AI 中文总结
本文构建了考虑长记忆与随机时间变化的单位一致精算框架，通过分数布朗运动与CIR过程建模温度异常，实现了温度指数保险的易处理定价，实证表明长记忆和随机时间变化会显著影响保费。
AI 中文摘要
本文构建了一个单位一致的精算框架，用于在长程依赖与随机变异性下对上限累积温度指数保险进行定价。每日温度异常被建模为分数布朗运动的增量，该分数布朗运动在由平稳归一化的Cox-Ingersoll-Ross（CIR）过程积分生成的操作时间上取值。我们证明，随机时间变化在保留增量的平稳性和长记忆协方差衰减特性的同时，还通过随机操作时钟引入了额外的变异性。累积温度指数具有条件高斯表示形式，这为上限止损合约导出了精确的条件指数核，并确保对于每个正的风险厌恶参数，熵保费均存在。因此，估值可简化为对累积CIR时间的外层蒙特卡罗期望，避免了分数布朗运动路径模拟和协方差矩阵的构建。我们进一步确立了保费关于风险厌恶和条件波动率的单调性性质。基于芝加哥温度数据的实证示例表明，与传统布朗运动和分数布朗运动基准相比，长记忆和随机时间变化均会对保险保费产生实质性影响，其中赫斯特参数在估值不确定性中发挥重要作用。因此，所提出的框架为将持续依赖、随机变异性和有界保险损失纳入气候指数定价提供了一种易处理的方法。
英文摘要
This paper develops a unit-consistent actuarial framework for pricing capped cumulative temperature-index insurance under long-range dependence and stochastic variability. Daily temperature anomalies are modeled as increments of fractional Brownian motion evaluated at an operational time generated by the integral of a stationary normalized Cox--Ingersoll--Ross process. We show that the stochastic time change preserves stationarity and the long-memory covariance decay of the increments while introducing additional variability through the random operational clock. The cumulative temperature index admits a conditionally Gaussian representation, which leads to an exact conditional exponential kernel for capped stop-loss contracts and ensures existence of the entropic premium for every positive risk-aversion parameter. Consequently, valuation reduces to an outer Monte Carlo expectation over the accumulated CIR time, avoiding fractional Brownian path simulation and covariance-matrix construction. We further establish monotonicity properties of the premium with respect to risk aversion and conditional volatility. An empirical illustration based on Chicago temperature data shows that both long memory and stochastic time change can materially affect insurance premiums relative to conventional Brownian and fractional Brownian benchmarks, with the Hurst parameter playing an important role in valuation uncertainty. The proposed framework therefore provides a tractable approach for incorporating persistent dependence, stochastic variability, and bounded insurance losses into climate-index pricing.
Comments43 pages, 6 figures
