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Research Article

Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments

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Pages 2269-2289 | Received 22 Nov 2022, Accepted 12 Oct 2023, Published online: 10 Nov 2023

References

  • G. Adomian and R. Rach, Nonlinear stochastic differential delay equations, J. Math. Anal. Appl. 91(1) (1983), pp. 94–101.
  • E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math. 125(1–2) (2000), pp. 297–307.
  • T.H.B. Christopher and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math. 3 (2000), pp. 315–335.
  • Y. Geng, M. Song, Y. Lu, and M. Liu, Convergence and stability of the truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments, Numer. Math. Theory Methods Appl. 14(1) (2021), pp. 194–218.
  • Q. Guo, W. Liu, X. Mao, and R. Yue, The partially truncated Euler-Maruyama method and its stability and boundedness, Appl. Numer. Math. 115 (2017), pp. 235–251.
  • D.J. Higham, X. Mao, and A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40 (2002), pp. 1041–1063.
  • M. Hutzenthaler, A. Jentzen, and P.E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467(2130) (2011), pp. 1563–1576.
  • M. Hutzenthaler, A. Jentzen, and P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22(4) (2012), pp. 1611–1641.
  • P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlog, Berlin, 1992.
  • V.B. Kolmanovskiĭ and M.A. Anatoliĭ, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1992.
  • X. Li, Existence and exponential stability of solutions for stochastic cellular neural networks with piecewise continuous argument, J. Appl. Math. 2014 (2014), pp. 1–11.
  • Y. Lu, M. Song, and M. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 317 (2017), pp. 55–71.
  • X. Mao, Existence and uniqueness of the solutions of delay stochastic integral equations, Stoch. Anal. Appl. 7(1) (1989), pp. 59–74.
  • X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood, Chichester, UK, 2007.
  • X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC 49(12) (2013), pp. 3677–3681.
  • X. Mao, The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), pp. 370–384.
  • X. Mao, W. Liu, L. Hu, Q. Luo, and J. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Syst. Control Lett. 73 (2014), pp. 88–95.
  • X. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics 85(1) (2013), pp. 144–171.
  • M. Milǒsević, Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay, J. Comput. Appl. Math. 280 (2015), pp. 248–264.
  • M. Milošević, The Euler-Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math. 298 (2016), pp. 1–12.
  • G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin, 2004.
  • S.E.A. Mohammed, Stochastic Functional Differential Equations, Pitman Advanced Publishing Program, Boston, MA, 1984.
  • D.T. Nguyen, S.L. Nguyen, T.A. Hoang, and G. Yin, Tamed-Euler method for hybrid stochastic differential equations with Markovian switching, Nonlinear Anal. Hybrid Syst. 30 (2018), pp. 14–30.
  • M. Obradović and M. Milǒsević, Almost sure exponential stability of the θ-Euler-Maruyama method, when θ∈(12,1), for neutral stochastic differential equations with time-dependent delay under nonlinear growth conditions, Calcolo 56(9) (2019), pp. 1–24.
  • S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18(47) (2013), pp. 1–10.
  • S. Sabanis, Euler approximations with varying coefficients, Ann. Appl. Probab. 26(4) (2016), pp. 2083–2105.
  • M. Song and L. Zhang, Numerical solutions of stochastic differential equations with piecewise continuous arguments under Khasminskii-type conditions, J. Appl. Math. 2012 (2012). Available at doi:10.1155/2012/696849.
  • M. Song, Y. Lu, and M. Liu, Convergence of the tamed Euler method for stochastic differential equations with piecewise continuous arguments under non-global Lipschitz continuous coefficients, Numer. Funct. Anal. Optim. 39(5) (2018), pp. 517–536.
  • X. Wang and S. Gan, The improved split-step backward Euler method for stochastic differential delay equations, Int. J. Comput. Math. 88(11) (2011), pp. 2359–2378.
  • B.S. White, Some limit theorems for stochastic delay-differential equations, Comm. Pure Appl. Math. 29(2) (1976), pp. 113–141.
  • H. Yang, M. Song, M. Liu, and H. Wang, Strong convergence of the tamed Euler method for stochastic differential equations with piecewise continuous arguments and poisson jumps, Filomat 31(12) (2017), pp. 3815–3836.
  • S. You, W. Liu, J. Lu, X. Mao, and Q. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim. 53(2) (2015), pp. 905–925.
  • W. Zhang, The truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments driven by Lévy noise, Int. J. Comput. Math. 98(2) (2021), pp. 389–413.
  • L. Zhang and M. Song, Convergence of the Euler method of stochastic differential equations with piecewise continuous arguments, Abstr. Appl. Anal. 2012 (2012), pp. 1–16.

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