Journal of Integral Equations and Applications

Split-step collocation methods for stochastic Volterra integral equations

Y. Xiao, J.N. Shi, and Z.W. Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, a split-step collocation method is proposed for solving linear stochastic Volterra integral equations (SVIEs) with smooth kernels. The H\"older condition and the conditional expectations of the exact solutions are investigated. The solvability and mean-square boundedness of numerical solutions are proved and the strong convergence orders of collocation solutions and iterated collocation solutions are also shown. In addition, numerical experiments are provided to verify the conclusions.

Article information

J. Integral Equations Applications, Volume 30, Number 1 (2018), 197-218.

First available in Project Euclid: 10 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K05: General theory 35B33: Critical exponents

Stochastic Volterra integral equations split-step collocation methods split-step backward Euler method strong convergence order


Xiao, Y.; Shi, J.N.; Yang, Z.W. Split-step collocation methods for stochastic Volterra integral equations. J. Integral Equations Applications 30 (2018), no. 1, 197--218. doi:10.1216/JIE-2018-30-1-197.

Export citation


  • H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge University Press, New York, 2004.
  • L.M. Delves and J.L. Mohamed, Computational methods for integral equations, Cambridge University Press, New York, 1985.
  • X. Ding, Q. Ma and L. Zhang, Convergence and stability of the split-step $\theta$-method for stochastic differential equations, Comp. Math. Appl. 60 (2010), 1310–1321.
  • F.A. Hendi and A.M. Albugami, Numerical solution for Fredholm-Volterra integral equation of the second kind by using collocation and Galerkin methods, J. King Saud Univ. Sci. 22 (2010), 37–40.
  • M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini and C. Cattani, A computational method for solving stochastic Itô-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comp. Phys. 270 (2014), 402–415.
  • 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), 1041–1063.
  • C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra integral equations of the second kind, J. Comp. Appl. Math. 230 (2009), 59–68.
  • K. Ito, On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type, Kodai Math. J. 2 (1979), 158–170.
  • S. Jankovic and D. Ilic, One linear analytic approximation for stochastic integro-differential equations, Acta Math. Sci. 30 (2010), 1073–1085.
  • M. Khodabin, K. Maleknejad, M. Rostami and M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comp. Math. Appl. 64 (2012), 1903–1913.
  • M. Khodabin, K. Maleknejad and F.H. Shekarabi, Application of triangular functions to numerical solution of stochastic Volterra integral equations, Iaeng Int. J. Appl. Math. 43 (2013), 1–9.
  • K. Maleknejad, M. Khodabin and, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comp. Model. 55 (2012), 791–800.
  • ––––, A numerical method for solving $m$-dimensional stochastic Ito-Volterra integral equations by stochastic operational matrix, Comp. Math. Appl. 63 (2012), 133–143.
  • K. Maleknejad, S. Sohrabi and Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. Comp. 188 (2007), 123–128.
  • X. Mao, Stochastic differential equations and their applications, Horwoof Publishing, Chichester, 1997.
  • G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo 4 (1955), 48–90.
  • F. Mirzaee and E. Hadadiyan, A collocation technique for solving nonlinear stochastic Itô-Volterra integral equations, Appl. Math. Comp. 247 (2014), 1011–1020.
  • Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Stat. Prob. Lett. 78 (2008), 1062–1071.
  • Y. Xiao and H.Y. Zhang, A note on convergence of semi-implicit Euler method for stochastic pantograph equations, Comp. Math. Appl. 59 (2010), 1419–1424.
  • X. Zhang, Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Diff. Eqs. 244 (2008), 2226–2250.
  • ––––, Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, Acta J. Funct. Anal. 258 (2010), 1361–1425.