Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching

Zhang, Wen; Ye, Jun; Li, Haibo

doi:10.1016/j.spl.2014.04.024

Cited by 8 publications

(3 citation statements)

References 19 publications

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“…Thus, it is very necessary to introduce SDDEs with jumps [4,5]. The theory of SDDEs with jumps has made great progress and has been widely used in various fields [23,15,25,11,32]. A common model is SDDEs driven by Poisson random measure.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

Chang

Chen

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th(<inline-formula><tex-math id="M3">\begin{document}$ p\geq2 $\end{document}</tex-math></inline-formula>) for stochastic delay differential equations with Markovian switching driven by L<inline-formula><tex-math id="M4">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.</p>

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Section: Introductionmentioning

confidence: 99%

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

Chang

Chen

2022

DCDS-B

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“…Tan and Wang [5] investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps. Zhang et al [8] derived some criteria on pth moment stability and almost sure stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching. Li et al [19] constructed compensated stochastic theta methods (CSTM) for stochastic differential delay equations with Poisson jumps, and got the mean square P-stability by some restriction on time stepsize.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

Rouz¹,

Ahmadian²,

Akbarfam³

et al. 2017

Int. J. of Pure and Appl. Math.

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In this paper we consider the Milstein method to investigate the almost sure exponential stability for nonlinear stochastic delay differential equations (SDDEs) with jump diffusion, that is, processes that change only by jumps. The class of jump-diffusion SDDEs that admits explicit solutions is rather limited. In this regard, we employ the continuous and discrete semimartingale convergence theorem. It is shown that the Milstein method reproduces the almost sure exponential stability under an additional linear growth condition on drift and the function L 1 g. In the numerical section we present the results by simulating the method on kind of equations.

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“…Nevertheless, not all SDDEs are stable with exponential decay rate, there also exist a great number of SDDEs are actually stable but obeying a certain lower decay rate except exponential decay rate. We mention here [2,29,34,33,3,43,18,16,30,45] among others. For example, Zhang and Chen [43] and Liu et al [18] discussed the pth moment and almost sure stability with general decay rate of highly nonlinear hybrid SDDEs and hybrid neutral SDDEs (NSDDEs) with time-varying coefficients, respectively.…”

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confidence: 99%

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

Zhang

Gao

2022

DCDS-B

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<p style='text-indent:20px;'>This paper focuses on the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, <inline-formula><tex-math id="M4">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.</p>

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Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching

Cited by 8 publications

References 19 publications

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise

A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

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