Stability of the analytic solution and the partially truncated Euler–Maruyama method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

References

• T. Burton, Volterra Integral and Differential Equations, Academic Press, London, 1983.
   Google Scholar
• Q. Guo, W. Liu, X. Mao, and R. Yue, The partially truncated Euler–Maruyama method and its stability and boundedness, Appl. Numer. Math. 115 (2003), pp. 235–251.
   Google Scholar
• T. Hara, T. Yoneyama, and R. Miyazaki, Some refinements of Razumikhin's method and their applications, Funkcial. Ekvac. Ser. Int. 35(2) (1992), pp. 279–305.
   Google Scholar
• 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(3) (2002), pp. 1041–1063.
   Web of Science ®Google Scholar
• P. Hu and C.M. Huang, Stability of Euler–Maruyama method for linear stochastic delay integro-differential equations, Math. Numer. Sin. 32(1) (2010), pp. 105–112.
   Google Scholar
• P. Hu and C.M. Huang, Stability of stochastic θ-methods for stochastic delay integro-differential equations, Int. J. Comput. Math. 88(7) (2011), pp. 1417–1429.
   Web of Science ®Google Scholar
• P. Hu and C.M. Huang, The stochastic Θ-method for nonlinear stochastic Volterra integro-differential equations, Abstr. Appl. Anal. 2014(1) (2014), pp. 137–151.
   Google Scholar
• 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. Roy. Soc. A-Math. Phys. 467(2130) (2011), pp. 1563–1576.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1992.
   Google Scholar
• H. Liang, Z.W. Yang, and J.F. Gao, Strong superconvergence of Euler–Maruyama method for linear stochastic Volterral integral equations, J. Comput. Appl. Math. 317 (2017), pp. 447–457.
   Web of Science ®Google Scholar
• J. Liao, W. Liu, and X. Wang, Truncated Milstein method for non-autonomous stochastic differential equations and its modification, J. Comput. Appl. Math. 402 (2022), pp. 113817.
   Google Scholar
• X. Mao, Stability of stochastic integro-differential equations, Stoch. Anal. Appl. 18(6) (2000), pp. 1005–1017.
   Web of Science ®Google Scholar
• X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood, Chichester (UK), 2007.
   Google Scholar
• X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), pp. 370–384.
   Web of Science ®Google Scholar
• X. Mao, Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), pp. 362–375.
   Web of Science ®Google Scholar
• X. Mao and M. Riedle, Mean square stability of stochastic Volterra integro-differential equations, Syst. Control Lett. 55(6) (2006), pp. 459–465.
   Web of Science ®Google Scholar
• X. Mao, F. Wei, and T. Wiriyakraikul, Positivity preserving truncated Euler–Maruyama method for stochastic Lotka–Volterra competition model, J. Comput. Appl. Math. 394(3) (2021), pp. 113566.
   Google Scholar
• M. Song, L. Hu, X. Mao, and L. Zhang, Khasminskii-type theorems for stochastic functional differential equations, Discrete Contin. Dyn. Syst. Ser. B. 18(6) (2013), pp. 1697–1714.
   Web of Science ®Google Scholar
• R. Subramaniam and K. Balachandran, Existence of solutions of a class of stochastic Volterra integral equations with applications to chemotherapy, J. Aust. Math. Soc. Series B. Appl. Math 41(1) (1999), pp. 93–104.
   Google Scholar
• D. Szynal and S. Wedrychowicz, On solutions of a stochastic integral equation of the Volterra type with applications for chemotherapy, J. Appl. Probab. 25 (1988), pp. 257–267.
   Web of Science ®Google Scholar
• M.V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51(6) (2013), pp. 3135–3162.
   Web of Science ®Google Scholar
• C.P. Tsokos and W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974.
   Google Scholar
• C.H. Wen and T.S. Zhang, Rectangular method on stochastic Volterra equations, Int. J. Appl. Math. Stat. 14(J09) (2009), pp. 12–26.
   Google Scholar
• C.H. Wen and T.S. Zhang, Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math. 235(8) (2011), pp. 2492–2501.
   Web of Science ®Google Scholar
• Q. Wu, L. Hu, and Z. Zhang, Convergence and stability of balanced methods for stochastic delay integro-differential equations, Appl. Math. Comput. 237(11) (2014), pp. 446–460.
   Google Scholar
• W. Zhang, Convergence of the balanced Euler method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients, Appl. Numer. Math.154 (2020), pp. 17–35.
   Web of Science ®Google Scholar
• W. Zhang, Theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients, Appl. Numer. Math. 147 (2020), pp. 254–276.
   Web of Science ®Google Scholar
• W. Zhang, H. Liang, and Jianfang Gao, Theoretical and numerical analysis of the Euler–Maruyama method for generalized stochastic Volterra integro-differential equations, J. Comput. Appl. Math. 365 (2020), pp. 112364.
   Google Scholar
• W. Zhang, M.H. Song, and M.Z. Liu, Strong convergence of the partially truncated Euler–Maruyama method for a class of stochastic differential delay equations, J. Comput. Appl. Math. 335 (2018), pp. 114–128.
   Web of Science ®Google Scholar
• Z.Q. Zhang and H.P. Ma, Order-preserving strong scheme for SDEs with locally Lipschitz coefficients, Appl. Numer. Math. 112 (2017), pp. 1–16.
   Web of Science ®Google Scholar