Differential Quantile-Based Sensitivity in Discontinuous Models

ArXiv ID: 2310.06151 “View on arXiv”

Authors: Unknown

Abstract

Differential sensitivity measures provide valuable tools for interpreting complex computational models used in applications ranging from simulation to algorithmic prediction. Taking the derivative of the model output in direction of a model parameter can reveal input-output relations and the relative importance of model parameters and input variables. Nonetheless, it is unclear how such derivatives should be taken when the model function has discontinuities and/or input variables are discrete. We present a general framework for addressing such problems, considering derivatives of quantile-based output risk measures, with respect to distortions to random input variables (risk factors), which impact the model output through step-functions. We prove that, subject to weak technical conditions, the derivatives are well-defined and derive the corresponding formulas. We apply our results to the sensitivity analysis of compound risk models and to a numerical study of reinsurance credit risk in a multi-line insurance portfolio.

Keywords: Differential sensitivity measures, Risk factors, Quantile-based output risk measures, Derivatives, Compound risk models, Insurance / Risk Management

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper presents a high-level theoretical framework with rigorous mathematical proofs involving generalised functions, quantile differentiation, and weak convergence, which is dense and advanced. While it includes numerical studies on reinsurance credit risk and compound models, the focus is on theoretical derivations rather than detailed backtesting protocols or public datasets.

  flowchart TD
    A["Research Goal: Derive differential sensitivity for discontinuous models"] --> B["Key Methodology: Quantile-based risk measures with step-function inputs"]
    B --> C["Key Methodology: Derivatives w.r.t. random input distortions"]
    C --> D["Input: Compound Risk Models &<br/>Multi-line Insurance Portfolio"]
    D --> E["Computational Process: Verify well-defined derivatives<br/>& derive sensitivity formulas"]
    E --> F["Outcome: General framework for sensitivity analysis"]
    F --> G["Outcome: Application to reinsurance credit risk"]