Exponential stability of numerical solution to neutral stochastic functional differential equation

Zhou, Shan

doi:10.1016/j.amc.2015.05.041

Cited by 16 publications

(9 citation statements)

References 14 publications

(14 reference statements)

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“…It should be highlighted that more attention has been paid to giving better conditions for stability of the analytic solution to the equation and its numerical solution. With the linear growth condition on drift coefficient, the EM numerical solution was shown to reproduce the exponential stability of the analytic solution for neutral stochastic functional (or delay) differential equation 18,19 . Several works showed that the linear growth condition was necessary to ensure the stability of the EM method 17,18,20,21 .…”

Section: Introductionmentioning

confidence: 99%

Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

Liu

Xing

et al. 2020

Math Methods in App Sciences

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The aim of this work is to investigate the exponential mean‐square stability for neutral stochastic differential equations with time‐varying delay and Poisson jumps. When all the drift, diffusion, and jumps coefficients are allowed to be nonlinear, the exponential mean‐square stability of the analytic solution to the equation is obtained. It is revealed that the implicit backward Euler–Maruyama numerical solution can reproduce the corresponding stability of the analytic solution under some given nonlinear conditions. It is different from the explicit Euler–Maruyama numerical solution whose stability depends on the linear growth condition. With some requirements related to the delayed function and the property of compensated Poisson process, we deal with time‐varying delay and Poisson jumps. One highly nonlinear example is given to confirm the effectiveness of our theory.

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Section: Introductionmentioning

confidence: 99%

Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

Liu

Xing

et al. 2020

Math Methods in App Sciences

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“…The theoretical results obtained have been analyzed in detail and some sufficient conditions have been presented. Some existing results, for example [8,21,[23][24][25][26][27][28], have been generalized. Meanwhile, the strong convergence for the theoretical solution and the numerical solution of such equations has been considered.…”

Section: Resultsmentioning

confidence: 99%

“…Recently, the exponential stability of the EM method for NSFDEs with jumps was analyzed in [16]. The almost surely and mean square exponential stability of numerical solutions for NSFDEs were considered in [23,27]. In [28], Zong et al analyzed the mean square exponential stability of the numerical solutions for NSFDEs.…”

Section: Introductionmentioning

confidence: 99%

“…Most of these papers discussed in [16,23,27,28] are related with the asymptotic stability in mean square or in probability, which means that the numerical solution will tend to zero in mean square or in probability as time t approaches infinity. Nevertheless, the asymptotic behavior sometimes is too strong, and under these circumstances to know if the numerical solution will converge in distribution or not is very useful.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solutions of neutral stochastic functional differential equations with Markovian switching

Chen

2019

Adv Differ Equ

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Until now, the theories about the convergence analysis, the almost surely and mean square exponential stability of the numerical solution for neutral stochastic functional differential equations with Markovian switching (NSFDEwMSs) have been well established, but there are very few research works concentrating on the stability in distribution of numerical solution. This paper will pay attention to the stability in distribution of numerical solution of NSFDEwMSs. The strong mean square convergence analysis is also discussed.

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“…Increasingly real-world systems are modeled by SFDEs of neutral type, as they represent systems which evolve in a random environment and whose evolution depends on the past states and derivatives of states of the systems through either memory or time delay. In the last decade, for SFDEs of neutral type, there are a large number of papers on, e.g., stochastic stability (see, e.g., [12,13,26]), on large fluctuations (see, e.g., [1]), on large deviation principle (see, e.g., [6]), on transportation inequality (see, e.g., [3]), to name a few.…”

Section: Introductionmentioning

confidence: 99%

Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

2021

J Inequal Appl

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In this paper, we are concerned with the convergence rate of Euler–Maruyama (EM) scheme for stochastic differential delay equations (SDDEs) of neutral type, where the neutral, drift, and diffusion terms are allowed to be of polynomial growth. More precisely, for SDDEs of neutral type driven by Brownian motions, we reveal that the convergence rate of the corresponding EM scheme is one-half; Whereas for SDDEs of neutral type driven by pure jump processes, we show that the best convergence rate of the associated EM scheme is slower than one-half. As a result, the convergence rate of general SDDEs of neutral type, which is dominated by pure jump process, is slower than one-half.

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Exponential stability of numerical solution to neutral stochastic functional differential equation

Cited by 16 publications

References 14 publications

Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

Numerical solutions of neutral stochastic functional differential equations with Markovian switching

Convergence rate of Euler–Maruyama scheme for SDDEs of neutral type

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