# American Institute of Mathematical Sciences

February  2019, 24(2): 587-613. doi: 10.3934/dcdsb.2018198

## Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

 1 College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China 2 School of mathematics and information technology, Jiangsu Second Normal University, Nanjing 210013, China 3 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 4 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK

* Corresponding author: Liangjian Hu

Received  May 2017 Revised  January 2018 Published  February 2019 Early access  June 2018

Fund Project: The author Wei Mao is supported by the National Natural Science Foundation of China (11401261) and "333 High-level Personnel Training Project" of Jiangsu Province. The author Liangjian Hu is supported by the National Natural Science Foundation of China (11471071). The author Xuerong Mao is supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1).

In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the $p$th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

Citation: Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198
##### References:
 [1] S. Albeverio, Z. Brzezniak and J. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl., 371 (2010), 309-322.  doi: 10.1016/j.jmaa.2010.05.039. [2] W. J. Anderson, Continuous-Time Markov Chains, Springer, Berlin, 1991. doi: 10.1007/978-1-4612-3038-0. [3] D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge University Press, 2004. doi: 10.1017/CBO9780511755323. [4] D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Levy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692. [5] J. Bao, B. Bottcher, X. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011), 119-131.  doi: 10.1016/j.cam.2011.05.043. [6] M. Baran, Approximations for solutions of Levy-Type Stochastic Differential Equations, Stochastic Analysis and Applications., 27 (2009), 924-961.  doi: 10.1080/07362990903136447. [7] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007), 982-1001.  doi: 10.1016/j.cam.2006.03.040. [8] A. Gardon, The order of approximations for solutions of Ito-type stochastic differential equations with jumps, Stoch. Anal. Appl., 22 (2004), 679-699.  doi: 10.1081/SAP-120030451. [9] D. J. Higham and P. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8. [10] L. Hu, X. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control. Lett., 62 (2013), 178-187.  doi: 10.1016/j.sysconle.2012.11.009. [11] L. Hu, X. Mao and L. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58 (2013), 2319-2332.  doi: 10.1109/TAC.2013.2256014. [12] J. Jakubowski and M. Nieweglowski, Jump-diffusion processes in random environments, J. Differential Equations., 257 (2014), 2671-2703.  doi: 10.1016/j.jde.2014.05.052. [13] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Stijhoff and Noordhoff, Alphen, 1980. [14] H. Kunita, Stochastic diffrential equations based on Lévy processes and stochastic flows of diffomorphisms in Real and Stochastic Analysis, New Perspectives, Berlin, (2004), 305-373. [15] X. Li, X. Mao and Y. Shen, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Difference Equ. Appl., 16 (2010), 195-207.  doi: 10.1080/10236190802695456. [16] L. Liu, Y. Shen and F. Jiang, The almost sure asymptotic stability and pth moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automa. Control., 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970. [17] J. Luo and K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl., 118 (2008), 864-895.  doi: 10.1016/j.spa.2007.06.009. [18] X. Mao, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369.  doi: 10.1006/jmaa.1999.6435. [19] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142.  doi: 10.1006/jmaa.2001.7803. [20] X. Mao and M. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069.  doi: 10.1080/07362990500118637. [21] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London, 2006. doi: 10.1142/p473. [22] X. Mao, Stochastic Differential Equations and their Applications, Horwood, Chichester, 1997. [23] X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.  doi: 10.1016/j.amc.2010.12.023. [24] G. Marion, X. Mao and E. Renshaw, Convergence of the Euler shceme for a class of stochastic Differential Equations, International Mathematical Journal., 1 (2002), 9-22. [25] M. Milosevic, Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation, Appl. Math. Comput., 237 (2014), 672-685.  doi: 10.1016/j.amc.2014.03.132. [26] E. Mordecki, A. Szepessy and R. Tempone, Adaptive weak approximation of diffusions with jumps, SIAM Journal on Numerical Analysis., 46 (2008), 1732-1768.  doi: 10.1137/060669632. [27] B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2005. [28] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [29] V. Popov, Hyperstability of control system, Springer, Berlin, 1973. [30] S. T. Rong, Theory of Stochastic Differential Equations with Jumps and Applications, Springer, Berlin, 2005. [31] M. Song, L. Hu and X. Mao, Khasminskii-Type theorems for stochastic functional differential equations, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1697-1714.  doi: 10.3934/dcdsb.2013.18.1697. [32] I. S. Wee, Stability for multidimensional jump-diffusion processes, Stochastic Process. Appl., 80 (1999), 193-209.  doi: 10.1016/S0304-4149(98)00078-7. [33] F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108. [34] F. Wu and S. Hu, Stochastic suppression and stabilization of delay differential systems, International Journal of Robust and Nonlinear Control., 21 (2011), 488-500.  doi: 10.1002/rnc.1606. [35] F. Wu and S. Hu, The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, 32 (2012), 1065-1094. [36] F. Xi, On the stability of a jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018. [37] F. Xi, Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Process. Appl., 119 (2009), 2198-2221.  doi: 10.1016/j.spa.2008.11.001. [38] F. Xi and G. Yin, Almost sure stability and instability for switching-jump-diffusion systems with state-dependent switching, J. Math. Anal. Appl., 400 (2013), 460-474.  doi: 10.1016/j.jmaa.2012.10.062. [39] Z. Yang and G. Yin, Stability of nonlinear regime-switching jump diffusion, Nonlinear Anal., 75 (2012), 3854-3873.  doi: 10.1016/j.na.2012.02.007. [40] G. Yin and C. Zhu, Hybrid Switching Diffusion: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [41] G. Yin and F. Xi, Stablity of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010), 4525-4549.  doi: 10.1137/080738301. [42] S. You, W. Mao, X. Mao and L. Hu, Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015), 73-83.  doi: 10.1016/j.amc.2015.04.022. [43] C. Yuan and W. Glover, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Comput. Appl. Math., 194 (2006), 207-226.  doi: 10.1016/j.cam.2005.07.004. [44] C. Yuan and J. Bao, On the exponential stability of switching-diffusion processes with jumps, Quart. Appl. Math., 71 (2013), 311-329.  doi: 10.1090/S0033-569X-2012-01292-8. [45] S. Zhou, M. Xue and F. Wu, Robustness of hybrid neutral differential systems perturbed by noise, Journal of Systems Science and Complexity, 27 (2014), 1138-1157.  doi: 10.1007/s11424-014-2037-9. [46] Q. Zhu, Asymptotic stability in the pth moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.

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##### References:
 [1] S. Albeverio, Z. Brzezniak and J. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl., 371 (2010), 309-322.  doi: 10.1016/j.jmaa.2010.05.039. [2] W. J. Anderson, Continuous-Time Markov Chains, Springer, Berlin, 1991. doi: 10.1007/978-1-4612-3038-0. [3] D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge University Press, 2004. doi: 10.1017/CBO9780511755323. [4] D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Levy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692. [5] J. Bao, B. Bottcher, X. Mao and C. Yuan, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011), 119-131.  doi: 10.1016/j.cam.2011.05.043. [6] M. Baran, Approximations for solutions of Levy-Type Stochastic Differential Equations, Stochastic Analysis and Applications., 27 (2009), 924-961.  doi: 10.1080/07362990903136447. [7] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007), 982-1001.  doi: 10.1016/j.cam.2006.03.040. [8] A. Gardon, The order of approximations for solutions of Ito-type stochastic differential equations with jumps, Stoch. Anal. Appl., 22 (2004), 679-699.  doi: 10.1081/SAP-120030451. [9] D. J. Higham and P. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8. [10] L. Hu, X. Mao and Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control. Lett., 62 (2013), 178-187.  doi: 10.1016/j.sysconle.2012.11.009. [11] L. Hu, X. Mao and L. Zhang, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58 (2013), 2319-2332.  doi: 10.1109/TAC.2013.2256014. [12] J. Jakubowski and M. Nieweglowski, Jump-diffusion processes in random environments, J. Differential Equations., 257 (2014), 2671-2703.  doi: 10.1016/j.jde.2014.05.052. [13] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Stijhoff and Noordhoff, Alphen, 1980. [14] H. Kunita, Stochastic diffrential equations based on Lévy processes and stochastic flows of diffomorphisms in Real and Stochastic Analysis, New Perspectives, Berlin, (2004), 305-373. [15] X. Li, X. Mao and Y. Shen, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Difference Equ. Appl., 16 (2010), 195-207.  doi: 10.1080/10236190802695456. [16] L. Liu, Y. Shen and F. Jiang, The almost sure asymptotic stability and pth moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automa. Control., 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970. [17] J. Luo and K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl., 118 (2008), 864-895.  doi: 10.1016/j.spa.2007.06.009. [18] X. Mao, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369.  doi: 10.1006/jmaa.1999.6435. [19] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142.  doi: 10.1006/jmaa.2001.7803. [20] X. Mao and M. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069.  doi: 10.1080/07362990500118637. [21] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London, 2006. doi: 10.1142/p473. [22] X. Mao, Stochastic Differential Equations and their Applications, Horwood, Chichester, 1997. [23] X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.  doi: 10.1016/j.amc.2010.12.023. [24] G. Marion, X. Mao and E. Renshaw, Convergence of the Euler shceme for a class of stochastic Differential Equations, International Mathematical Journal., 1 (2002), 9-22. [25] M. Milosevic, Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation, Appl. Math. Comput., 237 (2014), 672-685.  doi: 10.1016/j.amc.2014.03.132. [26] E. Mordecki, A. Szepessy and R. Tempone, Adaptive weak approximation of diffusions with jumps, SIAM Journal on Numerical Analysis., 46 (2008), 1732-1768.  doi: 10.1137/060669632. [27] B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2005. [28] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [29] V. Popov, Hyperstability of control system, Springer, Berlin, 1973. [30] S. T. Rong, Theory of Stochastic Differential Equations with Jumps and Applications, Springer, Berlin, 2005. [31] M. Song, L. Hu and X. Mao, Khasminskii-Type theorems for stochastic functional differential equations, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 1697-1714.  doi: 10.3934/dcdsb.2013.18.1697. [32] I. S. Wee, Stability for multidimensional jump-diffusion processes, Stochastic Process. Appl., 80 (1999), 193-209.  doi: 10.1016/S0304-4149(98)00078-7. [33] F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control., 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108. [34] F. Wu and S. Hu, Stochastic suppression and stabilization of delay differential systems, International Journal of Robust and Nonlinear Control., 21 (2011), 488-500.  doi: 10.1002/rnc.1606. [35] F. Wu and S. Hu, The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, 32 (2012), 1065-1094. [36] F. Xi, On the stability of a jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008), 588-600.  doi: 10.1016/j.jmaa.2007.10.018. [37] F. Xi, Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Process. Appl., 119 (2009), 2198-2221.  doi: 10.1016/j.spa.2008.11.001. [38] F. Xi and G. Yin, Almost sure stability and instability for switching-jump-diffusion systems with state-dependent switching, J. Math. Anal. Appl., 400 (2013), 460-474.  doi: 10.1016/j.jmaa.2012.10.062. [39] Z. Yang and G. Yin, Stability of nonlinear regime-switching jump diffusion, Nonlinear Anal., 75 (2012), 3854-3873.  doi: 10.1016/j.na.2012.02.007. [40] G. Yin and C. Zhu, Hybrid Switching Diffusion: Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6. [41] G. Yin and F. Xi, Stablity of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010), 4525-4549.  doi: 10.1137/080738301. [42] S. You, W. Mao, X. Mao and L. Hu, Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015), 73-83.  doi: 10.1016/j.amc.2015.04.022. [43] C. Yuan and W. Glover, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Comput. Appl. Math., 194 (2006), 207-226.  doi: 10.1016/j.cam.2005.07.004. [44] C. Yuan and J. Bao, On the exponential stability of switching-diffusion processes with jumps, Quart. Appl. Math., 71 (2013), 311-329.  doi: 10.1090/S0033-569X-2012-01292-8. [45] S. Zhou, M. Xue and F. Wu, Robustness of hybrid neutral differential systems perturbed by noise, Journal of Systems Science and Complexity, 27 (2014), 1138-1157.  doi: 10.1007/s11424-014-2037-9. [46] Q. Zhu, Asymptotic stability in the pth moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.
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