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The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including H∞$$ {H}_{\infty } $$ stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of d x a (s) + g (s , x a (s - ω (s))) = I x a (s) + f (s , x a (s - ϱ (s))) d s + ς (s) d ϖ H (s) , 0 ≤ s ≤ T , x a (s) = ζ (s) , - ρ ≤ s ≤ 0. The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. [ABSTRACT FROM AUTHOR]

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay. [ABSTRACT FROM AUTHOR]

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples. [ABSTRACT FROM AUTHOR]

This work addresses existence and stabilization problem for a hybrid neutral stochastic delay differential equations with Lévy noise (HNSDDELN). The coefficients of such systems do not satisfy the conventional linear growth conditions, but are subject to high nonlinearity. We first prove the existence and uniqueness of the solution. We then design a delay feedback controller to make an unstable HNSDDELN |$H_{\infty }$| and asymptotically stable in |${\mathbb{L}}^{p}$|⁠. We end up with a numerical example that corroborates our theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, we consider conformable fractional stochastic differential equations (CFSDEs) driven by fBm with infinite delay via measures of noncompactness (MNC). As far as we know, there are few papers considering this issue. First, by virtue of a Mönch fixed point theorem and MNC, we explore the existence of solutions for CFSDEs. Subsequently, with the aid of Jensen inequality, Hölder inequality, stochastic analysis techniques, and semigroup theory, the controllability for this considered CFSDEs is investigated by employing a Mönch fixed point theorem. Thereafter, the controllability of CFSDEs with nonlocal conditions is discussed. Finally, the theoretical result is supported through an example. [ABSTRACT FROM AUTHOR]

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C ∞ -smooth density. [ABSTRACT FROM AUTHOR]

This article focuses on the delay‐dependent stability of highly nonlinear hybrid neutral stochastic functional differential equations (NSFDEs). The delay dependent stability criteria for a class of highly nonlinear hybrid NSFDEs are derived via the Lyapunov functional. The stabilities discussed in this article include H∞$$ {H}_{\infty } $$ stability, asymptotically stability and exponential stability. A numerical example is given to illustrate the criteria established. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by the control function, the infinite time delay, and the highly nonlinear coefficients of the systems itself, which makes our problem more sophisticated. Different from the classical Khasminskii‐type conditions in the literature, we provide some sufficient conditions for the mean square and almost sure exponential stability for controlled HNSFDEswID by using the M$$ M $$‐matrix technique and a suitable Lyapunov functional. In this sense, the main contributions of our results, compared with those in the literature, are the infinite time delay, the neutral term, and the new Lyapunov functions. Finally, we illustrate our results by a numerical example. [ABSTRACT FROM AUTHOR]

Using the idea of step method, we discassed the existence and uniqueness of solutions of fractional neutral stochastic delay differential equations in the interval [0,τ],[τ,2τ],…,[（n-1）τ,nτ]. Combining Picard iterative method and integral operator theory, we proved that the solutions of fractional neutral stochastic delay differential equations are in existence and unique. The conclusions in existing literature are generalized and improved. [ABSTRACT FROM AUTHOR]

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for \nabla P_{t}f as t\rightarrow \infty are derived. [ABSTRACT FROM AUTHOR]

It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear stochastic differential equations with delay and a high order of nonlinearity can be extended to nonlinear mathematical models of a much more general form. An important feature is the combination of the method of Lyapunov functionals with the method of Linear Matrix Inequalities (LMIs). Some examples of applications of the proposed method of stability research to known mathematical models are given. [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]

In this article, we study the η-stability in qth moment (η.s.q.m) of hybrid neutral stochastic differential equations with infinite delay (HNSDEID) using the Lyapunov techniques and the method of M-matrix. Finally, we apply the main result to some examples. [ABSTRACT FROM AUTHOR]

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales. [ABSTRACT FROM AUTHOR]

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]

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the 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. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out. [ABSTRACT FROM AUTHOR]

Summary: In this article, we study the delay‐dependent stability of a class of hybrid neutral stochastic differential equations (NSDEs) with multiple delays which is highly nonlinear. Some novel stability criteria are investigated according to Lyapunov functions and generalized Itô's formula. In particular, we generalize and improve the results in Shen et al, Systems & Control Letters (2018) from a single delay to multiple delays. [ABSTRACT FROM AUTHOR]

In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo, Y., Xu, C., & Wu, J. [(2017). Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle. International Journal of Control, 90, 1555–1560]. Finally, an example is exhibited to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper studies a class of fractional stochastic delay differential equations driven by a Wiener process. The sufficient conditions for the existence and uniqueness of the solution of fractional stochastic delay differential equations are obtained by means of the Picard iteration method. Moreover, the upper bound of the error using the Picard approximation is estimated. Numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ (1/2, 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result. [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]

Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index H ∈ (1 2 , 1) {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result. [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]

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results. [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]

Abstract: We prove existence and uniqueness of solutions for a class of infinitely delayed stochastic evolution equations with multiplicative noise term where A is the generator of an analytic semigroup on a UMD Banach space E and F and G are functions from the history of the system satisfying Lipschitz conditions. This paper is based on recent work of van Neerven et al., developing the theory of abstract stochastic evolution equations in UMD Banach spaces. [Copyright &y& Elsevier]

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]

The class of deterministic 'Daphnia' models treated by Diekmann et al. (J Math Biol 61:277-318, ) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23:114-135, ) and Diekmann et al. (Nieuw Archief voor Wiskunde 4:82-109, ). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and through their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay. [Copyright &y& Elsevier]

Abstract: The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment -bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example. [Copyright &y& Elsevier]

In this article, we investigate the strong convergence of the Euler-Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established. [ABSTRACT FROM AUTHOR]

We study a semi-discretisation scheme for stochastic optimal control problems whose dynamics are given by controlled stochastic delay (or functional) differential equations with bounded memory. Performance is measured in terms of expected costs. By discretising time in two steps, we construct a sequence of approximating finite-dimensional Markovian optimal control problems in discrete time. The corresponding value functions converge to the value function of the original problem, and we derive an upper bound on the discretisation error or, equivalently, a worst-case estimate for the rate of convergence. [ABSTRACT FROM AUTHOR]

The subject of the present paper is a simplified model for a symmetric bistable system with memory or delay, the reference model, which in the presence of noise exhibits a phenomenon similar to what is known as stochastic resonance. The reference model is given by a one-dimensional parametrized stochastic differential equation with point delay; the basic properties of which we check. With a view to capturing the effective dynamics and, in particular, the resonance-like behavior of the reference model, we construct a simplified or reduced model, the two-state model, first in discrete time, then in the limit of discrete time tending to continuous time. The main advantage of the reduced model is that it enables us to explicitly calculate the distribution of residence times which in turn can be used to characterize the phenomenon of noise-induced resonance. Drawing on what has been proposed in the physics literature, we outline a heuristic method for establishing the link between the two-state model and the reference model. The resonance characteristics developed for the reduced model can thus be applied to the original model. [ABSTRACT FROM AUTHOR]

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kolmanovskii, V.B. and Nosov, V.R., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous-time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and stability. [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]

This paper is motivated by Lan and Yuan (J Comput Appl Math 285:230–242, 2015). The main aim of this paper is to establish certain results for the θ -Euler–Maruyama method for a class of neutral stochastic differential equations with time-dependent delay. The method is defined such that, in general case, it is implicit in both drift and neutral term. The drift and neutral term are both parameterized by θ in a way which guarantees that for θ = 0 and θ = 1 , the method reduces to the Euler–Maruyama method and backward Euler method, respectively, which can be found in the literature. The one-sided Lipschitz conditions in the present-state and delayed arguments of the drift coefficient of this class of equations for any θ ∈ [ 0 , 1 ] are employed in order to guarantee the existence and uniqueness of the appropriate θ -Euler–Maruyama approximate solution. The main result of this paper is almost sure exponential stability of the θ -Euler–Maruyama method, for θ ∈ (1 2 , 1) , under nonlinear growth conditions. Some comments and conclusions are presented for the corresponding deterministic case. An example and numerical simulations are provided to support the main results of the paper. [ABSTRACT FROM AUTHOR]