Split-step double balanced approximation methods for stiff stochastic differential equations

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.     

Convergence and stability of impulsive stochastic differential equations

In this paper, we consider impulsive stochastic differential equations. We show that these equations are the exponentially stable in the mean-square sense under Lipschitz conditions. We also construct the numerical method and prove the method is strongly convergent and exponentially stable in the mean-square sense. Moreover, we give some examples in order to illustrate the main results.     

Five-stage Milstein methods for SDEs

In this paper, we consider the problem of computing numerical solutions for Itô stochastic differential equations (SDEs). The five-stage Milstein (FSM) methods are constructed for solving SDEs driven by an m-dimensional Wiener process. The FSM methods are fully explicit methods. It is proved that the FSM methods are convergent with strong order 1 for SDEs driven by an m-dimensional Wiener process. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the methods proposed in this paper are larger than the Milstein method and three-stage Milstein methods.     

Delay-dependent exponential stability of the backward Euler method for nonlinear stochastic delay differential equations

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.     

The improved split-step backward Euler method for stochastic differential delay equations

A new, improved split-step backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.     

Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations

Numerical methods preserving a conserved quantity for stochastic differential equations are considered. A class of discrete gradient methods based on the skew-gradient form is constructed, and the sufficient condition of convergence order 1 in the mean-square sense is given. Then a class of linear projection methods is constructed. The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods. Numerical experiments verify our theory and show the efficiency of proposed numerical methods.     

分布时滞随机偏微分系统的均方指数稳定性

针对一类同时具有分布时滞和维纳过程的随机偏微分系统, 首先基于It?o微分公式, 通过计算弱无穷小算 子, 得到了随机微分导数; 其次利用Green公式和积分不等式及Schur补引理对矩阵不等式进行处理; 然后对微分两 边积分并同时取数学期望处理随机交叉项; 获得了分布时滞随机偏微分系统是均方指数稳定的充分条件. 在此基础 上, 进一步考虑了离散变时滞和分布变时滞在一定约束情形下的分布时滞随机偏微分系统的均方指数稳定性问题. 最后给出仿真实例, 仿真结果表明所获得的线性矩阵不等式条件保证了系统的稳定性, 验证了所得结论的有效性.     

Moment exponential stability analysis of Markovian jump stochastic differential equations with uncertain transition jump rates

This paper is concerned with the moment exponential stability analysis of Markovian jump stochastic differential equations. The equations under consideration are more general, whose transition jump rates matrix Q is not precisely known. Sufficient conditions for testing the stability of such equations are established, and some numerical examples to illustrate the effectiveness of our results are presented.     

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

Moment stability of fractional stochastic evolution equations with Poisson jumps

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of fractional stochastic partial differential equations with Poisson jumps. Sufficient conditions for the existence and asymptotic stability in pth moment of mild solutions are derived by employing the Banach fixed point principle. Further, we extend the result to study the asymptotic stability of fractional systems with Poisson jumps. An example is provided to illustrate the effectiveness of the proposed results.     

Stability of linear stochastic delay differential equations with infinite Markovian switchings

This paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.     

p-th moment exponential stability of stochastic differential equations with impulse effect

The p-th moment exponential stability of stochastic differential equations with impulse effect is addressed.By employing the method of vector Lyapunov functions,some sufficient conditions for the p-th moment exponential stability are established.In addition,the usual restriction of the growth rate of Lyapunov function is replaced by the condition of the drift and diffusion coefficients to study the p-th moment exponential stability.Several examples are also discussed to illustrate the effectiremess of the r...     

Strong convergence of split-step theta methods for non-autonomous stochastic differential equations

In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments.     

The improved stability analysis of the backward Euler method for neutral stochastic delay differential equations

The aim of this paper is to improve some results obtained in our earlier paper [Z. Yu and M. Liu, Almost surely asymptotic stability of numerical solutions for neutral stochastic delay differential equations, Discrete Dyn. Nat. Soc. 2011 (2011), article id 217672, 11 p., doi:10.1155/2011/217672]. In this paper, we establish an improved theorem and show that the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to neutral stochastic delay differential equations. To obtain the desired result, some new proof techniques are adopted.     

Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations

In this paper, we investigate the almost sure and mean square exponential stability of the Euler method and the backward Euler method for neutral stochastic functional differential equations (NSFDEs). Moreover, the almost sure and pth moment exponential stability of exact solutions for NSFDEs are considered. It is shown that the Euler method and the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to NSFDEs under suitable conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.     

Exponential stability for second-order neutral stochastic differential equations with impulses

In this paper, we investigate the problem on the exponential stability of mild solution for the second-order neutral stochastic partial differential equations with impulses by utilising the cosine function theory. A set of novel sufficient conditions is derived by establishing an impulsive integral inequality. As a final point, an example is given to illustrate the effectiveness of the obtained theory.