An analysis of stability of milstein method for stochastic differential equations with delay

Wang, Zhiyong; Zhang, Chengjian

doi:10.1016/j.camwa.2006.01.004

Cited by 50 publications

(18 citation statements)

References 8 publications

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“…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.

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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.

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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.

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“…The convergence and stability of the semi-implicit Euler method for a linear stochastic delay differential equation was derived by Liu et al [3]. The stability of the Milstein method for stochastic delay differential equations was studied by Wang et al [4]. Rathinasamy and Balachandran analysed the mean-square stability of the semi-implicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching in [5].…”

Section: Introductionmentioning

confidence: 99%

-stability of the split-step -methods for linear stochastic delay integro-differential equations

Rathinasamy

Balachandran

2011

Nonlinear Analysis: Hybrid Systems

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“…order 1. Liu et al [13] studied the convergence and stability of the semi-implicit Euler method and Wang and Zhang [18] analyzed the stability of the Milstein method for linear SDDEs. Wang [17] studied the convergence and stability of some numerical methods for nonlinear SDDEs.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Gan

2009

Journal of Computational and Applied Mathematics

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MSC: 60H10 65C20Keywords: Stochastic delay differential equation Split-step backward Euler method Mean-square stability General mean-square stability Finite-time convergence Numerical solution a b s t r a c t In this paper, the numerical approximation of solutions of linear stochastic delay differential equations (SDDEs) in the Itô sense is considered. We construct split-step backward Euler (SSBE) method for solving linear SDDEs and develop the fundamental numerical analysis concerning its strong convergence and mean-square stability. It is proved that the SSBE method is convergent with strong order γ = 1 2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the meansquare stability of the SSBE method.

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An analysis of stability of milstein method for stochastic differential equations with delay

Cited by 50 publications

References 8 publications

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

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

-stability of the split-step -methods for linear stochastic delay integro-differential equations

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

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