Search Results (56)

The well-known effect of stabilization by noise for Ito’s scalar linear stochastic differential equation was proven by R.Z. Khasminskii more than 50 years ago. Here, a similar statement is obtained for a system of linear stochastic differential equations. The obtained result is illustrated on the system of two linear stochastic differential equations via several special examples with numerical simulations and figures. Full article

This paper designs a novel aperiodic intermittent control (AIC) strategy using discrete-time observation information. It can stabilize unstable hybrid McKean–Vlasov stochastic differential equations and reduce control consumption effectively. Key contributions include the following: (1) Lévy noise is introduced into the hybrid McKean–Vlasov framework to describe discontinuous disturbances. We further derive the existence, uniqueness and generalized Itô formula for the above system. (2) A new distribution-dependent Lyapunov functional to prove moment finiteness, mean square, and asymptotic exponential stability is constructed. (3) We derive explicit ranges for the AIC time rate and observation intervals. By tightening the state error bound via an innovative technique, the control design constraints are effectively relaxed. (4) We prove the equivalence of exponential stability between the controlled system and its particle approximation. This approach avoids the computational intractability of the exact probability distribution. Finally, the efficacy of our method is demonstrated through a numerical example. Full article

This paper addresses the stabilization problem of hybrid stochastic McKean–Vlasov differential equations via a discrete-time state observation feedback control strategy. Utilizing the coupling method and particle system approximation, Itô’s formula for Markovian switching stochastic McKean–Vlasov differential equations is established. Based on the derived formula, we construct two novel Lyapunov functionals that incorporate state processes, probability distributions, and Markovian switching signals. Using the proposed Lyapunov functionals, we further analyze three stability properties of the closed-loop system, including H_∞ stability, asymptotic stability, and mean-square exponential stability. Due to the time-varying characteristics of system distributions, numerical simulation lacks fixed reference benchmarks and faces considerable difficulties. To overcome this challenge, this paper introduces a particle system approximation scheme. We further prove the exponential stability equivalence between the controlled McKean–Vlasov system and its corresponding particle system. This equivalence relation provides an effective new approach for the stability analysis of such controlled hybrid stochastic systems. Finally, an illustrative example is given to verify our theory results. Full article

This study develops a stochastic degradation-based reliability framework for mechanical systems subject to interacting operational stresses and imperfect maintenance. The degradation dynamics are formulated in cumulative damage space and modeled using a geometric Itô diffusion process, in which the drift term incorporates a multiplicative degradation kernel representing the combined influence of load, speed, misalignment, and environmental exposure. Imperfect maintenance is represented through a continuous attenuation functional embedded within the drift structure, allowing maintenance actions to reduce degradation growth without restoring the system to an as-good-as-new condition. Using a logarithmic transformation, the multiplicative stochastic differential equation is converted into an additive diffusion process, enabling analytical treatment via Itô’s lemma. A closed-form reliability expression is then obtained through first-passage analysis, yielding a lognormal survival function governed directly by the degradation dynamics. Numerical evaluation demonstrates physically consistent wear-out behavior and confirms the stability of the derived reliability formulation. The model further enables reliability-based maintenance optimization through preventive replacement analysis. Sensitivity results indicate that system reliability is strongly influenced by the degradation growth parameter governing the stochastic drift. The proposed framework provides a mathematically tractable connection between stochastic degradation modeling, reliability theory, and maintenance optimization. Beyond its application to conveyor belt systems, the formulation offers a general analytical structure for reliability assessment of degrading engineering systems governed by multiplicative stochastic dynamics. Full article

Because the introduction of a vibro-impact structure can widen the bandwidth and improve the harvesting efficiency of the vibration energy harvesting (VEH) systems, an analytical method for a VEH system based on vibro-impact is proposed to employ the stochastic response and stability. Firstly, the piezoelectric control equation is decoupled by the generalized harmonic transformation, which obtains an uncoupled equivalent system. Secondly, the Itô stochastic differential equation with amplitude is analytically derived by applying the proposed analytical method. Furthermore, the influence of crucial parameters on the mean square voltage (MSV) and the mean output power is explored, such as the coupling factors and restitution coefficient. Finally, the top Lyapunov exponent (TLE) can be derived based on the linearized averaged Itô equations and the condition for the stability with probability one is obtained. It turned out that restitution coefficient r and time constant ratio $\mu$ have remarkable effects on the system’s stability. Full article

This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The Itô or Stratonovich stochastic differential equations with the Wiener component describe dynamic systems, and the manifold is implicitly defined by a differentiable function. A convenient implementation of the algorithm for forming invariant stochastic differential systems within symbolic computation environments characterizes the proposed method. It is based on determining a basis associated with a tangent hyperplane to the manifold. This article discusses the problem of basis degeneration and examines variants that allow for the simple construction of a basis that does not degenerate. Examples of invariant stochastic differential systems are given, and numerical simulations are performed for them. Full article

Empirical debates about a “crisis of trust” highlight long-lived pockets of high trust and deep distrust in institutions, as well as abrupt, shock-induced shifts between the two. We propose a probabilistic model in which such phenomena emerge endogenously from social learning on hierarchical networks. Starting from a discrete model on a directed acyclic graph, where each agent makes a binary adoption decision about a single assertion, we derive an effective influence kernel that maps individual priors to stationary adoption probabilities. A continuum limit along hierarchical depth yields a degenerate, non-conservative logistic–diffusion equation for the adoption probability $u(x,t)$, in which diffusion is modulated by $(1-u)$ and increases the integral of u rather than preserving it. To account for micro-level uncertainty, we perturb these dynamics by multiplicative Stratonovich noise with amplitude proportional to $u(1-u)$, strongest in internally polarised layers and vanishing at consensus. At the level of a single depth layer, Stratonovich–Itô conversion and Fokker–Planck analysis show that the noise induces an effective double-well potential with two robust stochastic phases, $u\approx 0$ and $u\approx 1$, corresponding to persistent distrust and trust. Coupled along depth, this local bistability and degenerate diffusion generate extended domains of trust and distrust separated by fronts, as well as rare, Kramers-type transitions between them. We also formulate the associated stochastic partial differential equation in Martin–Siggia–Rose–Janssen–De Dominicis form, providing a field-theoretic basis for future large-deviation and data-informed analyses of trust landscapes in hierarchical societies. Full article

This study examines the synchronization of Markovian switching stochastic delayed complex dynamical networks (MSSDCDNs). MSSDCDNs have general structures and coupling forms, which are influenced by Markovian switching, random disturbances, and time delays. Simultaneously, relevant controllers are incorporated into certain nodes. Utilizing the theory of stochastic differential equations, we establish adequate requirements to guarantee exponential synchronization in mean square inside the network by formulating suitable Lyapunov functions and employing general Itô formula and inequality approaches. Lastly, numerical examples and simulations are used to verify the validity of the derived theoretical results. Full article

This study introduces Itô-RMSProp, a novel extension of the RMSProp optimizer inspired by Itô stochastic calculus, which integrates adaptive Gaussian noise into the update rule to enhance exploration and mitigate overfitting during training. We embed this optimizer within Gated Recurrent Unit (GRU) networks for stock price forecasting, leveraging the GRU’s strength in modeling long-range temporal dependencies under nonstationary and noisy conditions. Extensive experiments on real-world financial datasets, including a detailed sensitivity analysis over a wide range of noise scaling parameters ($\epsilon$), reveal that Itô-RMSProp-GRU consistently achieves superior convergence stability and predictive accuracy compared to classical RMSProp. Notably, the optimizer demonstrates remarkable robustness across all tested configurations, maintaining stable performance even under volatile market dynamics. These findings suggest that the synergy between stochastic differential equation frameworks and gated architectures provides a powerful paradigm for financial time series modeling. The paper also presents theoretical justifications and implementation details to facilitate reproducibility and future extensions. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. 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

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

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

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

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result with a solution of stochastic ordinary differential equations. By applying the extended tanh function method, we obtain the soliton solutions for the deterministic counterparts of the graphene sheets model. Because graphene sheets are important in many fields, such as electronics, photonics, and energy storage, the solutions of the stochastic graphene sheets model are beneficial for understanding several fascinating scientific phenomena. Using the MATLAB program, we exhibit several 3D graphs that illustrate the impact of multiplicative noise on the exact solutions of the SGSM. By incorporating stochastic elements into the equations that govern the evolution of graphene sheets, researchers can gain insights into how these fluctuations impact the behavior of the material over time. Full article