金融研究arXiv q-fin8 Oct, 12:00 pm SGTSIGNAL 33AFBB

Optimal Transport Offers a New Mathematical Language for Distributional Comparison and Stress Testing in Actuarial Science

ORIGINAL / Optimal Transport for Actuarial Science

These lecture notes introduce optimal transport as a mathematical language for actuarial science, treating losses, premiums, reserves, and climate losses as probability measures to be compared, transported, averaged, stressed, and interpolated. The novelty lies in systematically connecting optimal transport tools with actuarial problems such as risk measures, Wasserstein robustness, pricing, capital allocation, and longevity risk, while emphasizing the interpretation of model

01 ABSTRACT

The notes argue that optimal transport can serve as a unified mathematical language for actuarial science, handling distributional objects. They are divided into two parts: first introducing tools such as couplings, Kantorovich problems, duality, Wasserstein distances, quantile transport, barycenters, and entropic regularization; then applying them to risk measures, robustness, pricing, capital, reserving, climate prevention, reinsurance, dependence uncertainty, and longevity risk. The author stresses that transport maps and couplings are distributional objects, not causal claims unless additional assumptions are imposed.

02 KEY FINDINGS

Optimal transport models losses, premiums, reserves, climate losses, and lifetime distributions as probability measures, enabling comparison, averaging, stressing, and interpolation.
Core tools include couplings, push-forwards, discrete and continuous Kantorovich problems, duality, Wasserstein distances, quantile transport, barycenters, and entropic regularization.
Applications cover risk measures and Wasserstein robustness, pricing and capital, portfolio drift, reserving cash flows, climate-prevention diagnostics, reinsurance, dependence uncertainty, capital allocation, and longevity risk.

03 RELEVANCE

For actuarial research, these notes provide a unified distributional framework that helps formalize uncertainty in robustness, fairness diagnostics, and stress testing. They highlight the impact of modeling choices, cautioning users to consider assumptions such as state space, cost function, and reference distribution, rather than interpreting transport results directly as causal effects.

04 EVIDENCE BOUNDARY

These are lecture notes and do not provide new empirical results or causal identification. All applications depend on modeling choices (state space, ground cost, ambiguity radius, reference distribution) and additional assumptions; transport maps and couplings represent distributional relationships and cannot automatically be interpreted as causal effects. Later chapters and appendices discuss counterfactual transport, dynamic formulations, unbalanced transport, and Schrödinger bridges, but the evidence boundary remains confined to methodological introduction and computational labs.

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