Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

Buckwar, Evelyn; Horváth-Bokor, Rosza; Winkler, Renate

doi:10.1007/s10543-006-0060-5

Cited by 38 publications

(28 citation statements)

References 14 publications

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“…In this work, we consider long time dynamics, and in particular focus on exponential mean-square stability. Our work therefore builds on the well known and highly informative analysis for deterministic problems and its more recent extension to SDEs [2,[4][5][6][7][8]10,11,[14][15][16].…”

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confidence: 99%

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

2007

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Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler-Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.

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confidence: 99%

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

2007

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“…Theorem 4.4 shows that the LSTM scheme can not only share the exponential mean square stability of the exact solution, but also preserve the bound of Lyapunov exponent for sufficient small stepsize. The existed works [5,27] are devoted to the asymptotic mean square stability of the linear two-step Maruyama schemes for the linear SODEs. For nonlinear SODEs (the delay vanishes), Theorem 4.4 further takes the Lyapunov exponent of mean square stability into consideration.…”

Section: Exponential Mean Square Stabilitymentioning

confidence: 99%

“…Stability analysis is another research interest of the multi-step methods. By Lyapunov-type functionals, the asymptotic meansquare stability of the two-step Maruyama methods for linear SODEs was investigated in [5]. Necessary and sufficient conditions in terms of the parameters of the two-step Maruyama schemes guaranteeing their mean square stability were derived for linear SODEs in [27].…”

Section: Introductionmentioning

confidence: 99%

Two-step Maruyama schemes for nonlinear stochastic differential delay equations

Lei¹,

Zong²,

Hu³

2017

J. Nonlinear Sci. Appl.

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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).

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“…where , ∈ R. Mean-square (MS) stability conditions for several numerical methods have been derived (see [8][9][10]). Saito and Mitsui [10] proposed the concept of the MS stability for a numerical methods solving (2).…”

Section: Introductionmentioning

confidence: 99%

Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

Eissa

Zhang

Xiao

2018

Mathematical Problems in Engineering

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The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSS M) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSS M methods is investigated. Furthermore, the stability regions of the DSS M methods are compared with those of test equation, and it is proved that the methods with ≥ 3/2 are stochastically A-stable. Second, the nonlinear stability of DSS M methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with > 1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

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Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

Cited by 38 publications

References 14 publications

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Two-step Maruyama schemes for nonlinear stochastic differential delay equations

Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

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