Skip to main content


Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations

  • 858 Accesses

  • 10 Citations


We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.



  1. 1.

    Afanas'ev VN, Kolmanovskii VB, Nosov VR: Mathematical Theory of Control Systems Design, Mathematics and Its Applications. Volume 341. Kluwer Academic, Dordrecht; 1996:xxiv+668.

  2. 2.

    Brunner H, Lambert JD: Stability of numerical methods for Volterra integro-differential equations. Computing (Arch. Elektron. Rechnen) 1974,12(1):75–89.

  3. 3.

    Brunner H, van der Houwen PJ: The Numerical Solution of Volterra Equations, CWI Monographs. Volume 3. North-Holland, Amsterdam; 1986:xvi+588.

  4. 4.

    Busenberg S, Cooke KL: The effect of integral conditions in certain equations modelling epidemics and population growth. J. Math. Biol. 1980,10(1):13–32. 10.1007/BF00276393

  5. 5.

    Drozdov A: Explicit stability conditions for stochastic integro-differential equations with non-selfadjoint operator coefficients. Stochastic Anal. Appl. 1999,17(1):23–41. 10.1080/07362999908809586

  6. 6.

    Edwards JT, Ford NJ, Roberts JA: The numerical simulation of the qualitative behaviour of Volterra integro-differential equations. In Proceedings of Algorithms for Approximation IV (Huddersfield, 2001), 2002, Huddersfield. Edited by: Levesley J, Anderson IJ, Mason JC. University of Huddersfield; 86–93.

  7. 7.

    Edwards JT, Ford NJ, Roberts JA, Shakhet LE: Stability of a discrete nonlinear integro-differential equation of convolution type. Stab. Control Theory Appl. 2000,3(1):24–37.

  8. 8.

    Elaydi S, Sivasundaram S: A unified approach to stability in integrodifferential equations via Liapunov functions. J. Math. Anal. Appl. 1989,144(2):503–531. 10.1016/0022-247X(89)90349-1

  9. 9.

    Ford NJ, Baker CTH, Roberts JA: Nonlinear Volterra integro-differential equations—stability and numerical stability of θ -methods. J. Integral Equations Appl. 1998,10(4):397–416. 10.1216/jiea/1181074246

  10. 10.

    Gihman II, Skorokhod AV: Stochastic Differential Equations. Izdat. Naukova Dumka, Kiev; 1968.

  11. 11.

    Golec J, Sathananthan S: Sample path approximation for stochastic integro-differential equations. Stochastic Anal. Appl. 1999,17(4):579–588. 10.1080/07362999908809621

  12. 12.

    Golec J, Sathananthan S: Strong approximations of stochastic integro-differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 2001,8(1):139–151.

  13. 13.

    Higham DJ, Mao XR, Stuart AM: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 2002,40(3):1041–1063. 10.1137/S0036142901389530

  14. 14.

    Higham DJ, Mao XR, Stuart AM: Exponential mean-square stability of numerical solutions to stochastic differential equations. LMS J. Comput. Math. 2003, 6: 297–313.

  15. 15.

    Kloeden PE, Platen E: Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York). Volume 23. Springer, Berlin; 1992:xxxvi+632.

  16. 16.

    Kolmanovskii VB, Myshkis A: Applied Theory of Functional-Differential Equations, Mathematics and Its Applications (Soviet Series). Volume 85. Kluwer Academic, Dordrecht; 1992:xvi+234.

  17. 17.

    Kolmanovskii VB, Shaikhet LE: A method for constructing Lyapunov functionals for stochastic systems with aftereffect. Differ. Uravn. 1993,29(11):1909–1920, 2022. translation in Differential Equations 29 (1993), no. 11, 1657–1666 (1994)

  18. 18.

    Kolmanovskii VB, Shaikhet LE: New results in stability theory for stochastic functional-differential equations (SFDEs) and their applications. In Proceedings of Dynamic Systems and Applications, Vol. 1 (Atlanta, GA, 1993), 1994, Georgia. Dynamic; 167–171.

  19. 19.

    Kolmanovskii VB, Shaikhet LE: A method for constructing Lyapunov functionals for stochastic differential equations of neutral type. Differ. Uravn. 1995,31(11):1851–1857, 1941. translation in Differential Equations 31 (1995), no. 11, 1819–1825 (1996)

  20. 20.

    Kolmanovskii VB, Shaikhet LE: General method of Lyapunov functionals construction for stability investigation of stochastic difference equations. In Dynamical Systems and Applications, World Sci. Ser. Appl. Anal.. Volume 4. World Scientific, New Jersey; 1995:397–439.

  21. 21.

    Kolmanovskii VB, Shaikhet LE: Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Math. Comput. Modelling 2002,36(6):691–716. 10.1016/S0895-7177(02)00168-1

  22. 22.

    Kolmanovskii VB, Shaikhet LE: Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett. 2002,15(3):355–360. 10.1016/S0893-9659(01)00143-4

  23. 23.

    Kolmanovskii VB, Shaikhet LE: About one application of the general method of Lyapunov functionals construction. Internat. J. Robust Nonlinear Control 2003,13(9):805–818. Special issue on Time-Delay Systems, RNC 10.1002/rnc.846

  24. 24.

    Lambert JD: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley & Sons, Chichester; 1991:x+293.

  25. 25.

    Levin JJ, Nohel JA: Note on a nonlinear Volterra equation. Proc. Amer. Math. Soc. 1963, 14: 924–929. 10.1090/S0002-9939-1963-0157201-7

  26. 26.

    Mao XR: Stability of stochastic integro-differential equations. Stochastic Anal. Appl. 2000,18(6):1005–1017. 10.1080/07362990008809708

  27. 27.

    Øksendal B: Stochastic Differential Equations: An Introduction with Applications, Universitext. 5th edition. Springer, Berlin; 1998:xx+324.

  28. 28.

    Saito Y, Mitsui T: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 1996,33(6):2254–2267. 10.1137/S0036142992228409

  29. 29.

    Shaikhet LE: Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. Appl. Math. Lett. 1997,10(3):111–115. 10.1016/S0893-9659(97)00045-1

  30. 30.

    Shaikhet LE: Numerical simulation and stability of stochastic systems with Markovian switching. Neural Parallel Sci. Comput. 2002,10(2):199–208.

  31. 31.

    Shaikhet LE: About Lyapunov functionals construction for difference equations with continuous time. Appl. Math. Lett. 2004,17(8):985–991. 10.1016/j.aml.2003.06.011

  32. 32.

    Shaikhet LE: Construction of Lyapunov functionals for stochastic difference equations with continuous time. Math. Comput. Simulation 2004,66(6):509–521. 10.1016/j.matcom.2004.03.006

  33. 33.

    Shaikhet LE: Lyapunov functionals construction for stochastic difference second-kind Volterra equations with continuous time. Adv. Differ. Equ. 2004,2004(1):67–91. 10.1155/S1687183904308022

Download references

Author information

Correspondence to Leonid E Shaikhet.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shaikhet, L.E., Roberts, J.A. Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations. Adv Differ Equ 2006, 073897 (2006).

Download citation


  • Numerical Experiment
  • Functional Equation
  • Original Problem
  • Asymptotic Stability
  • Stability Property