Representation and Simulation of Ambit Fields

  • Ole E. Barndorff-Nielsen
  • Fred Espen Benth
  • Almut E. D. Veraart
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 88)


In this chapter we present methods for simulating ambit fields based on extending the Fourier approach presented in the temporal case in Chap.  2. In principle the approach is analogous to the simulation of volatility modulated Volterra processes, however, becoming much more technical due to the additional spatial dimension. In this technical chapter, we provide a full-blown theory for simulation using Fourier expansion with proofs. The Fourier method is based on a particular series expansion of the ambit field along a set of basis functions. We expand on this idea and view ambit fields as stochastic processes in a separable Hilbert space, where we establish a series representation of the fields as a countable sum of volatility modulated Volterra processes scaled by basis functions.


  1. Barndorff-Nielsen, O. E. & Basse-O’Connor, A. (2011), ‘Quasi Ornstein–Uhlenbeck Processes’, Bernoulli 17, 916–941.MathSciNetCrossRefGoogle Scholar
  2. Benth, F. E. & Eyjolfsson, H. (2017), ‘Representation and approximation of ambit fields in Hilbert space’, Stochastics 89(1), 311–347.MathSciNetCrossRefGoogle Scholar
  3. Chen, B., Chong, C. & Klüppelberg, C. (2016), Simulation of stochastic Volterra equations driven by space-time Lévy noise, in M. Podolskij, R. Stelzer, S. Thorbjørnsen & A. E. D. Veraart, eds, ‘The Fascination of Probability, Statistics and their Applications’, Springer, pp. 209–229.CrossRefGoogle Scholar
  4. Eyjolfsson, H. (2015), ‘Approximating ambit fields via Fourier methods’, Stochastics 87(5), 885–917.MathSciNetCrossRefGoogle Scholar
  5. Folland, G. B. (1984), Real Analysis – Modern Techniques and their Applications, John Wiley & Sons.zbMATHGoogle Scholar
  6. Peszat, S. & Zabczyk, J. (2007), Stochastic partial differential equations with Lévy noise, Vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge.Google Scholar
  7. Protter, P. E. (2005), Stochastic integration and differential equations, Vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin. Second edition. Version 2.1, Corrected third printing.Google Scholar
  8. Sato, K. (1999), Lévy processes and infinitely divisible distributions, Vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge. Translated from the 1990 Japanese original, Revised by the author.Google Scholar
  9. Yosida, K. (1995), Functional Analysis, Reprint of the 1980 Edition, Springer Verlag Berlin Heidelberg New York.Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Fred Espen Benth
    • 2
  • Almut E. D. Veraart
    • 3
  1. 1.Department of MathematicsUniversity of AarhusAarhusDenmark
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.Department of MathematicsImperial College LondonLondonUK

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