change of measure

This study addresses the lack of systematic investigation into the statistical properties and testing performance of Jensen–Shannon divergence (JSD) and Kullback–Leibler (KL) divergence in credit risk model monitoring. It derives, for the first time, chi-squared asymptotic reference distributions for both divergences under distributional shift using asymptotic theory, and conducts a comprehensive Monte Carlo simulation to evaluate their Type I error control and statistical power relative to the Population Stability Index (PSI). The results demonstrate that JSD exhibits superior Type I error control, closely attaining the nominal 5% level, yet shows limited power (27%) in small samples (n=200); in contrast, KL divergence and PSI achieve higher power (32%). These findings provide both theoretical grounding and empirical guidance for selecting appropriate divergence metrics in practical model monitoring.

Standard empirical risk minimization (ERM) suffers from inaccurate risk assessment in high-confidence subregions—specifically, large-margin regions in classification and low-variance regions in regression. Method: This paper proposes a weighted empirical risk minimization (WERM) framework that employs data-dependent weighting functions to prioritize samples based on local confidence. Contribution/Results: We establish, for the first time under a general “balanceable” Bernstein condition, that WERM achieves subregion-adaptive superiority: its conditional risk bound incorporates a data-dependent constant term, strictly improving upon standard ERM. Theoretical analysis demonstrates that WERM selectively enhances risk control accuracy in high-confidence subregions. Synthetic experiments validate the theory, showing significant improvements in both generalization error and risk estimation within these critical subregions.

This study clarifies the informational nature of the Kullback–Leibler (KL) divergence difference (Δ_KL) between discrete empirical distributions and corrects the common misinterpretation of its sign as indicating support set inclusion or coverage. By analytically decomposing the mathematical structure of Δ_KL, the work reveals that it fundamentally serves as an asymmetric contrastive measure of weighted log-probability ratios across categories, rather than a metric of distributional breadth or set containment. The theoretical insights are substantiated through a bibliometric case study examining topic distributions in COVID-19 preprints, offering both intuitive interpretation and empirical validation. This research advances the information-theoretic understanding of asymmetric distributional discrepancies and provides a rigorous foundation—supported by concrete examples—for the proper interpretation of Δ_KL in practical applications.

In credit risk management, binary classifiers often suffer from miscalibrated probability estimates due to target prior distribution shift. To address this, we propose two novel calibration methods: CSPD (Calibration via Separable Parametric Modeling of Covariate and Posterior Drift) and ROC-based QMM (Quantile Moment Matching under ROC constraints). First, we systematically uncover and formalize the distribution-invariance assumption implicitly encoded in the AUC metric, integrating it into the calibration framework design. CSPD enables interpretable, parametric calibration by decoupling covariate shift from posterior shift. QMM delivers conservative estimates of concave objectives—such as credit risk-weighted metrics—under ROC curve constraints. Experiments on real-world financial datasets demonstrate that QMM significantly outperforms standard baselines—including Platt scaling and isotonic regression—while maintaining high robustness and practicality under unknown prior distributions.

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 work addresses why the Kullback–Leibler (KL) divergence is uniquely suited for inference by formalizing inference as the selection of a minimal element within a preorder of positive measures, where divergences serve merely as numerical representations. Building on the axiom of reconstruction invariance—which requires that inference outcomes remain unchanged under equivalent problem formulations—the authors show that KL divergence emerges uniquely without invoking additional assumptions. This framework unifies maximum entropy, Bayesian updating, and exponential family estimation, extending classical axiomatic characterizations from finite alphabets to general measurable spaces. By integrating category theory, f-divergence theory, preorder structures, and Čencov’s category of statistical models, the paper establishes a rigorous mathematical foundation wherein inference operators arise naturally as covariant functors, applicable uniformly across both discrete and continuous settings.

This study investigates principles for dynamic financial risk measurement that depend solely on the distribution of random variables and the associated information revelation process. Addressing time-consistent dynamic risk measures, it introduces “adapted law invariance” as the dynamic counterpart to static law invariance, thereby overcoming the limitations of terminal law invariance. By employing techniques such as Fatou regularity, conditional extensions, and backward composition, the paper establishes an adapted Kusuoka representation in the coherent setting and extends the Kupper–Schachermayer theorem. The main contribution lies in proving that time-consistent risk measures satisfying adapted law invariance can be recursively generated from static law-invariant risk measures, and in fully characterizing their structural properties.

This study addresses the complexity and limited applicability of existing characterizations of Uniformly Weighted Divergence Preferences (UWDP) by proposing a simpler, computationally tractable representation. By constructing a translation-invariant envelope of state-independent expected utility over the L⁰ space and leveraging tools from convex analysis, duality theory, and variational methods in spaces of probability measures, the paper establishes—for the first time under full generality—the equivalence between UWDP and this envelope. The resulting formulation not only unveils the intrinsic structure of UWDP but also substantially enhances its operationality and interpretability, yielding several important theoretical implications.

This study quantifies the cost of holding a suboptimal portfolio relative to the Kelly-optimal portfolio by introducing both the true and subjective probability measures. The discrepancy between these measures is characterized via Kullback–Leibler (KL) divergence: the forward KL divergence corresponds to wealth loss, while the reverse KL divergence reflects apparent excess returns. Drawing on information theory, measure change techniques, and the Kelly criterion, the work establishes an exact duality between the cost of suboptimality and information entropy, yielding a precise analytical relationship. This result provides the first rigorous linkage between portfolio suboptimality and the framework of information geometry, offering new theoretical insights into the interplay between investment performance and informational inefficiency.

This study aims to unify the theoretical framework of probability measures in asset pricing by addressing how market prices can be represented through measure changes. It systematically traces the conceptual evolution from state prices and risk-neutral measures to stochastic discount factors (pricing kernels), emphasizing that asset pricing fundamentally relies on equivalent measures—adjusted via discounting, numéraire normalization, or utility weighting—rather than the original physical probability measure. The work innovatively incorporates data-driven information such as textual content, attention metrics, and sentiment into the measure transformation process, thereby extending the learning paradigm for pricing kernels in incomplete markets. By integrating stochastic discount factors, Radon–Nikodym derivatives, Girsanov transformations, implied densities, and machine learning techniques, the paper constructs a cohesive framework that bridges classical asset pricing theory with modern empirical methodologies, offering both theoretical grounding and practical guidance.