Global Impact Journal: Pure & Applied Mathematics

Efficient Calibration for Option Pricing via a Physics-Informed Chebyshev Kolmogorov–Arnold Network

Sumei Zhang — 2026-05-19

Efficient calibration is essential for the practical application of option pricing models. The Fractional Stochastic Volatility Jump Diffusion (FVSJ) model can reproduce several stylized features observed in option markets, including the volatility smile, volatility clustering, and long-memory effects. However, its multiple stochastic components make conventional calibration computationally expensive. This paper proposes a two-step calibration framework that combines a neural network with a differential evolution (DE) algorithm. In the first step, we construct a Physics-Informed Kolmogorov–Arnold Network (PCKAN) to approximate the FVSJ pricing map. Specifically, we replace the B-spline basis in KAN with second-kind Chebyshev polynomials and incorporate a Black–Scholes PDE residual as an additional penalty term in the training objective, aiming to improve global approximation and enhance numerical stability and interpretability. In the second step, the trained PCKAN is used as a fast surrogate pricer within the DE algorithm to accelerate parameter estimation. Empirical results show that the proposed method achieves calibration accuracy comparable to direct pricing while substantially reducing computational time.

Unsupervised Clustering of Cities Using Commercial Air Traffic: A Proxy for Economic Connectivity

Sergio Iglesias-Perez — 2026-08-29

This paper proposes a data-driven framework to identify and rank economically connected cities by using commercial air traffic as a proxy for urban economic connectivity. The study is motivated by the limitation of traditional city classifications, which often rely on costly and multidimensional socioeconomic indicators, and by the need for scalable alternatives based on open mobility data. Using daily flight frequencies between 213 cities included in the GaWC classification for the year 2022, we built a time series for each origin–destination pair and unsupervised clustering these temporal profiles. The resulting clusters were used to define the layers of a multiplex network, where each layer represents a different pattern of flight connectivity. City importance was then estimated through Multiplex PageRank, which allows for temporal behavior and multilayer network structure to be combined in a single ranking scheme. Rather than introducing a new standalone algorithm, this paper contributes a reproducible analytical pipeline that integrates time-series clustering with multiplex centrality analysis using open aviation data. The results show that the ranking obtained is broadly aligned with established classifications such as GaWC, supporting the idea that commercial flight dynamics can provide a useful proxy for economic interconnectedness. The proposed approach offers a simple and replicable tool for comparative urban analysis, although the results should be interpreted with caution given the limited post-pandemic period covered by the data.

Existence and Maximal Regularity of Solutions for a Class of Third-Order Differential Equations with an Unbounded Coefficient

Sabit Igissinov — 2026-08-29

This paper investigates a class of third-order partial differential equations with an unbounded lower-order coefficient in the Hilbert space L₂(R²). The study is motivated by the wide use of third-order equations, particularly of Korteweg–de Vries type, in mathematical physics and wave theory, as well as by the limited development of the corresponding theory in the presence of unbounded coefficients. The main focus is on the existence, uniqueness, and maximal regularity of solutions. Within a functional-analytic framework, the well-posedness of the problem is established in natural function spaces under minimal assumptions on the coefficients. In particular, a priori estimates ensuring maximal regularity are derived.

Predefined-Time Neural Adaptive Control for Distributed Formation Control of Nonlinear Multiagent Systems with Full-State Constraints

Yuehua Fang — 2026-08-29

This paper investigates the distributed formation control problem for nonlinear multiagent systems subject to full-state constraints and proposes a predefined-time neural adaptive control scheme based on a nonlinear mapping technique. To handle the time-varying asymmetric constraints on system states, a smooth and invertible nonlinear mapping function is introduced to transform the original constrained states into unconstrained variables, thereby eliminating the dependence on initial conditions typically required by traditional barrier Lyapunov functions. Within this transformed framework, a predefined-time distributed formation control law is developed, which guarantees that all followers converge to the desired formation configuration and track the leader’s trajectory within a user-specified time upper bound, independent of the initial states. Radial basis function neural networks are employed to approximate the unknown nonlinear dynamics of each agent, and adaptive laws are designed to update the network weights online. Theoretical analysis shows that all closed-loop signals remain bounded, the original system states strictly stay within their prescribed constraint boundaries at all times, and the formation tracking errors converge to a small neighborhood of the origin within the predefined time. Numerical simulations validate the effectiveness of the proposed method, demonstrating faster convergence, higher steady-state accuracy, and improved robustness to initial conditions compared to existing control approaches.

The Operational Efficiency Measurement of China’s Top 100 Digital Economy Firms: An Approach Based on DEA and Kernel Density Estimation

Linyan Zhang — 2026-08-29

In recent years, China’s digital economy has become a key engine for high-quality development. Assessing the operational efficiency of leading digital enterprises is crucial for optimizing resource allocation and promoting sectoral growth. However, existing research largely remains at regional or industry levels and typically reports efficiency scores without diagnosing the root sources of inefficiency. To fill this gap, this study measures the operational efficiency of 99 firms selected from China’s Top 100 Digital Economy list (2017–2022) using the BCC-DEA model, and analyzes their dynamic evolution via kernel density estimation. The findings reveal a fluctuating upward trend in overall efficiency, and that the gap in overall technical efficiency primarily originates from scale efficiency rather than pure technical efficiency. The kernel density peak exhibits a “rise–decline–rise” pattern, indicating existing but narrowing efficiency differences among firms. By decomposing efficiency, this study further classifies firms into four types, revealing that inefficiency is heterogeneous. This paper makes three main contributions. First, it identifies scale efficiency as the main source of efficiency gaps. Second, it classifies firms into four types, revealing that inefficiency is heterogeneous. Third, it uses kernel density estimation to track the dynamic evolution of efficiency, showing a narrowing efficiency gap but a persistent superstar effect. Two policy implications follow: firms with low pure technical efficiency should focus on management training and technology adoption, while firms with low scale efficiency should pursue scale expansion through mergers or partnerships.