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In this paper, we consider the global asymptotical stability of stochastic functional differential equations with impulsive effects. First, by constructing the Lyapunov function, some stability criteria of impulsive stochastic functional differential equations are established. Second, we propose an application to investigate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results. [ABSTRACT FROM AUTHOR]

Motivated by the error analysis of numerical solution for stochastic delay differential equations, a new result indicating the dependency of convergence rate of the scheme on two separate quantities p and q , where the former is the global order of underlying scheme and the latter is that of interpolation approximating the history of system, is presented. As an objective instance, the stochastic Runge–Kutta (SRK) schemes of global order 0.5, 1 and 1.5, in the strong convergence sense are utilized to formulate three one-step schemes to approximate the solution of the problem. Moreover, the mean-square stability of these schemes will be asymptotically developed. Finally, the theoretical results will be confirmed by some test problems. [ABSTRACT FROM AUTHOR]

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a CAR(∞) representation. Furthermore, we show how the CAR(∞) representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations. [ABSTRACT FROM AUTHOR]

This paper considers a class of neutral-type stochastic functional differential equations with infinite delay (IDNSFDEs) and highly nonlinear coefficients (i.e., coefficients do not satisfy the linear growth condition). These systems are often unstable. Main aim of this paper is to design a delay feedback control to make them become stable with general decay rate. This general decay stability contains the exponential stability and the polynomial stability. Finally, to illustrate our results more clearly, as examples, this paper also introduces unstable scalar neutral-type stochastic integro-differential equations and discusses their exponential and polynomial stabilisation by delay feedback controls, respectively. [ABSTRACT FROM AUTHOR]

Neutral-type's stochastic functional differential equations with infinite delay (NSFDEwID) have received increasing attention in the last decades. Recently, Asker (2021) studied the stability in distribution of NSFDEwID. In this paper, by using the Euler–Maruyama (EM) approximate solution we examine stability in distribution of numerical solution of NSFDEwID. [ABSTRACT FROM AUTHOR]

Stability of the solution to stochastic delay differential equations (SDDEs) have received a great deal of attention, but there is so far little work on stability of numerical solutions to nonlinear stochastic functional differential equations (SFDEs). To close the gap, this paper proposes and studies a numerical scheme called complete backward Euler numerical scheme for general SFDEs. In the paper, we come up with a more general polynomial growth condition with Lyapunov function. Under the generalized polynomial growth condition, the almost sure exponential stability of the underlying continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Besides, the solvability of the implicit scheme is studied specially by introducing the concept of the generalized monotone vectorial functions. To establish the stability criteria for the nonlinear continuous model and the implicit scheme, some necessary lemmas have been established at first. At the end of the paper, a numerical example with simulation is proposed to illustrate the conditions, method and conclusions of the paper. • Model. We consider more general model which is described by the Lyapunov function. • Scheme. The backward Euler numerical scheme is a better approximation than the mixed Euler scheme to general SFDEs. • Methods. We establish some useful lemmas, which are very convenient for our use and can be applied for further investigations. • Results. The stability criterion for it is step size independent. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider exponential stability (ES) for a type of neutral stochastic delay differential equations with delay-dependent impulses (NSDDEWDI). By means of the average dwell time (ADT) condition, a new sufficient criterion for ES is given. The infinitesimal operator of Lyapunov functions is controlled by two time-variant coefficients. Furthermore, these new conditions are calculated through an example so as to verify the validity of our results. [ABSTRACT FROM AUTHOR]

Given an unstable hybrid neutral stochastic differential equation (NSDE), can we design a delay feedback control to make the controlled hybrid NSDE become stable? It has been proved that this is possible under the linear growth condition. However, there is no answer to the question if the drift and diffusion coefficients of the given unstable NSDE satisfy highly nonlinear growth condition. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid NSDEs whose coefficients satisfy the polynomial growth condition. [ABSTRACT FROM AUTHOR]

In this paper we investigate the stability in distribution for a class of stochastic functional differential equations (SFDEs), which include stochastic differential delay equations (SDDEs). Although stability in distribution has been studied by several authors recently, there is so far no stability-in-distribution criterion on SFDEs where the terms involved the delay components are highly nonlinear (not bounded by linear functions). In this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear SFDEs. Two examples will be given to illustrate our new results. We also explain the reason why the existing stability-in-distribution criteria are not applicable by these two examples. [ABSTRACT FROM AUTHOR]