Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation

Gan

Applied Numerical Mathematics

2015

Nonlinear stochastic differential delay equations Variable lag θ-Maruyama methods Strong convergence Exponential mean-square stability Mean-square stabilityIn this paper, mean-square convergence and mean-square stability of θ -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 1 2 , and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h > 0. Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ -Maruyama method with θ = 1 can preserve the exponential mean-square stability for any step size. Additionally, the θ -Maruyama method with 1 2 ≤ θ ≤ 1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.

Section: Introductionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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θ-Maruyama methods for nonlinear stochastic differential delay equations

Gan

Applied Numerical Mathematics

2015

“…For SDEs, two very natural concepts are mean-square stability and asymptotic stability. Mean-square stability is more amenable to analyse, and hence this property dominates in the literature, for example, see [11][12][13][14][15]. Asymptotic stability has received some attention in the case of non-jump SDEs [16][17][18][19].…”

Section: Dx(t) = F T X(t) X(t − τ ) Dt + G T X(t) X(t − τ ) Dw (T)mentioning

confidence: 99%

Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

Gan

2010

J. Appl. Math. Comput.

This paper deals with the almost sure exponential stability of the Eulertype methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.

J. Nonlinear Sci. Appl.

“…Abundant achievements have been made in the research of the strong convergence and exponential mean square stability of the one-step schemes. The linear mean square stability of thetaEuler methods and theta-Milstein schemes was investigated in [4,11,19,24] and the strong convergence and exponential mean square stability of nonlinear SDEs with different Lipschitz conditions were well studied in [13, 15-17, 28, 29, 31-33].…”

Section: Introductionmentioning

confidence: 99%

Two-step Maruyama schemes for nonlinear stochastic differential delay equations

Lei¹,

Zong²,

Hu³

2017

This work concerns the two-step Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split two-step Maruyama scheme and linear two-step Maruyama scheme (including Adams-Bashforth and Adams-Moulton schemes) for nonlinear SDDEs with highly nonlinear delay variables, then we investigate the exponential mean square stability and exponential decay rates of the two classes of two-step Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the non-global Lipschitz condition; second, these stability results show that these two-step Maruyama schemes can not only reproduce the exponential mean square stability, but also preserve the bound of Lyapunov exponent for sufficient small stepsize; third, they are also suitable for the corresponding two-step Maruyama methods of stochastic ordinary differential equations (SODEs).