Search Results (16)

In this paper, we propose the exponentially fitted midpoint scheme for the stochastic oscillator. This scheme is first-order strongly convergent and it preserves symplectic. It can effectively simulate the oscillatory behavior of stochastic oscillators, and its second moment grows linearly with time. In addition, we also propose a two-parameter estimation method by analyzing the expectation and variance in the discrete scheme. Numerical experiments are given to verify effectiveness of the exponential fitting method and parameter estimation methods based on this scheme. Full article

In this paper, we propose two methods for parameter estimation in stochastic Korteweg–de Vries (KdV) equations with unknown parameters. Both methods are based on the numerical discretization of the stochastic KdV equation. Moreover, we further propose an extrapolation-based approach to improve the accuracy of parameter estimation. In addition, for the deterministic case, the convergence and conservation of the fully discrete schemes are analyzed. Both our theoretical analysis and numerical tests indicate the efficiency of the proposed methods for the KdV equations considered. Full article

This article solves the anomalies’ operational detection in the network traffic problem for cyber police units by developing an adaptive neural network platform combining a variational autoencoder with continuous stochastic dynamics of the latent space (integration according to the Euler–Maruyama scheme), a continuous–discrete Kalman filter for latent state estimation, and Hotelling’s T² statistical criterion for deviation detection. This paper implements an online learning mechanism (“on the fly”) via the Euler Euclidean gradient step. Verification includes variational autoencoder training and validation, ROC/PR and confusion matrix analysis, latent representation projections (PCA), and latency measurements during streaming processing. The model’s stable convergence and anomalies’ precise detection with the metrics precision is ≈0.83, recall is ≈0.83, the F1-score is ≈0.83, and the end-to-end delay of 1.5–6.5 ms under 100–1000 sessions/s load was demonstrated experimentally. The computational estimate for typical model parameters is ≈5152 operations for a forward pass and ≈38,944 operations, taking into account batch updating. At the same time, the main bottleneck, the O(m³) term in the Kalman step, was identified. The obtained results’ practical significance lies in the possibility of the developed adaptive neural network platform integrating into cyber police units (integration with Kafka, Spark, or Flink; exporting incidents to SIEM or SOAR; monitoring via Prometheus or Grafana) and in proposing applied optimisation paths for embedded and high-load systems. Full article

This paper examines temporal symmetry breaking and structural duality in a class of time-changed McKean–Vlasov neutral stochastic differential equations. The system features super-linear drift coefficients satisfying a one-sided local Lipschitz condition and incorporates a fundamental duality: one drift component evolves under a random time change $E_t$, while the other progresses in regular time t. Within the symmetric framework of mean-field interacting particle systems, where particles exhibit permutation invariance, we establish strong convergence of the tamed Euler–Maruyama method over finite time intervals. By replacing the one-sided local condition with a globally symmetric Lipschitz assumption, we derive an explicit convergence rate for the numerical scheme. Two numerical examples validate the theoretical results. Full article

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the $L^2$ sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the $L^2$ sense. Full article

The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise. Full article

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis. Full article

For a family of stochastic differential equations driven by additive Gaussian noise, we study the asymptotic behaviors of its corresponding Euler–Maruyama scheme by deriving its convergence rate in terms of relative entropy. Our results for the convergence rate in terms of relative entropy complement the conventional ones in the strong and weak sense and induce some other properties of the Euler–Maruyama scheme. For example, the convergence in terms of the total variation distance can be implied by Pinsker’s inequality directly. Moreover, when the drift is $\beta$($0<\beta <1$)-Hölder continuous in the spatial variable, the convergence rate in terms of the weighted variation distance is also established. Both of these convergence results do not seem to be directly obtained from any other convergence results of the Euler–Maruyama scheme. The main tool this paper relies on is the Girsanov transform. Full article

This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in ${\mathcal{L}}^p(p\ge 2)$ sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in ${\mathcal{L}}^p(0<p<2)$ sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples. Full article

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the idea is to transform the original SDEJs to standard SDEJs without singularity by using a deterministic real-valued function that satisfies a second-order differential equation. The Euler–Maruyama scheme is used to approximate the solution to the equations. It is shown that the rate of convergence is $\frac{1}{2}$. Numerically, two different methods are used to approximate solutions for this class of SDEJs. The first method is the direct approximation of the original equation using the Euler–Maruyama scheme with specific tests for the evaluation of the singular part at simulated values of the solution. The second method consists of taking the inverse of the Euler–Maruyama approximation for Zvonkin’s transformed SDEJ, which is free of singular terms. Comparative analysis of the two numerical methods is carried out. Theoretical results are illustrated and proved by means of an example. Full article

The main aim of this contribution is to construct a numerical scheme for solving stochastic time-dependent partial differential equations (PDEs). This has the advantage of solving problems with positive solutions. The scheme provides conditions for obtaining positive solutions, which the existing Euler–Maruyama method cannot do. In addition, it is more accurate than the existing stochastic non-standard finite difference (NSFD) method. Theoretically, the suggested scheme is more accurate than the current NSFD method, and its stability and consistency analysis are also shown. The scheme is applied to the linear scalar stochastic time-dependent parabolic equation and the nonlinear auto-catalytic Brusselator model. The deficiency of the NSFD in terms of accuracy is also shown by providing different graphs. Many observable occurrences in the physical world can be traced back to certain chemical concentrations. Examining and understanding the inter-diffusion between chemical concentrations is important, especially when they coincide. The Brusselator model is the gold standard for describing the relationship between chemical concentrations and other variables in chemical systems. A computational code for the proposed model scheme may be made available to readers upon request for convenience. Full article

The object under investigation is a controllable linear stochastic differential system affected by some external statistically uncertain piecewise continuous disturbances. They are directly unobservable but assumed to be a continuous-time Markov chain. The problem is to stabilize the system output concerning a quadratic optimality criterion. As is known, the separation theorem holds for the system. The goal of the paper is performance analysis of various numerical schemes applied to the filtering of the external Markov input for system stabilization purposes. The paper briefly presents the theoretical solution to the considered problem of optimal stabilization for systems with the Markov jump external disturbances: the conditions providing the separation theorem, the equations of optimal control, and the ones defining the Wonham filter. It also contains a complex of the stable numerical approximations of the filter, designed for the time-discretized observations, along with their accuracy characteristics. The approximations of orders $\frac{1}{2}$, 1, and 2 along with the classical Euler–Maruyama scheme are chosen for the comparison of the Wonham filter numerical realization. The filtering estimates are used in the practical stabilization of the various linear systems of the second order. The numerical experiments confirm the significant influence of the filtering precision on the stabilization performance and superiority of the proposed stable schemes of numerical filtering. Full article

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus. Full article

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases. Full article