215
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Convergence and stability of exponential integrators for semi-linear stochastic variable delay integro-differential equations

Pages 903-932 | Received 01 Nov 2019, Accepted 10 Jun 2020, Published online: 20 Jul 2020

References

  • X. Wang, J. Wu, and B. Dong, Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition, BIT Numer. Math. 65(2) (2020). doi:10.1007/s10543-019-00793-0.
  • J. Arnulf, P.E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, SIAM Press, Philadelphia, 2011.
  • C.T.H. Baker and E. Buckwar, Exponential stability in pth mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math. 184 (2005), pp. 404–427.
  • J. Bao, G. Yin, and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal. 98 (2014), pp. 66–82.
  • A. Andersson, R. Kruse, Mean-square convergence of the BDF2-Maruyama and backward Eulerschemes for SDE satisfying a global monotonicity condition. BIT Numer. Math. 57(1) (2017), pp. 21–53.
  • W. Cao, M. Liu, and Z. Fan, MS-stability of the Euler–Maruyama method for stochastic differential delay equations, Appl. Math. Comput. 159 (2004), pp. 127–135.
  • K. Dekker and J Verwer, Stability of Runge–Kutta methods for stiff nonlinear differential equations, CWI Monographs, 2, North-Holland, 1984.
  • X. Ding, K. Wu, and M. Liu, Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations, Int. J. Comput. Math. 83 (2006), pp. 753–763.
  • W.H. Enright, Continuous numerical methods for ODEs with defect control, J. Comput. Appl. Math.125 (2000), pp. 159–170.
  • U. Erdoğan and G.J. Lord, A new class of exponential integrators for stochastic differential equations with multiplicative noise, IMA J. Numer. Anal. 39(2) (2018), pp. 820–846.
  • A. Friedman, Stochastic Differential Equations and Applications, Academic Press, Cambridge, MA1975.
  • D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001), pp. 525–546.
  • D. Higham, X. Mao, and A. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS J. Comput. Math. 6 (2003), pp. 297–313.
  • M. Hochbruck and C. Lubich, Exponential integrators for large systems of differential equations, SIAM J. SCI. Comput. 19(5) (1998), pp. 1552–1574.
  • M. Hochbruck and A. Ostermann, Exponent. int., Acta Numer. 19 (2010), pp. 209–286.
  • F. Jiang, Y. Shen, and J. Hu, Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching, Commun. Nonlinear Sci. Numer. Simul. 16(2) (2011), pp. 814–821.
  • P.E. Kloeden, G.J. Lord, A. Neuenkirch, and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, J. Comput. Appl. Math. 235 (2011), pp. 1245–1260.
  • X. Mao, Almost sure exponential stability for delay stochastic differential equations with respect to semimartingales. Stoch. Anal.  Appl. 9(2) (1991), pp. 177–194.
  • J.D. Lawson, Generalized Runge–Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal. 4(3) (1967), pp. 372–380.
  • Q. Li and S. Gan, Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations, J. Appl. Math. Comput. 39(1–2) (2012), pp. 69–87.
  • G.J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise, IMA J. Numer. Anal. 33 (2013), pp. 515–543.
  • V.T. Luan and A. Ostermann, Exponential B-series: The stff case, SIAM J. Numer. Anal. 51(6) (2013), pp. 3431–3445.
  • X. Mao, The LaSalle-type theorems for stochastic differential equations, Nonlinear Stud. 7(2) (2000), pp. 307–328.
  • X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl. 268(1) (2002), pp. 125–142.
  • X. Mao, Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations, J. Comput. Appl. Math. 200 (2007), pp. 297–316.
  • X. Mao, Stochastic Differential Equations and Applications. 2nd ed. Horwood Publishing Limited, Chichester, 2008.
  • X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput. 217 (2011), pp. 5512–5524.
  • X. Mao, The truncated Euler Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015a), pp. 370–384.
  • X. Mao, Almost sure exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 53 (2015b), pp. 370–389.
  • X. Mao, Convergence rates of the truncated Euler Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), pp. 362–375.
  • S. Maset and M. Zennaro, Stability properties of explicit exponential Runge–Kutta methods, IMA J. Numer. Anal. 33(1) (2013), pp. 111–135.
  • B. Minchev and W. Wright, A review of exponential integrators for first order semi-linear problems, Tech. Rep. Norwegian University of Science and Technology, 2005.
  • F. Mirzaee and E. Hadadiyan, A collocation technique for solving nonlinear Stochastic It oˆ-Volterra integral equations, Appl. Math. Comput. 247 (2014), pp. 1011–1020.
  • F. Mirzaee and N. Samadyar, Numerical solution of nonlinear Stochastic It oˆ-Volterra integral equations driven by fractional Brownian motion, Math. Method Appl. Sci. 41(4) (2018), pp. 1410–1423.
  • J.D. Mukama and A. Tambue, Optimal strong convergence rates of numerical methods for semilinear parabolic SPDE driven by Gaussian noise and Poisson random measure, Comput. Math. Appl. 77(10) (2019), pp. 2786–2803.
  • A. Rathinasamy and K. Balachandran, Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations, Nonlinear Anal. Hybrid Syst. 2(4) (2008), pp. 1256–1263.
  • Y. Ren, Y. Qin, and R. Sakthivel, Existence results for fractional order semilinear integro-Differential evolution equations with infinite delay, Integr. Equ. Oper. Theory 67 (2010), pp. 33–49.
  • L.E. Shaikhet and J.A. Roberts, Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations, Adv. Differ. Equ. 2016 (2006), pp. 1–23.
  • A. Tambue, An exponential integrator for finite volume discretization of a reaction–advection–diffusion equation, Comput. Math. Appl. 71(9) (2016), pp. 1875–1897.
  • A. Tambue, G.J. Lord, and S. Geiger, An exponential integrator for advection-dominated reactive transport in heterogeneous porous media, J. Comput. Phys. 229(10) (2010), pp. 3957–3969.
  • X. Wang, Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without malliavin calculus, Discr. Cont. Dyn. S. 36(1) (2016), pp. 481–497.
  • X. Wang and R. Qi, A note on an accelerated exponential Euler method for parabolic SPDEs with additive noise, Appl. Math. Lett. 46 (2015), pp. 31–37.
  • F. Wu and X. Mao, Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Appl. Math. Comput. 228 (2014), pp. 240–250.
  • M. Zennaro, Natural continuous extensions of Runge–Kutta methods, Math. Comput. 46 (1986), pp. 119–133.
  • L. Zhang, Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations, J. Inequal. Appl. 2017(249) (2017), pp. 249–268.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.