the unconditional ruin probability $\psi \left(u\right)$ in this setting where the premium is either updated according to incoming information about the claim distribution or computed by the expected value principle. This is in turn studied via the conditional ruin probability ${\psi }_{\zeta }\left(u\right)$, for which large deviations estimates are available. Rigorous proofs are given only for the case of the $F_{\zeta }$ forming a scale parameter family, including the classical case of gamma claims with a gamma prior. However, the analysis readily suggests what should be the behaviour of $\psi \left(u\right)$ in different models for the claims.

摘要

在保险公司不完全了解索赔规模X的完整分布的情况下，考虑离散时间风险过程。相反，它假设给定的X的分布$Z=\zeta$ 是 $F_{\zeta }$其中Z是一些结构随机变量，其先验是可用的。论文的主要重点是无条件破产概率$\psi \left(你\right)$在这种情况下，保险费要么根据有关索赔分布的传入信息更新，要么根据期望值原则计算。这反过来又通过条件破产概率来研究${\psi }_{\zeta }\left(你\right)$，对于大偏差估计是可用的。严格的证明只针对以下情况$F_{\zeta }$形成一个尺度参数族，包括具有伽马先验的伽马声明的经典情况。然而，分析很容易表明应该是什么行为$\psi \left(你\right)$ 在不同的索赔模型中。