Simulating Stress Laws under Extremal Dependence: Characterizing What Generative Models Must Preserve

2026-08-13

AI Summary

This study addresses the challenge of generating extreme joint loss scenarios and characterizing conditional distributions in multivariate heavy-tailed risk factor systems. It proposes a Self-Similar Generative Estimation (SSGEN) framework that models extremal dependence via Pareto radial components, learning from intermediate exceedances to extrapolate reliably to rarer events. The key insight is that both the conditional stress distribution and the most likely stress configuration are governed by a common limiting tail law, enabling a generative approach that ensures convergence even when the target event is absent from observed samples. The method accurately recovers rare-event probabilities and scaling laws for conditional stress scenarios, delivering a data-driven inverse stress solution with theoretical guarantees on convergence rates.

Abstract

We study stress-scenario generation for systems driven by multivariate heavy-tailed risk factors. Within regions where several financial losses are simultaneously extreme, stress analysis concerns both the conditional law of the risk factors and the most plausible configurations producing those losses. We show that both are governed by the same limiting tail law. Preserving its measure recovers rare-event probabilities and scaled conditional stress laws, while misspecifying extremal dependence distorts some regular joint-stress probability. Its density governs reverse-stress optimization, whose maximizers identify the most plausible stress configurations. To exploit this common structure in finite samples, we develop SSGEN (Self-Similar Generative Estimation), which learns extremal dependence from intermediate exceedances and extrapolates to rarer levels using a Pareto radial component. Even when the target event is absent from the sample, we establish convergence rates for the generated conditional law, and data-driven reverse-stress solutions.

Problem

• extremal dependence
• stress scenarios
• heavy-tailed risks
• conditional law
• rare events

Innovation

• extremal dependence
• stress scenario generation
• heavy-tailed distributions
• self-similar generative estimation
• reverse-stress optimization