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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]

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]

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]

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]