AI 中文总结
研究具有随机游走结构且潜在概率律未知的保险风险模型，决策者需根据统计量\(Q_n\)选择资本缓冲\(C_n = nf(c,Q_n)\)。分析朴素插件规则不足，确定有良好性质的\(f^*\)，并通过示例展示框架适用性及不同指数包络对资本缓冲的影响。
AI 中文摘要
我们考虑一个具有随机游走结构的保险风险模型，其潜在概率律未知。决策者观察从n个历史观测值计算出的统计量\(Q_n\)，然后需要选择形式为\(C_n = nf(c,Q_n)\)的资本缓冲，其中\(c = \log(1/\delta)/n\)平衡数据量和容忍的破产概率\(\delta\)。经典的安全资本缓冲与调整系数\(\gamma\)成反比，当概率律未知时，该系数必须从数据中推断。函数\(f\)将观察非典型历史统计量的统计成本与决策导致的未来破产指数联系起来。我们研究未来最大值超过数据依赖缓冲的联合事前概率（在历史样本和独立未来风险过程上）。我们表明朴素的插件规则未能达到规定的对数衰减指数\(c\)，说明了在罕见事件约束下决策时模型不确定性的不利影响。然后我们确定了一个具有吸引人性质的函数\(f^*\)：其下半连续主函数是安全的，而严格低于\(f^*\)的正则连续规则在温和条件下是不安全的。我们通过开发三个参数示例说明了我们框架的潜在适用性。在非参数设置中，我们表明单个指数包络导致退化的资本缓冲，而两级指数包络产生非退化的资本缓冲，我们证明其是安全的。
英文摘要
We consider an insurance risk model with a random walk structure in which the underlying probability law is unknown. A decision maker observes a statistic $Q_n$ computed from $n$ historical observations and then needs to choose a capital buffer of the form $C_n=n f(c,Q_n)$, where $c=\log(1/δ)/n$ balances the amount of data and the tolerated ruin probability $δ$. The classical safe capital buffer is inversely proportional to the adjustment coefficient $γ$, the exponential rate at which the ruin probability decays; when the law is unknown, this coefficient must be inferred from the data. The profile $f$ couples the statistical cost of observing an atypical historical statistic with the future ruin exponent induced by the resulting decision. We study the joint ex ante probability (over both the historical sample and an independent future risk process) that the future maximum exceeds the data-dependent buffer. We show that the naive plug-in rule fails to achieve the prescribed logarithmic decay exponent $c$, illustrating the adverse impact of model uncertainty when making decisions under rare-event constraints. We then identify a profile $f^*=f^*(c,Q_n)$ with the following appealing properties: its lower semicontinuous majorants are safe, while regular continuous rules that fall strictly below $f^*$ are, under mild conditions, unsafe. We illustrate the potential applicability of our framework by developing three parametric examples. In nonparametric settings, we show that a single exponential envelope leads to degenerate capital buffers, whereas a two-level exponential envelope yields a nondegenerate capital buffer which we prove to be safe.
