Abstract
This paper studies both the non-autonomous stochastic differential equations and stochastic differential delay equations with Markovian switching. A new result on almost sure stability of stochastic differential equations is given. Moreover, we provide new conditions for tightness and almost sure stability of stochastic differential equations.
Similar content being viewed by others
References
Dang, N.H.: A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching. Nonlinear Anal. 95, 625–631 (2014)
Dang, N.H., Du, N.H., Yin, G.: Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise. J. Differ. Equ. 257, 2078–2101 (2014)
Du, N.H., Dang, N.H.: Dynamics of Kolmogorov systems of competitive type under the telegraph noise. J. Differ. Equ. 250, 386–409 (2011)
Du, N.H., Dang, N.H., Dieu, N.T.: On stability in distribution of stochastic differential delay equations with Markovian switching. Syst. Control Lett. 65, 43–49 (2014)
Ji, Y., Chizeck, H.J.: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Autom. Control 35, 777–788 (1990)
Kac, I.Ya: Method of Lyapunov Functions in Problems of Stability and Stabilization of Systems of Stochastic Structure. Ural State Academy of Railway Communication. Ekaterinburg (1998). (in Russian)
Kac, I.Ya., Krasovskii: About stability of systems with stochastic parameters. Prikl. Mat. Mekhanika 24, 809–823 (1960). (in Russian)
Luo, Q., Mao, X.: Stochastic population dynamics under regime switching. J. Math. Anal. Appl. 334, 69–84 (2007)
Luo, J., Zou, J., Hou, Z.: Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching. Sci. China Ser. A 46, 129–138 (2003)
Mao, X., Shaikhet, L.: Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching. Stab. Control Theory Appl. 3, 88–102 (2000)
Mao, X.: Stability of stochastic differential equations with Markovian switching. Stochastic Proc. Appl. 79, 45–67 (1999)
Mao, X.: Asymptotic stability for stochastic differential delay equations with Markovian switching. Funct. Differ. Equ. 9, 201–220 (2002)
Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker, New York (1990)
Rathinasamy, A., Balachandran, K.: Mean square stability of semi-implicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching. Appl. Math. Comput. 206, 968–979 (2008)
Rathinasamy, A., Balachandran, K.: Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations. Nonlinear Anal. Hybrid Syst. 2, 1256–1263 (2008)
Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching. Theory Stoch. Proc. 2(18), 180–185 (1996)
Shaikhet, L.: Numerical simulation and stability of stochastic systems with Markovian switching. Neural Parallel Sci. Comput. 10, 199–208 (2002)
Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Switzerland (2013)
Sethi, S.P., Zhang, Q.: Hierarchical Decision Making in Stochastic Manufacturing Systems . Basel, Birkhäuser (1994)
Sethi, S.P., Zhang, H.-Q., Zhang, Q.: Average-Cost Control of Stochastic Manufacturing Systems. Springer, New York (2005)
Wang, L., Wu, F.: Existence, uniqueness and asymptotic properties of a class of nonlinear stochastic differential delay equations with Markovian switching. Stoch. Dyn. 9, 253–275 (2009)
Yin, G., Dey, S.: Weak convergence of hybrid filtering problems involving nearly completely decomposable hidden Markov chains. SIAM J. Control Optim. 41, 1820–1842 (2003)
Yin, G., Liu, R.H., Zhang, Q.: Recursive algorithms for stock liquidation: a stochastic optimization approach. SIAM J. Optim. 13, 240–263 (2002)
Yuan, C.: Stability in terms of two measures for stochastic differential equations with Markovian switching. Stoch. Anal. Appl. 23, 1259–1276 (2005)
Yuan, C., Lygeros, J.: Stabilization of a class of stochastic differential equations with Markovian switching. Syst. Control Lett. 54, 819–833 (2005)
Yuan, C., Mao, X.: Robust stability and controllability of stochastic differential delay equations with Markovian switching. Automatica 40, 343–354 (2004)
Yuan, C., Mao, X.: Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching. Math. Comput. Simul. 64, 223–235 (2004)
Yuan, C., Zou, J., Mao, X.: Stability in distribution of stochastic differential delay equations with Markovian switching. Syst. Control Lett. 50, 195–207 (2003)
Zhang, Q.: Stock trading: an optimal selling rule. SIAM J. Control Optim. 40, 64–87 (2001)
Zhu, C., Yin, G.: On competitive Lotka–Volterra model in random environments. J. Math. Anal. Appl. 357, 154–170 (2009)
Acknowledgements
Author would like to thank anonymous reviewers for their valuable comments which helped to improve the manuscript. This research was supported in part by the Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training. No. B2015-27-15 and Vietnam National Foundation for Science and Technology Development (NAFOSTED) no. 101.03-2014.58. This work was finished during the author’s postdoctoral fellowship at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He is grateful for the support and hospitality of VIASM.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dieu, N.T. Some Results on Almost Sure Stability of Non-Autonomous Stochastic Differential Equations with Markovian Switching. Vietnam J. Math. 44, 665–677 (2016). https://doi.org/10.1007/s10013-015-0181-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-015-0181-8