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doi: 10.3934/dcdsb.2021301
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## Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy 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 Lévy noise. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021301
##### References:
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Math. China, 16 (2021), 395-423.  doi: 10.1007/s11464-021-0914-9. [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. [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. [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. [9] D. J. Higham, X. 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. [10] J. Hull and A. 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Franklin Inst., 350 (2013), 1848-1864.  doi: 10.1016/j.jfranklin.2013.05.009. [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. [16] H. Li, G. Yin and J. Ye, Asymptotic stability of switching diffusions having sub-exponential rates of decay, Dynam. Systems Appl., 22 (2013), 65-94. [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. [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. [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. [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. [21] X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastic Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102. [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. [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. [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. [25] Y. Shen, Q. 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. [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. [27] S. Torkamani, E. Samiei, O. 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. [28] J. Xiong, S. 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. [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. [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. [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. [32] W. Zhang, J. 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. [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. [34] Q. X. Zhu, F. Jiang, H. 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.

show all references

##### References:
 [1] B. Berrhazi, M. E. Fatini, A. Hilbert, N. Mrhardy and R. Petterson, Reflected backward doubly stochastic differential equations with discontinuous barrier, Stochastics, 92 (2020), 1100-1124.  doi: 10.1080/17442508.2019.1691207. [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. [3] H. Chen, P. 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. [4] S. N. Deng, W. Y. Fei, W. 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. [5] S. Gao, J. Hu, T. 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. [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. [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. [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. [9] D. J. Higham, X. 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. [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. [11] F. Jiang, Y. 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. [12] O. I. Klesov, I. 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. [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. [14] B. Li, D. 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. [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. [16] H. Li, G. Yin and J. Ye, Asymptotic stability of switching diffusions having sub-exponential rates of decay, Dynam. Systems Appl., 22 (2013), 65-94. [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. [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. [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. [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. [21] X. Mao and A. Shah, Exponential stability of stochastic differential delay equations, Stochastics Stochastic Rep., 60 (1997), 135-153.  doi: 10.1080/17442509708834102. [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. [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. [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. [25] Y. Shen, Q. 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. [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. [27] S. Torkamani, E. Samiei, O. 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. [28] J. Xiong, S. 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. [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. [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. [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. [32] W. Zhang, J. 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. [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. [34] Q. X. Zhu, F. Jiang, H. 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.
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