Search Results (374)

This study introduces a novel approach to correlation-based feature selection and dimensionality reduction in high-dimensional data structures. To this end, a customized scoring function is proposed, designed as a dual-objective structure that simultaneously maximizes the correlation with the target variable while penalizing redundant information among features. The method is built upon three main components: correlation-based preliminary assessment, feature selection via the tailored scoring function, and integration of the selection results into a t-SNE visualization guided by Rel/Red ratios. Initially, features are ranked according to their Pearson correlation with the target, and then redundancy is assessed through pairwise correlations among features. A priority scheme is defined using a scoring function composed of relevance and redundancy components. To enhance the selection process, an optimization framework based on stochastic differential equations (SDEs) is introduced. Throughout this process, feature weights are updated using both gradient information and diffusion dynamics, enabling the identification of subsets that maximize overall correlation. In the final stage, the t-SNE dimensionality reduction technique is applied with weights derived from the Rel/Red scores. In conclusion, this study redefines the feature selection process by integrating correlation-maximizing objectives with stochastic modeling. The proposed approach offers a more comprehensive and effective alternative to conventional methods, particularly in terms of explainability, interpretability, and generalizability. The method demonstrates strong potential for application in advanced machine learning systems, such as credit scoring, and in broader dimensionality reduction tasks. Full article

Accurate modeling of event sequences is valuable in domains like electronic health records, financial risk management, and social networks. Random time intervals in these sequences contain key dynamic information, and temporal point processes (TPPs) are widely used to analyze event triggering mechanisms and probability evolution patterns in asynchronous sequences. Neural TPPs (NTPPs) enhanced by deep learning improve modeling capabilities, but most suffer from limited interpretability due to predefined functional structures. This study proposes KANJDP (Kolmogorov–Arnold Neural Jump-Diffusion Process), a novel event sequence modeling method: it decomposes the intensity function via stochastic differential equations (SDEs), with each component parameterized by learnable spline functions. By analyzing each component’s contribution to event occurrence, KANJDP quantitatively reveals core event generation mechanisms. Experiments on real-world and synthetic datasets show that KANJDP achieves higher prediction accuracy with fewer trainable parameters. Full article

As a crucial modeling tool for stochastic financial markets, the Lévy risk model effectively characterizes the evolution of risks during enterprise operations. Through dynamic evaluation and quantitative analysis of risk indicators under specific dividend- distribution strategies, this model can provide theoretical foundations for optimizing corporate capital allocation. Addressing the inadequate adaptability of traditional single-period threshold strategies in time-varying market environments, this paper proposes a dividend strategy based on multiperiod dynamic threshold adjustments. By implementing periodic modifications of threshold parameters, this strategy enhances the risk model’s dynamic responsiveness to market fluctuations and temporal variations. Within the framework of the spectrally negative Lévy risk model, this paper constructs a stochastic control model for multiperiod threshold dividend strategies. We derive the integro-differential equations for the expected present value of aggregate dividend payments before ruin and the Gerber–Shiu function, respectively. Combining the methodologies of the discounted increment density, the operator introduced by Dickson and Hipp, and the inverse Laplace transforms, we derive the explicit solutions to these integro-differential equations. Finally, numerical simulations of the related results are conducted using given examples, thereby demonstrating the feasibility of the analytical method proposed in this paper. Full article

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved. Full article

The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the time-varying coefficient $\theta \left(t\right)$ in the following evolution equation:$$\left\{\begin{array}{c}{\displaystyle du(t,x)=(A_0+\theta \left(t\right)A_1)u(t,x)dt+d{\xi }^H(t,x),x\in [0,1],t\in (0,T],}\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill \end{array}\right.$$ where ${\xi }^H(t,x)$ is a cylindrical sub-fractional Brownian motion in ${\mathbb{L}}^2\left([0,T]\times [0,1]\right)$, and $A_0+\theta \left(t\right)A_1$ is a strongly elliptic differential operator. We obtain the asymptotic mean square error and the limiting distribution of the proposed estimator. These results are proved under some standard conditions on the kernel and some mild conditions on the model. Finally, we give an application for the confidence interval construction. Full article

This paper focuses on exploring an impulsive stochastic differential variational inequality (ISDVI), which combines an impulsive stochastic differential equation and a stochastic variational inequality. Innovatively, our work incorporates two key aspects: first, our stochastic differential equation contains an impulsive term, enabling better handling of sudden event impacts; second, we utilize a non-local condition $z\left(0\right)={\chi }_0+\vartheta \left(z\right)$ that integrates measurements from multiple locations to construct superior models. Methodologically, we commence our analysis by using the projection method, which ensures the existence and uniqueness of the solution to ISDVI. Subsequently, we showcase the practical applicability of our theoretical findings by implementing them in the investigation of a stochastic consumption process and electrical circuit model. Full article

Background: Rock–blast design is a canonical inverse problem that joins elastodynamic partial differential equations (PDEs), fracture mechanics, and stochastic heterogeneity. Objective: Guided by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol, a systematic review of mathematical methods for geomechanically informed blast modelling and optimisation is provided. Methods: A Scopus–Web of Science search (2000–2025) retrieved 2415 records; semantic filtering and expert screening reduced the corpus to 97 studies. Topic modelling with Bidirectional Encoder Representations from Transformers Topic (BERTOPIC) and bibliometrics organised them into (i) finite-element and finite–discrete element simulations, including arbitrary Lagrangian–Eulerian (ALE) formulations; (ii) geomechanics-enhanced empirical laws; and (iii) machine-learning surrogates and multi-objective optimisers. Results: High-fidelity simulations delimit blast-induced damage with ≤0.2 m mean absolute error; extensions of the Kuznetsov–Ram equation cut median-size mean absolute percentage error (MAPE) from 27% to 15%; Gaussian-process and ensemble learners reach a coefficient of determination ($R^2>0.95$) while providing closed-form uncertainty; Pareto optimisers lower peak particle velocity (PPV) by up to 48% without productivity loss. Synthesis: Four themes emerge—surrogate-assisted PDE-constrained optimisation, probabilistic domain adaptation, Bayesian model fusion for digital-twin updating, and entropy-based energy metrics. Conclusions: Persisting challenges in scalable uncertainty quantification, coupled discrete–continuous fracture solvers, and rigorous fusion of physics-informed and data-driven models position blast design as a fertile test bed for advances in applied mathematics, numerical analysis, and machine-learning theory. Full article

Alkaline carbonatite complexes are formed by magmatic, hydrothermal, and weathering geological events, which modify the minerals present in the rocks, resulting in ores with varied metallurgical behavior. To better spatially distinguish ores with distinct plant responses, creating a 3D geometallurgical block model was necessary. To establish the clusters, four different algorithms were tested: K-Means, Hierarchical Agglomerative Clustering, dual-space clustering (DSC), and clustering by autocorrelation statistics. The chosen method was DSC, which can consider the multivariate and spatial aspects of data simultaneously. To better understand each cluster’s mineralogy, an XRD analysis was conducted, shedding light on why each cluster performs differently in the plant: cluster 0 contains high magnetite content, explaining its strong magnetic yield; cluster 3 has low pyrochlore, resulting in reduced flotation yield; cluster 2 shows high pyrochlore and low gangue minerals, leading to the best overall performance; cluster 1 contains significant quartz and monazite, indicating relevance for rare earth elements. A hierarchical indicator kriging workflow incorporating a stochastic partial differential equation (SPDE) trend model was applied to spatially map these domains. This improved the deposit’s circular geometry reproduction and better represented the lithological distribution. The elaborated model allowed the identification of four geometallurgical zones with distinct mineralogical profiles and processing behaviors, leading to a more robust model for operational decision-making. Full article

We present a formal framework for modeling neural network dynamics using Category Theory, specifically through Markov categories. In this setting, neural states are represented as objects and state transitions as Markov kernels, i.e., morphisms in the category. This categorical perspective offers an algebraic alternative to traditional approaches based on stochastic differential equations, enabling a rigorous and structured approach to studying neural dynamics as a stochastic process with topological insights. By abstracting neural states as submeasurable spaces and transitions as kernels, our framework bridges biological complexity with formal mathematical structure, providing a foundation for analyzing emergent behavior. As part of this approach, we incorporate concepts from Interacting Particle Systems and employ mean-field approximations to construct Markov kernels, which are then used to simulate neural dynamics via the Ising model. Our simulations reveal a shift from unimodal to multimodal transition distributions near critical temperatures, reinforcing the connection between emergent behavior and abrupt changes in system dynamics. Full article

The simulation of wind power, electricity load, and natural gas prices will allow commodity traders to see the future movement of prices in a more probabilistic manner. The ability to observe possible paths for wind power, electricity load, and natural gas prices enables traders to obtain valuable insights for placing their trades on electricity prices. Since the above processes involve a seasonality factor, the seasonality component was modeled using a truncated Fourier series, and the random component was modeled using stochastic differential equations (SDE). It is evident from the literature that all the above processes are mean-reverting processes; thus, three mean-reverting Ornstein–Uhlenbeck (OU) processes were considered the model for wind power, the electricity load, and natural gas prices. Industry experts believe there is a correlation between wind power, the electricity load, and natural gas prices. For example, when wind power is higher and the electricity load is lower, natural gas prices are relatively low. The novelty of this study is the incorporation of the correlation structure between processes into the mean-reverting OU process using a copula function. Thus, the study utilized a vine copula and integrated it into the simulation. The study was conducted for the Texas energy market and used daily time scales for the simulations, and it was able to conclude that the proposed novel mean-reverting OU process outperforms the classical mean-reverting process in the case of wind power and the electricity load. Full article

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields. Full article

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. In contrast to earlier studies that imposed static or overly simplified regime-switching assumptions, our work presents a fully integrated framework—combining optimal control theory, a regime-dependent system of elliptic PDEs, and comprehensive numerical and sensitivity analyses—to more accurately capture the complex stochastic dynamics of production planning and thereby deliver enhanced, actionable insights for modern manufacturing environments. Full article

We investigate the bridge problem for stochastic processes, that is, we analyze the statistical properties of trajectories constrained to begin and terminate at a fixed position within a time interval $\tau$. Our primary focus is the time-reversal symmetry of these trajectories: under which conditions do the statistical properties remain invariant under the transformation $t\to \tau -t$? To address this question, we compare the stochastic differential equation describing the bridge, derived equivalently via Doob’s transform or stochastic optimal control, with the corresponding equation for the time-reversed bridge. We aim to provide a concise overview of these well-established derivation techniques and subsequently obtain a local condition for the time-reversal asymmetry that is specifically valid for the bridge. We are specifically interested in cases in which detailed balance is not satisfied and aim to eventually quantify the bridge asymmetry and understand how to use it to derive useful information about the underlying out-of-equilibrium dynamics. To this end, we derived a necessary condition for time-reversal symmetry, expressed in terms of the current velocity of the original stochastic process and a quantity linked to detailed balance. As expected, this formulation demonstrates that the bridge is symmetric when detailed balance holds, a sufficient condition that was already known. However, it also suggests that a bridge can exhibit symmetry even when the underlying process violates detailed balance. While we did not identify a specific instance of complete symmetry under broken detailed balance, we present an example of partial symmetry. In this case, some, but not all, components of the bridge display time-reversal symmetry. This example is drawn from a minimal non-equilibrium model, namely Brownian Gyrators, that are linear stochastic processes. We examined non-equilibrium systems driven by a "mechanical” force, specifically those in which the linear drift cannot be expressed as the gradient of a potential. While Gaussian processes like Brownian Gyrators offer valuable insights, it is known that they can be overly simplistic, even in their time-reversal properties. Therefore, we transformed the model into polar coordinates, obtaining a non-Gaussian process representing the squared modulus of the original process. Despite this increased complexity and the violation of detailed balance in the full process, we demonstrate through exact calculations that the bridge of the squared modulus in the isotropic case, constrained to start and end at the origin, exhibits perfect time-reversal symmetry. Full article

In response to the limitations of traditional static reliability analysis methods in characterizing the reliability changes of the Integrated Power System, this paper proposes a time-varying reliability analysis framework based on a Dynamic Bayesian Network. By embedding a multi-physics coupled degradation model into the conditional probability nodes of the Dynamic Bayesian Network, a joint stochastic differential equation for the degradation process was constructed, and the dynamic correlation between continuous degradation and discrete fault events throughout the entire life cycle was achieved. A modified method for modeling continuous degradation systems was proposed, which effectively solves the numerical stability problem of modeling complex degradation systems. Finally, the applicability and correctness of the model were verified through numerical examples, and the results showed that the analysis framework can be effectively applied to time-varying reliability assessment and dynamic health management of complex equipment systems such as the Integrated Power System. Full article

In this paper, we investigate the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. Based on the center manifold theorem and stochastic averaging method, a reduced-order dynamic model of the system is established, with a focus on analyzing the influence of time delay parameters and stochastic excitation on the system’s Hopf bifurcation characteristics. By introducing stochastic differential equation theory, the system is transformed into the form of an Itô equation, revealing bifurcation phenomena in the parameter space. Numerical simulation results demonstrate that decreasing the time delay and increasing the time delay feedback gain can effectively enhance system stability, whereas increasing the time delay and decreasing the time delay feedback gain significantly improves dynamic performance. Additionally, it is observed that Gaussian white noise intensity modulates the bifurcation threshold. This study provides a novel theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems and offers a theoretical basis for subsequent research. Full article