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Approximation and simulation of infinite-dimensional Lévy processes

  • Andrea Barth
  • Andreas SteinEmail author
Article
  • 209 Downloads

Abstract

In this paper approximation methods for infinite-dimensional Lévy processes, also called (time-dependent) Lévy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen–Loève expansion a set of dependent but uncorrelated one-dimensional Lévy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Lévy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, \(L^p\)-convergence rates can be obtained and, under certain regularity assumptions, mean square and \(L^p\)-convergence of the approximated field is proved. Further, a class of (time-dependent) Lévy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.

Keywords

Infinite-dimensional Lévy processes Karhunen–Loève expansion Subordinated processes Stochastic partial differential equations 

References

  1. 1.
    Applebaum, D., Riedle, M.: Cylindrical Lévy processes in Banach spaces. Proc. Lond. Math. Soc. 101, 697–726 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis, vol. 57. Springer, Berlin (2007)zbMATHGoogle Scholar
  3. 3.
    Asmussen, S., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38, 482–493 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 353, 401–419 (1977)CrossRefGoogle Scholar
  5. 5.
    Barndorff-Nielsen, O.E.: Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat. 5, 151–157 (1978)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Barndorff-Nielsen, O.E., Halgreen, C.: Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 38, 309–311 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barndorff-Nielsen, O.E., Kent, J., Sørensen, M.: Normal variance–mean mixtures and z distributions. Int. Stat. Rev/Rev. Int. Stat. 50, 145–159 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I.: Lévy Processes: Theory and Applications. Springer, Berlin (2012)zbMATHGoogle Scholar
  9. 9.
    Barth, A., Benth, F.E.: The forward dynamics in energy markets—infinite-dimensional modelling and simulation. Stoch. Int. J. Probab. Stoch. Process. 86, 932–966 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Barth, A., Lang, A.: Milstein approximation for advection–diffusion equations driven by multiplicative noncontinuous martingale noises. Appl. Math. Optim. 66, 387–413 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Barth, A., Lang, A.: Simulation of stochastic partial differential equations using finite element methods. Stochastics 84, 217–231 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Benth, F.E., Krühner, P.: Integrability of multivariate subordinated Lévy processes in Hilbert space. Stochastics 87, 458–476 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Blæsild, P.: The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s bean data. Biometrika 68, 251–263 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blæsild, P., Jensen, J.L.: Multivariate Distributions of Hyperbolic Type, pp. 45–66. Springer, Dordrecht (1981)zbMATHGoogle Scholar
  15. 15.
    Brzeźniak, Z., Zabczyk, J.: Regularity of Ornstein–Uhlenbeck processes driven by a Lévy white noise. Potential Anal. 32, 153–188 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cherny, A., Shiryaev, A.: Change of Time and Measure for Lévy Processes, Lectures for the Summer School “From Lévy Processes to Semimartingales—Recent Theoretical Developments and Applications to Finance” (2002)Google Scholar
  17. 17.
    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, 2873–2896 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Eberlein, E.: Application of generalized hyperbolic Lévy motions to finance. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes: Theory and Applications, pp. 319–336. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  19. 19.
    Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ernst, O.G., Powell, C.E., Silvester, D.J., Ullmann, E.: Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31, 1424–1447 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fournier, N.: Simulation and approximation of Lévy-driven stochastic differential equations. ESAIM Probab. Stat. 15, 233–248 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gil-Pelaez, J.: Note on the inversion theorem. Biometrika 38, 481–482 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hoyle, E., Hughston, L.P., Macrina, A.: Lévy random bridges and the modelling of financial information. Stoch. Process. Appl. 121, 856–884 (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Hughett, P.: Error bounds for numerical inversion of a probability characteristic function. SIAM J. Numer. Anal. 35, 1368–1392 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002)zbMATHGoogle Scholar
  26. 26.
    Kelker, D.: Infinite divisibility and variance mixtures of the normal distribution. Ann. Math. Stat. 42, 802–808 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kim, K.-K., Kim, S.: Simulation of tempered stable Lévy bridges and its applications. Oper. Res. 64, 495–509 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Maller, R., Szimayer, A.: Finite approximation schemes for Lévy processes, and their application to optimal stopping problems. Stoch. Process. Appl. 117, 1422–1447 (2007)CrossRefzbMATHGoogle Scholar
  29. 29.
    Mansuy, R., Yor, M.: Harnesses, Lévy bridges and Monsieur Jourdain. Stoch. Process. Appl. 115, 329–338 (2005)CrossRefzbMATHGoogle Scholar
  30. 30.
    Masuda, H.: On multidimensional Ornstein–Uhlenbeck processes driven by a general Lévy process. Bernoulli 10, 97–120 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2006)zbMATHGoogle Scholar
  32. 32.
    Øigård, T.A., Hanssen, A., Hansen, R.E.: The multivariate normal inverse Gaussian distribution: EM-estimation and analysis of synthetic aperture sonar data. In: Signal Processing Conference, 2004 12th European, pp. 1433–1436 (2004)Google Scholar
  33. 33.
    Olver, F.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  34. 34.
    Pérez-Abreu, V., Rocha-Arteaga, A.: Covariance-parameter Lévy processes in the space of trace class operators. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 08, 33–54 (2005)CrossRefzbMATHGoogle Scholar
  35. 35.
    Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise, Volume 113 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2007)Google Scholar
  36. 36.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, vol. 2. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  37. 37.
    Rasmussen, C.E., Williams, C.K.: Gaussian Processes for Machine Learning, vol. 2, p. 4. The MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  38. 38.
    Ribeiro, C., Webber, N.: A Monte Carlo Method for the Normal Inverse Gaussian Option Valuation Model Using an Inverse Gaussian Bridge, Preprint. City University (2003)Google Scholar
  39. 39.
    Rydberg, T.H.: The normal inverse Gaussian Lévy process: simulation and approximation. Commun. Stat. Stoch. Models 13, 887–910 (1997). Heavy tails and highly volatile phenomenaGoogle Scholar
  40. 40.
    Sato, K.: Lévy Processes and Infinite Divisibility. Cambridge University Press, Cambridge (1999)Google Scholar
  41. 41.
    Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York (2003)CrossRefGoogle Scholar
  42. 42.
    Veillette, M.S., Taqqu, M.S.: A technique for computing the PDFs and CDFs of nonnegative infinitely divisible random variables. J. Appl. Probab. 48, 217237 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhang, D., Kang, Q.: Pore scale simulation of solute transport in fractured porous media. Geophys. Res. Lett. 31, L12504 (2004)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.SimTechUniversity of StuttgartStuttgartGermany

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