发表机构
Université du Québec à Montréal (UQAM); Kyoto University（魁北克大学蒙特利尔分校; 京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文以最优传输为数学框架，系统介绍其在精算科学中的应用，涵盖风险度量、定价、准备金等核心工具与方法。
AI 中文摘要
这些讲义将最优传输引入为精算科学的数学语言。它们将损失、保费、评分、准备金、资本情景、气候损失和寿命分布视为可比较、可传输、可平均、可压力测试和可插值的概率测度。第一部分开发主要工具：耦合、推前映射、离散和连续Kantorovich问题、对偶性、Wasserstein距离、分位数传输、重心、熵正则化和统计最优传输。第二部分将这些工具应用于风险度量、Wasserstein稳健性、定价与资本、投资组合漂移、现金流分布准备金、气候预防诊断、再保险、依赖不确定性、资本配置、分布公平性诊断和长寿风险。后续章节和附录讨论反事实传输、动态公式、非平衡传输、Schrödinger桥、成本工程和R语言计算实验室。重点在于精算建模选择：状态空间、地面成本、模糊半径、参考分布和传输计划的解释。传输映射和耦合被用作分布对象，除非施加额外假设，否则不视为因果主张。
英文摘要
These lecture notes introduce optimal transport as a mathematical language for actuarial science. They treat losses, premiums, scores, reserves, capital scenarios, climate losses and lifetime distributions as probability measures that can be compared, transported, averaged, stressed and interpolated. The first part develops the main tools: couplings, push-forwards, discrete and continuous Kantorovich problems, duality, Wasserstein distances, quantile transport, barycenters, entropic regularization and statistical optimal transport. The second part applies these tools to risk measures, Wasserstein robustness, pricing and capital, portfolio drift, reserving cash-flow distributions, climate-prevention diagnostics, reinsurance, dependence uncertainty, capital allocation, distributional fairness diagnostics and longevity risk. Later chapters and appendices discuss counterfactual transport, dynamic formulations, unbalanced transport, Schrödinger bridges, cost engineering and computational labs in R. The emphasis is on actuarial modelling choices: the state space, the ground cost, the ambiguity radius, the reference distribution and the interpretation of the transport plan. Transport maps and couplings are used as distributional objects, not as causal claims unless additional assumptions are imposed.
Comments17 lectures, appendices and R labs
