Mean-Square Convergence of Two-Step Milstein Methods for Nonlinear Stochastic Delay Differential Equations

Stochastic differential equations can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. So far there are not many results on the numerical stability of nonlinear neutral stochastic delay differential equations. The purpose of our work is to show that the Euler method applied to the nonlinear neutral stochastic delay differential equations is mean square stable under the condition which guarantees the stability of the analytical solution. The main aim of this paper is to establish new results on the numerical stability. It is proved that the Euler method is mean-square stable under suitable condition, i.e., assume the some conditions are satisfied, then, the Euler method applied to the nonlinear neutral stochastic delay differential equations with initial data is mean-square stable. Moreover, the theoretical result is also verified by a numerical example.

θ-Maruyama methods for nonlinear stochastic differential delay equations

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

In this paper, we investigate the numerical performance of a family ofP-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises. We also test the convergence of one of the schemes for a time-delayed Burgers’ equation with an additive white noise. Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.

In this paper, the strong convergence of numerical solutions of nonlinear hybrid stochastic delay differential equations is investigated. The coefficients of nonlinear hybrid stochastic delay differential equations satisfy the monotone conditions motivated by many finance and biology models. The strong convergence results is obtained by using stochastic θ-Euler Maruyama scheme.

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and $\mathbb {P}-1$ , are examined. Several numerical experiments are carried out to illustrate our results.

Stability of analytical and numerical solutions of nonlinear stochastic delay differential equations

In this paper, we consider stability and ergodicity of nonlinear stochastic delay differential equations with infinite delay on C r . Existing results depend on linear structure or finite delay. Under the local Lipschitz condition and modified dissipative condition, we obtain the stability in distribution of the solution map x t with nonlinear distribution delay, which also admits a unique invariant measure under slightly stronger conditions. We show that the convergence rate to the invariant measure is exponential under the Wasserstein distance.

Summary The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.

This paper investigates the delay-dependent stability of the split-step backward Euler method for nonlinear stochastic delay differential equations. Under a delay-dependent stability condition, in the case of fixed stepsize, it is proved that the split-step backward Euler method can reproduce the mean-square exponential stability of the exact solution under the restriction on the stepsize. Numerical experiments are also provided for demonstration.

Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

Ultimate boundedness and stability of highly nonlinear neutral stochastic delay differential equations with semi-Markovian switching signals

Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results.

Stochastic delayed modeling (stochastic differential equations (SDEs) with delay parameters) has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. Mumps is a viral disease specified with swollen jaws and inflated cheeks. Direct contact with saliva or respiratory drop less from the mouth is the major causes of its outbreak. According to the World Health Organization (WHO), still, 20% of young adult males develop mumps worldwide. No doubt, the vaccination of Mumps exists. The main cause is to study the transmission dynamics of Mumps through stochastic with delay approaches. How is the stochastic delay the best strategy to study the dynamics of disease in a population? For this, we consider the existing deterministic model in literature, with the whole population, divided as susceptible human population S(t), exposed human population E(t), symptomatic infectious I(t), asymptomatic infectious A(t), isolated and treated symptomatic Q(t), recovered humans R(t). After that, we extend the deterministic model into a stochastic delay model (Stochastic delay differential equations (SDDEs) by using the transition probabilities and non-parametric perturbation ways. The positivity, boundedness, extinction, and persistence of disease study with essential properties of reproduction number rigorously. The mump-free equilibrium (MFE) and mumps existing equilibrium (MEE) are two states, local, and global stability of second order and sensitivity analysis of parameters analyzed to verify the model validations. Due to the highly nonlinear stochastic delay differential equations of the model, we used both standard and nonstandard methods such as Euler Maryama, stochastic Euler, stochastic Runge-Kutta, and stochastic nonstandard finite difference with a delayed sense to visualization of results. In the end, the comparison of the methods is presented to support the efficiency of non-standard methods in the sense of stochastic with delay parameters.