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In the present paper, we developed a new explicit Milstein-type integrator for stiff SDEs. Theoretically, we indicate that the scheme converges to the true value with a strong order of 1.0. For linear scalar SDE, the asymptotic mean square stability of our method is investigated. For all time steps, we prove that the presented integrator is asymptotically mean square stable. In addition, the A-stability and L-stability of the scheme was discussed in the mean square sense. For dimension two of the submitted scheme, the asymptotic mean square stability of two test systems has been analyzed. Numerical simulations confirm the theoretical results.

As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size h > 0 . Numerical examples further illustrate the obtained theoretical results.

In this paper, we carry out a comprehensive stability analysis of discrete time-varying stochastic equations using the Lyapunov direct second method. By constructing an appropriate quadratic Lyapunov function, we successfully apply Lyapunov’s theorems to investigate the stochastic stability of the trivial solution of the system. Our analysis led to significant findings regarding the system’s p-stability, mean-square stability, and stochastic asymptotic stability in the large. The results of this research underscore the effectiveness and versatility of the Lyapunov direct second method in assessing the stability characteristics of complex stochastic systems. In particular, we establish that, under suitable conditions, the trivial solution satisfies key stability properties, namely p-stability, mean-square stability, and stochastic asymptotic stability. These findings are not only theoretically significant, but also have substantial practical relevance. The methods developed in this work are lastly illustrated through comprehensive examples.

This paper addresses the stability of stochastic delayed recurrent neural networks (SDRNNs), identifying challenges in existing scalar methods, which suffer from strong assumptions and limited applicability. Three key innovations are introduced: (1) weakening noise perturbation conditions by extending diagonal matrix assumptions to non-negative definite matrices; (2) establishing criteria for both mean-square exponential stability and almost sure exponential stability in the absence of input; (3) directly handling complex structures like time-varying delays through matrix analysis. Compared with prior studies, this approach yields broader stability conclusions under weaker conditions, with numerical simulations validating the theoretical effectiveness.

This work studies the sliding mode control (SMC) of semi-Markov switching systems (S-MSSs) with time delay in discrete domain. The time delay is first considered in studying discrete S-MSSs with SMC strategy. Owing to their excellent engineering background and advantages in complex system modeling, S-MSSs have a wide range of application prospects. By virtue of the delay-dependent Lyapunov-Krasovskill functional and the probability form of semi-Markov switching signal, a novel convex mean-square stability of the underlying system is provided through a new set of less conservative linear matrix inequalities and suitable semi-Markov conditions. Furthermore, an appropriate delayed SMC mechanism is built to drive the states onto the quasi-sliding mode and remain there for all subsequent time. Finally, an electronic throttle model is presented to validate the availability of the proposed method.

This paper proposes a stochastic linear parameter-varying (LPV) model approach to design a state feedback controller for three-phase, two-level inverters. To deal with the parameter changes, stochastic noise, and delays faced by the inverter, it is modeled as a stochastic LPV system with time delay. Stability analysis and control synthesis are conducted for the LPV system. With parameter-dependent Lyapunov functionals, a condition of sufficient stability for asymptotical mean-square stability is obtained. In addition, the slack matrix technique is employed to improve the feasibility and reduce the conservatism of the conditions. The obtained theoretical results are applied to the three-phase, two-level inverter, whose currents are treated as state variables and are controlled to reach the equilibrium point. The simulation results validate the effectiveness of the proposed theories and demonstrate the advantages of using the slack matrix.

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

This manuscript explores the stability theory of several stochastic/random models. It delves into analyzing the stability of equilibrium states in systems influenced by standard Brownian motion and exhibit random variable coefficients. By constructing appropriate Lyapunov functions, various types of stability are identified, each associated with distinct stability conditions. The manuscript establishes the necessary criteria for asymptotic mean-square stability, stability in probability, and stochastic global exponential stability for the equilibrium points within these models. Building upon this comprehensive stability investigation, the manuscript delves into two distinct fields. Firstly, it examines the dynamics of HIV/AIDS disease persistence, particularly emphasizing the stochastic global exponential stability of the endemic equilibrium point denoted as , where the underlying basic reproductive number is greater than one ( ). Secondly, the paper shifts its focus to finance, deriving sufficient conditions for both the stochastic market model and the random Ornstein–Uhlenbeck model. To enhance the validity of the theoretical findings, a series of numerical examples showcasing stability regions, alongside computer simulations that provide practical insights into the discussed concepts are provided.

Fractional stochastic differential equations with memory effects are fundamental in modeling phenomena across physics, biology, and finance, where long‐range dependencies and random fluctuations coexist, yet their stability analysis under non‐Lipschitz conditions remains a significant challenge, particularly for systems involving Riemann–Liouville fractional operators with stochastic perturbations. To address this challenge, this article establishes sufficient conditions for the existence, uniqueness, and Ulam–Hyers–Rassias (UHR) stability of mild solutions to nonlinear Riemann–Liouville fractional stochastic differential equations (RL‐FSDEs) driven by additive white noise. Through the application of a generalized Banach contraction principle within a carefully constructed weighted metric space, our methodology relaxes the restrictive Lipschitz constraints commonly found in existing literature, thereby significantly broadening the class of admissible nonlinearities. The UHR stability is rigorously established in the mean‐square sense by deriving explicit bounds that systematically relate approximate solutions to exact trajectories. Our theoretical findings are substantiated through four illustrative examples demonstrating that solutions remain consistently close to approximate trajectories under random fluctuations, thereby significantly advancing the stability theory of stochastic fractional systems and providing a robust analytical framework for applications in control theory, financial modeling, and engineering systems governed by memory effects and stochastic noise.

This paper studies the mean square stability and stabilization of linear parabolic stochastic partial difference systems, which contain space–time characteristics and stochastic noise. A definition of the mean square stability for this system is proposed. Using stochastic analysis and some mathematical analysis methods, a strict decreasing sequence is constructed to represent the expectation of the sum of squares of the state variable along the spatial dimension. The sufficient conditions of the mean square stability are established based on system parameters, and then the convergence along the time axis is rigorously proven by the Squeeze criterion. Moreover, some stabilization criteria are derived by designing a linear feedback controller. Finally, two examples are given to illustrate the effectiveness of the results.