Search Results (1,397)

We investigate a numerical method for approximating stochastic Caputo fractional differential equations driven by multiplicative noise. The nonlinear functions f and g are assumed to satisfy the global Lipschitz conditions as well as the linear growth conditions. The noise is approximated by a piecewise constant function, yielding a regularized stochastic fractional differential equation. We prove that the error between the exact solution and the solution of the regularized equation converges in the $L^2((0,T)\times \mathrm{\Omega })$ norm with an order of $O(\Delta t^{\alpha -1/2})$, where $\alpha \in (1/2,1]$ is the order of the Caputo fractional derivative, and $\Delta t$ is the time step size. Numerical experiments are provided to confirm that the simulation results are consistent with the theoretical convergence order. Full article

In this paper, we investigate the Malliavin differentiability and density smoothness of solutions to stochastic differential equations (SDEs) with non-Lipschitz coefficients. Specifically, we consider equations of the form $d{X}_t= b\left(X_t\right)dt + \sigma \left(X_t\right)d{W}_t, X_0= x_0$ where the drift b(·) and diffusion σ(·) may violate the global Lipschitz condition but satisfy weaker assumptions such as Hölder continuity, linear growth, and non-degeneracy. By employing Malliavin calculus theory, large deviation principles, and Fokker–Planck equations, we establish comprehensive results concerning the existence and uniqueness of solutions, their Malliavin differentiability, and the smoothness properties of density functions. Our main contributions include (1) proving the Malliavin differentiability of solutions under the standard linear growth condition combined with Hölder continuity; (2) establishing the existence and smoothness of density functions using Norris lemma and the Bismut–Elworthy–Li formula; and (3) providing optimal estimates for density functions through large deviation theory. These results have significant applications in financial mathematics (e.g., CIR, CEV, and Heston models), biological system modeling (e.g., stochastic population dynamics and neuronal and epidemiological models), and other scientific domains. Full article

The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form ${\displaystyle \frac{1}{2}}|y\left(T\right)-y_1{|}^2+\alpha {\int }_0^T{|u\left(t\right)|}_4^{4}dt$, where $y_1$ is the observed final state and $y=y^u$ is the solution to the state control system. Here, we prove the existence of a solution and obtain also the Euler–Lagrange optimality conditions which generate a gradient type algorithm for the above optimal control problem. A conceptual algorithm to compute the approximating optimal control and numerical implementation of this algorithm is discussed. Full article

Wind turbine power forecasting is crucial for optimising energy production, planning maintenance, and enhancing grid stability. This research focuses on predicting the output of a Senvion MM92 wind turbine at the Kelmarsh wind farm in the UK using SCADA data from 2020. Two approaches are explored: a hybrid model combining Stochastic Differential Equations (SDEs) with Neural Networks (NNs) and Deep Learning models, in particular, Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM), and the Combination of Convolutional Neural Networks (CNNs) and LSTM. Notably, while SDE-NN models are well suited for predictions in cases where data patterns are chaotic and lack consistent trends, incorporating stochastic processes increases the complexity of learning within SDE models. Moreover, it is worth mentioning that while SDE-NNs cannot be classified as purely “white box” models, they are also not entirely “black box” like traditional Neural Networks. Instead, they occupy a middle ground, offering improved interpretability over pure NNs while still leveraging the power of Deep Learning. This balance is precious in fields such as wind power prediction, where accuracy and understanding of the underlying physical processes are essential. The evaluation of the results demonstrates the effectiveness of the SDE-NNs compared to traditional Deep Learning models for wind power prediction. The SDE-NNs achieve slightly better accuracy than other Deep Learning models, highlighting their potential as a powerful alternative. Full article

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

This paper investigates the dispersion of atmospheric pollutants in urban environments using a fractional-order convection–diffusion-reaction model with dynamic line sources associated with vehicle traffic. The model includes Caputo fractional time derivatives and Riesz fractional space derivatives to account for memory effects and non-local transport phenomena characteristic of complex urban air flows. Vehicle trajectories are generated stochastically on the road network graph using Dijkstra’s algorithm, and each moving vehicle acts as a mobile line source of pollutant emissions. To reflect the daily variability of emissions, a time-dependent modulation function determined by unknown parameters is included in the source composition. These parameters are inferred by solving an inverse problem using synthetic concentration measurements from several fixed observation points throughout the area. The study presents two main contributions. Firstly, a detailed numerical analysis of how fractional derivatives affect pollutant dispersion under realistic time-varying mobile source conditions, and secondly, an evaluation of the performance of the proposed parameter estimation method for reconstructing time-varying emission rates. The results show that fractional-order models provide increased flexibility for representing anomalous transport and retention effects, and the proposed method allows for reliable recovery of emission dynamics from sparse measurements. Full article

The widely used Heun algorithm for the numerical integration of stochastic differential equations (SDEs) is critically re-examined. We discuss and evaluate several alternative implementations, motivated by the fact that the standard Heun scheme is constructed from a low-order integrator. The convergence, stability, and equilibrium properties of these alternatives are assessed through extensive numerical simulations. Our results confirm that the standard Heun scheme remains a benchmark integration algorithm for SDEs due to its robust performance. As a byproduct of this analysis, we also disprove a previous claim in the literature regarding the strong convergence of the Heun scheme. 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

Power flow analysis of low-voltage network (LVN) is one of the most crucial methods for achieving refined management of such networks. To accurately calculate the three-phase (TP) probabilistic power flow (PPF) distribution in LVN, this paper first draws on the injection-type Newton method; by leveraging TP power measurements relative to the neutral point obtained from smart meters, the injected power is expressed in terms of injected current equations, thereby establishing TP power flow models for various components within the low-voltage distribution transformer area grid. Subsequently, addressing the stochastic fluctuation models of load power and photovoltaic output, this paper employs conventional numerical methods and an improved Latin hypercube sampling technique. Utilizing linearized power flow equations and based on the improved semi-invariant method (SIM) and Gram–Charlier (GC) series fitting, a calculation method for three-phase PPF in low-voltage distribution transformer area grids using the improved semi-invariant is proposed. Finally, simulations of the proposed three-phase PPF method are conducted using the IEEE-13 node distribution system. The simulation results demonstrate that the proposed method can effectively perform three-phase PPF calculations for the distribution transformer area grid and accurately obtain probabilistic distribution information of the TP power flow within the grid. 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 paper presents a comprehensive dynamical analysis of a nonlinear oscillator subjected to both deterministic and stochastic excitations. Utilizing a diverse suite of analytical tools—including phase portraits, Poincaré sections, Lyapunov exponents, recurrence plots, Fokker–Planck equations, and sensitivity diagnostics—we investigate the transitions between quasi-periodicity, chaos, and stochastic disorder. The study reveals that quasi-periodic attractors exhibit robust topological structure under moderate noise but progressively disintegrate as stochastic intensity increases, leading to high-dimensional chaotic-like behavior. Recurrence quantification and Lyapunov spectra validate the transition from coherent dynamics to noise-dominated regimes. Poincaré maps and sensitivity analysis expose multistability and intricate basin geometries, while the Fokker–Planck formalism uncovers non-equilibrium steady states characterized by circulating probability currents. Together, these results provide a unified framework for understanding the geometry, statistics, and stability of noisy nonlinear systems. The findings have broad implications for systems ranging from mechanical oscillators to biological rhythms and offer a roadmap for future investigations into fractional dynamics, topological analysis, and data-driven modeling. 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

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well as a risky asset whose drift and volatility is influenced by a stochastic process indicating the probability of potential market downturns. We use a dynamic portfolio optimization approach in continuous time to maximize the expected utility of terminal wealth and solve the corresponding HJB equations for the general class of HARA utility functions. The resulting optimal strategy can be obtained in closed form. It corresponds to a CPPI strategy with a stochastic multiplier that depends on the information from the crisis indicator. In addition to the theoretical results, a performance analysis of the derived strategy is implemented. The specified model is fitted using historic market data and the performance is compared to the optimal portfolio strategy obtained in a Black–Scholes framework without crisis information. The new strategy clearly dominates the BS-based CPPI strategy with respect to the Sharpe Ratio and Adjusted Sharpe Ratio. Full article

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation. 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