Simulation of Stochastic Volterra Equations Driven by Space–Time Lévy Noise

  • Bohan Chen
  • Carsten Chong
  • Claudia KlüppelbergEmail author


In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space–time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods \(L^p\)- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.


Simulation of SPDEs Simulation of stochastic Volterra equations Space–time Lévy noise Stochastic heat equation Stochastic partial differential equation 



We take pleasure in thanking Jean Jacod for his valuable advice on this subject. The second author acknowledges support from the Studienstiftung des deutschen Volkes and the graduate programme TopMath at Technische Universität München.


  1. 1.
    Asmussen, S., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38(2), 482–493 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.E.D.: Ambit processes and stochastic partial differential equations. In: Nunno, G.D., Øksendal, B. (eds.) Advanced Mathematical Methods for Finance, pp. 35–74. Springer, Berlin (2011)CrossRefGoogle Scholar
  3. 3.
    Barth, A., Lang, A.: Simulation of stochastic partial differential equations using finite element methods. Stochastics 84(2–3), 217–231 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, B.: Simulation of stochastic partial differential equations. Master’s thesis, Technische Universität München (2014).
  5. 5.
    Chong, C.: Lévy-driven Volterra equations in space–time (2014). Preprint under arXiv:1407.8092 [math.PR]
  6. 6.
    Chong, C., Klüppelberg, C.: Integrability conditions for space–time stochastic integrals: theory and applications. Bernoulli 21(4), 2190–2216 (2015)Google Scholar
  7. 7.
    Davie, A.M., Gaines, J.G.: Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comput. 70(233), 121–134 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dunst, T., Hausenblas, E., Prohl, A.: Approximate Euler method for parabolic stochastic partial differential equations driven by space-time Lévy noise. SIAM J. Numer. Anal. 50(6), 2873–2896 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gyöngy, I.: Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal. 11(1), 1–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hausenblas, E.: Finite element approximation of stochastic partial differential equations driven by Poisson random measures of jump type. SIAM J. Numer. Anal. 46(1), 437–471 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hausenblas, E., Marchis, I.: A numerical approximation of parabolic stochastic differential equations driven by a Poisson random measure. BIT Numer. Math. 46(4), 773–811 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Knopp, K.: Theory and Application of Infinite Series. Dover, New York (1990)zbMATHGoogle Scholar
  14. 14.
    Protter, P.: Volterra equations driven by semimartingales. Ann. Probab. 13(2), 519–530 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rajput, B.S., Rosiński, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82(3), 451–487 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rosiński, J.: On path properties of certain infinitely divisible processes. Stoch. Process. Appl. 33(1), 73–87 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rosiński, J.: On series representations of infinitely divisible random vectors. Ann. Probab. 18(1), 405–430 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes, pp. 401–415. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  19. 19.
    Saint Loubert Bié: E.: Étude d’une EDPS conduite par un bruit poissonnien. Probab. Theory Relat. Fields 111(2), 287–321 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) École d’Été de Probabilités de Saint Flour XIV - 1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)CrossRefGoogle Scholar
  21. 21.
    Walsh, J.B.: Finite elment methods for parabolic stochastic PDE’s. Potential Anal. 23(1), 1–43 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, X.: Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J. Differ. Equ. 244(9), 2226–2250 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Bohan Chen
    • 1
  • Carsten Chong
    • 1
  • Claudia Klüppelberg
    • 1
    Email author
  1. 1.Technische Universität MünchenGarchingGermany

Personalised recommendations