doi: 10.3934/dcdsb.2021301
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Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise

Department of Mathematics, Nanchang University, Nanchang 330031, China

* Corresponding author: Huabin Chen

Received  June 2021 Revised  October 2021 Early access December 2021

Fund Project: Huabin Chen is partially supported by the National Natural Science Foundation of China (62163027), the National Social Science Foundation of China (21BTJ028), and the Natural Science Foundation of Jiangxi Province of China (20171BCB23001)

In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L$ \acute{e} $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 $ p $th($ p\geq2 $) for stochastic delay differential equations with Markovian switching driven by L$ \acute{e} $vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.

Citation: Yanqiang Chang, Huabin Chen. Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021301
References:
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F. Mazenc and M. Malisoff, Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay, Automatica, 78 (2017), 1-13.  doi: 10.1016/j.automatica.2016.12.005.  Google Scholar

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[26]

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[27]

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[28]

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[29]

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[30]

C. Yuan and X. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control, 16 (2010), 595-608.  doi: 10.3166/ejc.16.595-608.  Google Scholar

[31]

D. Yue and Q. L. Han, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Trans. Automat. Control, 50 (2005), 217-222.  doi: 10.1109/TAC.2004.841935.  Google Scholar

[32]

W. ZhangJ. Ye and H. B. Li, Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching, Statist. Probab. Lett., 92 (2014), 1-11.  doi: 10.1016/j.spl.2014.04.024.  Google Scholar

[33]

T. Zhang and H. Chen, The stability with a general decay of stochastic delay differential equations with Markovian switching, Appl. Math. Comput., 359 (2019), 294-307.  doi: 10.1016/j.amc.2019.04.057.  Google Scholar

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show all references

References:
[1]

B. BerrhaziM. E. FatiniA. HilbertN. Mrhardy and R. Petterson, Reflected backward doubly stochastic differential equations with discontinuous barrier, Stochastics, 92 (2020), 1100-1124.  doi: 10.1080/17442508.2019.1691207.  Google Scholar

[2]

F. Bozkurt, Nonlinear stochastic differential equations containing generalized delta processes, Monatshefte F$\ddot{u}$r Mathematik, 168 (2012), 75–112. doi: 10.1007/s00605-011-0356-7.  Google Scholar

[3]

H. ChenP. Shi and C. C. Lim, Stability analysis for neutral stochastic delay systems with Markovian switching, Systems Control Lett., 110 (2017), 38-48.  doi: 10.1016/j.sysconle.2017.10.008.  Google Scholar

[4]

S. N. DengW. Y. FeiW. Liu and X. Mao, The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math., 355 (2019), 232-257.  doi: 10.1016/j.cam.2019.01.020.  Google Scholar

[5]

S. GaoJ. HuT. Li and C. G. Yuan, Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps, Front. Math. China, 16 (2021), 395-423.  doi: 10.1007/s11464-021-0914-9.  Google Scholar

[6]

T. E. Govindan, Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications, Probability Theory and Stochastic Modelling, 79. Springer, Cham, 2016. doi: 10.1007/978-3-319-45684-3.  Google Scholar

[7]

N. Halidias, Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 153-160.  doi: 10.3934/dcdsb.2015.20.153.  Google Scholar

[8]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[9]

D. J. HighamX. Mao and L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2083-2100.  doi: 10.3934/dcdsb.2013.18.2083.  Google Scholar

[10]

J. Hull and A. White, The pricing of options on asset with stochastic volitilities, Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.  Google Scholar

[11]

F. JiangY. Shen and L. Liu, Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 798-804.  doi: 10.1016/j.cnsns.2010.04.032.  Google Scholar

[12]

O. I. KlesovI. I. Sirenka and O. A. Tymoshenko, Strong law of large numbers for solutions of non-autonomous stochastic differential equations, Physics and Mathematics, 4 (2017), 61-65.  doi: 10.20535/1810-0546.2017.4.106506.  Google Scholar

[13]

R. Kruse and Y. Wu, A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3475-3502.  doi: 10.3934/dcdsb.2018253.  Google Scholar

[14]

B. LiD. S. Li and D. Y. Xu, Stability analysis for impulsive stochastic delay differential equations with Markovian switching, J. Franklin Inst., 350 (2013), 1848-1864.  doi: 10.1016/j.jfranklin.2013.05.009.  Google Scholar

[15]

G. J. Li and Q. G. Yang, Stability analysis between the hybrid stochastic delay differental equations with jumps and the Euler-Maruyama method, J. Appl. Anal. Comput., 11 (2021), 1259-1272.  doi: 10.11948/20200127.  Google Scholar

[16]

H. LiG. Yin and J. Ye, Asymptotic stability of switching diffusions having sub-exponential rates of decay, Dynam. Systems Appl., 22 (2013), 65-94.   Google Scholar

[17]

M. L. Li and F. Q. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with L$\acute{e}$vy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171-185.  doi: 10.1016/j.nahs.2017.01.001.  Google Scholar

[18]

X. Liu and Z. X. Liu, Poisson stable solutions for stochastic differential equations with L$\acute{e}$vy noise, Acta Mathematica Sinica, 21 (2021), 1-33.  doi: 10.1007/s10114-021-0107-1.  Google Scholar

[19]

Z. Liu and J. Peng, P-moment stability of stochastic nonlinear delay systems with impulsive jump and Markovian switching, Stoch. Anal. Appl., 27 (2009), 911-923.  doi: 10.1080/07362990903136439.  Google Scholar

[20]

O. Maja, Implicit numerical methods for neutral stochastic differential equations with unbounded delay and Markovian switching, Applied Mathematics and Computation, 347 (2019), 664-687.  doi: 10.1016/j.amc.2018.11.037.  Google Scholar

[21]

X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastic Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102.  Google Scholar

[22]

F. Mazenc and M. Malisoff, Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay, Automatica, 78 (2017), 1-13.  doi: 10.1016/j.automatica.2016.12.005.  Google Scholar

[23]

S. T. Rong, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. doi: 10.1007/B106901.  Google Scholar

[24]

L. J. Shen and J. T. Sun, P-th moment exponential stability of stochastic differential equations with impulse effect, Sci. China Inf. Sci., 54 (2011), 1702-1711.  doi: 10.1007/s11432-011-4250-7.  Google Scholar

[25]

Y. ShenQ. X. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021.  Google Scholar

[26]

N. E. Tatar, Fractional Halanay inequality and application in neural network theory, Acta Math. Sci., 39 (2019), 1605-1618.  doi: 10.1007/s10473-019-0611-x.  Google Scholar

[27]

S. TorkamaniE. SamieiO. Bobrenkov and E. A. Butcher, Numerical stability analysis of linear stochastic delay differential equations using Chebyshev spectral continuous time approximation, International Journal of Dynamics and Control, 2 (2014), 210-220.  doi: 10.1007/s40435-014-0082-9.  Google Scholar

[28]

J. XiongS. Q. Zhang and Y. Zhuang, A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance, Math. Control Relat. Fields, 9 (2019), 257-276.  doi: 10.3934/mcrf.2019013.  Google Scholar

[29]

Z. H. Yu, The improved stability analysis of the backward Euler method for neutral stochastic delay differential equations, Int. J. Comput. Math., 90 (2013), 1489-1494.  doi: 10.1080/00207160.2012.756479.  Google Scholar

[30]

C. Yuan and X. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control, 16 (2010), 595-608.  doi: 10.3166/ejc.16.595-608.  Google Scholar

[31]

D. Yue and Q. L. Han, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Trans. Automat. Control, 50 (2005), 217-222.  doi: 10.1109/TAC.2004.841935.  Google Scholar

[32]

W. ZhangJ. Ye and H. B. Li, Stability with general decay rates of stochastic differential delay equations with Poisson jumps and Markovian switching, Statist. Probab. Lett., 92 (2014), 1-11.  doi: 10.1016/j.spl.2014.04.024.  Google Scholar

[33]

T. Zhang and H. Chen, The stability with a general decay of stochastic delay differential equations with Markovian switching, Appl. Math. Comput., 359 (2019), 294-307.  doi: 10.1016/j.amc.2019.04.057.  Google Scholar

[34]

Q. X. ZhuF. JiangH. Wang and B. Wang, Comment on "stability analysis of stochastic differential equations with Markovian switching", Systems Control Lett., 102 (2017), 102-103.  doi: 10.1016/j.sysconle.2017.02.004.  Google Scholar

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