risk decomposition

To address scenario-based decision-making under uncertainty, this paper proposes a risk-controllable decision-making method based on scenario compression. To overcome the looseness and strong distributional assumptions inherent in existing risk upper bounds, we derive the first compression-size-dependent risk bound that requires no additional assumptions—integrating stochastic geometric analysis, compression set theory, and refined probabilistic inequalities with optimization. This bound significantly improves tightness: for identical numbers of scenarios and prescribed risk tolerance levels, the upper bound on decision failure probability is reduced by 20–40% on average. The method ensures theoretical rigor while maintaining broad applicability across diverse data-driven robust decision-making settings, thereby providing a more reliable risk-quantification framework for robust optimization under uncertainty.

This paper investigates the Pareto-optimal risk-sharing problem among multiple agents under nonmonotonic, nonconvex distortion-based risk measures—specifically, law-invariant, comonotonically additive functionals. Focusing on three canonical variability measures—Gini deviation, mean–median deviation, and interquantile range—we develop an analytical framework grounded in extremal negative dependence structures and Pareto optimization. We establish that, for Gini and mean–median deviations, the optimal allocation remains comonotonic; in contrast, the interquantile range induces a novel *pairwise antimonotonic mixed structure*—a departure from classical comonotonicity assumptions and the first identification of such extreme negative dependence patterns in optimal risk sharing. We derive closed-form optimal allocations for all three measures, thereby extending the theoretical scope of distortion risk measures and offering a new paradigm for insurance reserving and portfolio risk management.

This work addresses the lack of a unified theoretical foundation in traditional uncertainty quantification (UQ), which hinders systematic distinction between epistemic and aleatoric uncertainty. The authors propose a subjective risk decomposition framework that conceptualizes uncertainty as an emergent consequence of modeling choices. For the first time, this framework formally decomposes the two uncertainty types using strictly proper loss functions—such as reversed cross-entropy—grounded in rigorous statistical principles. By integrating information-theoretic analysis with excess risk decomposition from statistical learning theory, the approach not only recovers established uncertainty measures but also establishes novel theoretical connections between uncertainty quantification and the foundations of statistical learning.

This study addresses the challenge of effectively integrating heterogeneous risks across multiple scenarios in financial markets by proposing a Weighted Generalized Risk Measure (WGRM) and its associated Weighted Risk Quadrangle (WRQ), thereby extending the generalized risk measure and risk quadrangle framework to a weighted setting for the first time. Theoretically, the work establishes analytical characterizations of WGRM under both discrete and continuous settings, proving that its structural properties remain invariant and revealing intrinsic connections among risk, deviation, regret, and error under weighting. Computationally, it leverages convex analysis, stochastic optimization, and linear programming reformulation techniques to transform complex risk optimization problems into tractable linear programs. Empirical results demonstrate that portfolios constructed using WGRM significantly improve risk-adjusted returns, enhance downside resilience, and mitigate losses caused by misjudgments in individual scenarios on NASDAQ 100 and S&P 500 constituents.

In uncertainty quantification, existing risk measures face a tension between overly restrictive law invariance and insufficient probabilistic sufficiency. Method: We introduce the novel concept of “partial law invariance” to unify these two properties. We formally define partial and strong partial law invariance; establish a new theoretical bridge between Kusuoka representations and real-world uncertainty; and propose families of partially law-invariant risk measures—namely, expected shortfall and entropy-based measures. Contribution/Results: We derive necessary and sufficient conditions for compatibility of such risk measures and provide computationally tractable optimization formulations. Numerical experiments demonstrate that the proposed measures exhibit superior modeling flexibility and robustness under heterogeneous uncertainty. This work extends the foundational theory of risk measurement and furnishes new analytical tools for financial risk management and behavioral decision modeling.

This study addresses the challenge of dependency uncertainty in risk and decision models where marginal distributions are sparse and dependence structures are partially unknown, rendering traditional probability bounds analysis ineffective. The authors propose a black-box risk decision framework that integrates p-boxes, precise cumulative distribution functions, and fixed quantities as mixed inputs. Known dependencies are characterized using copulas, while unknown dependencies are propagated through Fréchet-type admissible coupling sets. For the first time, dependence sensitivity is incorporated into probability bounds analysis, enabling cross-dependence modeling between imprecise and precise variables and substantially enhancing the transparency and reasonableness of uncertainty propagation. Case studies demonstrate that neglecting dependence structure can severely underestimate tail risk, leading to overly optimistic decision assessments.

This study addresses the limitation of traditional risk contribution measures, which fail to distinguish whether an asset’s risk stems from its own volatility or its correlation with other assets. The authors propose a novel decomposition—based on a leave-one-out approach—that, for the first time, rigorously preserves additivity while separating total risk contribution into two economically interpretable components: intrinsic risk (capturing idiosyncratic volatility) and correlation risk (reflecting co-movement with the rest of the portfolio). Integrated with time-series analysis, this framework enables dynamic tracking of both risk components across varying market regimes. Empirical results demonstrate that the method reliably and transparently disentangles the effects of volatility shocks and correlation shifts on portfolio risk, offering practical utility for risk reporting, stress testing, performance attribution, and the identification of effective hedging instruments.

This work addresses the misalignment between conventional uncertainty quantification metrics—such as negative log-likelihood and expected calibration error—and the utility of downstream decision-making, which often renders them poor proxies for real-world decision value. To bridge this gap, the paper introduces a “decision-aligned” evaluation principle, systematically exposing the mismatch between widely used scoring rules and common decision tasks. Building on decision theory and proper scoring rules, the authors propose a class of prior-weighted utility-based metrics that directly reflect the impact of predictive uncertainty on decision outcomes. Empirical evaluations across multiple benchmarks and real-world scenarios demonstrate that the proposed metric consistently correlates strongly with actual decision utility, significantly outperforming traditional approaches and offering a principled foundation for decision-relevant uncertainty assessment.

This work proposes an interpretable risk scoring method that directly optimizes net benefit—the primary objective in clinical decision-making—rather than conventional accuracy metrics. By formulating the problem as a sparse integer linear program, the approach simultaneously optimizes net benefit across multiple decision thresholds and yields transparent scoring rules with integer coefficients. Integrating techniques from interpretable machine learning and model calibration, the method achieves significantly higher net benefit on multiple public and real-world clinical datasets while maintaining strong discriminative performance and calibration. Furthermore, the study establishes theoretical connections between net benefit and traditional evaluation metrics, offering new insights into the relationship between clinical utility and standard predictive performance measures.

This study addresses the issue of suboptimal, non-comonotonic risk sharing that arises when regulatory or contractual constraints undermine incentives for risk-averse agents. The paper introduces “quantile-convex order robustness” as a sufficient condition on the feasible set under which a comonotonic improvement exists for all preferences consistent with the convex order, thereby restoring the comonotonicity between Pareto-optimal allocations and aggregate losses. This condition encompasses common risk management constraints—such as value-at-risk (VaR) caps and individual deductibles—and is validated within the mean-variance framework. The result provides a unified and tractable theoretical foundation for constrained risk-sharing problems.