Research article

Stochastic invariance for hybrid stochastic differential equation with non-Lipschitz coefficients

  • Received: 23 December 2019 Accepted: 16 March 2020 Published: 14 April 2020
  • MSC : 60J27, 60H27, 60H28

  • In this paper, by using of the martingale property and positive maximum principle, we investigate the stochastic invariance for a class of hybrid stochastic differential equations (HSDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $\mathbb{R}^d$ with non-Lipschitz coefficients. Moreover, an example of the most probable phase portrait is given to illustrate the effectiveness of the main results.

    Citation: Chunhong Li, Sanxing Liu. Stochastic invariance for hybrid stochastic differential equation with non-Lipschitz coefficients[J]. AIMS Mathematics, 2020, 5(4): 3612-3633. doi: 10.3934/math.2020234

    Related Papers:

  • In this paper, by using of the martingale property and positive maximum principle, we investigate the stochastic invariance for a class of hybrid stochastic differential equations (HSDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $\mathbb{R}^d$ with non-Lipschitz coefficients. Moreover, an example of the most probable phase portrait is given to illustrate the effectiveness of the main results.


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    [1] X. R. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Appl., 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2
    [2] Y. Xu, Z. M. He, P. G. Wang, Pth monent asymptotic stability for neutral stochastic functional diferential equations with Lévy processes, Appl. Math. Comput., 269 (2015), 594-605.
    [3] F. Chen, M. X. Shen, W. Y. Fei, et al. Stability of highly nonlinear hybrid stochastic integrodifferential delay equations, Nonlinear Anal. Hybrid Syst., 31 (2019), 180-199. doi: 10.1016/j.nahs.2018.09.001
    [4] J. W. Luo, K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stoch. Proc. Appl., 118 (2008), 864-895. doi: 10.1016/j.spa.2007.06.009
    [5] A. V. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Providence: American Mathematical Society, 1989.
    [6] H. J. Wu, J. T. Sun, p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching, Automatica, 42 (2006), 1753-1759. doi: 10.1016/j.automatica.2006.05.009
    [7] E. W. Zhu, X. Tian, Y. H. Wang, On pth moment exponential stability of stochastic differential equations with Markovian switching and time-varying delay, J. Inequal. Appl., 1 (2015), 1-11.
    [8] X. R. Mao, C. G. Yuan, Stochastic differential equations with Markovian switching, London: Imperial College Press, 2006.
    [9] N. T. Dieu, Some results on almost sure stability of non-Autonomous stochastic differential equations with Markovian switching, Vietnam J. Math., 44 (2016), 1-13. doi: 10.1007/s10013-016-0187-x
    [10] L. G. Xu, Z. L. Dai, H. X. Hu, Almost sure and moment asymptotic boundedness of stochastic delay differential systems, Appl. Math. Comput., 361 (2019), 157-168. doi: 10.1016/j.cam.2019.04.001
    [11] A. E. Jaber, B. Bouchard, C. Illand, Stochastic invariance of closed sets with non-Lipschitz coefficients, Stoch. Proc. Appl., 129 (2019), 1726-1748. doi: 10.1016/j.spa.2018.06.003
    [12] D. Cao, C. Y. Sun, M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differ. Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020
    [13] D. Li, C. Y. Sun, Q. Q. Chang, Global attractor for degenerate damped hyperbolic equations, J. Math. Anal. Appl., 453 (2017), 1-19. doi: 10.1016/j.jmaa.2017.03.077
    [14] A. Friedman, Stochastic differential equations and applications, New York: Academic Press, 1975.
    [15] J. P. Aubin, G. D. Prato, Stochastic viability and invariance, Ann. Scuola. Norm-Sci., 17 (1990), 595-613.
    [16] Tappe, Stefan, Invariance of closed convex cones for stochastic partial differential equations, J. Math. Anal. Appl., 451 (2017), 1077-1122. doi: 10.1016/j.jmaa.2017.02.044
    [17] I. Chueshov, M. Scheutzow, Invariance and monotonicity for stochastic delay differential equations, Discrete Cont. Dyn-B., 18 (2013), 1533-1554.
    [18] B. Øksendal, Stochastic differential equations: An introduction with applications, 6 Eds., Bei Jing: World Publishing Corporation, 2003.
    [19] D. H. He, L. G. Xu, Boundedness analysis of stochastic integrodifferential systems with Lévy noise, J. Taibah Univ. Sci., 14 (2020), 87-93. doi: 10.1080/16583655.2019.1708540
    [20] S. E. A. Mohammed, Stochastic functional differential equations, Boston: Pitman Advanced Publishing Program, 1984.
    [21] R. Buckdahn, M. Quincampoix, C. Rainer, Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems, B. Sci. Math., 134 (2010), 207-214. doi: 10.1016/j.bulsci.2007.11.003
    [22] B. P. Cheridito, H. M. Soner, N. Touzi, Small time path behavior of double stochastic integrals and applications to stochastic control, Ann. Appl. Probab., 15 (2005), 2472-2495. doi: 10.1214/105051605000000557
    [23] R. T. Rockafellar, J. B. Wets, Variational analysis, New York: Springer, 1998.
    [24] G. T. Kurtz, Lectures on stochastic analysis, 2 Eds., Madison: University of Wisconsin-Madison, 2007.
    [25] S. N. Ethier, T. G. Kurtz, Markov processes: Characterization and convergence, New Jersey: John Wiley and Sons, 1986.
    [26] C. H. Li, J. W. Luo, Stochastic invariance for neutral functional differential equation with nonLipschitz coefficients, Discrete. Cont. Dyn-B., 24 (2019), 3299-3318.
    [27] X. R. Mao, Stochastic defferential equations and application, 2 Eds., Chichester: Woodhead Publishing, 2007.
    [28] F. K. Wu, S. G. Hu, C. M. Huang, Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay, Syst. Control. Lett., 59 (2010), 195-202. doi: 10.1016/j.sysconle.2010.01.004
    [29] C. G. Yuan, J. Lygeros, Stochastic markovian switching hybrid processes, Cambridge: University of Cambridge, 2004.
    [30] L. G. Xu, S. S. Ge, H. X. Hu, Boundedness and stability analysis for impulsive stochastic differential equations driven by G-Brownian motion, Int. J. Control, 92 (2017), 1-16.
    [31] B. Yang, Z. Zeng, L. Wang, Most probable phase portraits of stochastic differential equations and its numerical simulation, arXiv.org, 2017. Available from: https://arxiv.org/abs/1703.06789.
    [32] J. R. Magnus, H. Neudecker, Matrix differential calculus with applications in statistics and econometrics, 3 Eds., New Jersey: Wiley, 2007.
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