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Lévy Processes

  • Ernst Eberlein
  • Jan Kallsen
Chapter
Part of the Springer Finance book series (FINANCE)

Abstract

The continuous-time analogue of a random walk is called a Lévy process. One may also view Lévy processes as the stochastic counterpart of linear functions. Both viewpoints illustrate that these processes play a fundamental role.

References

  1. 1.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 (U.S. Government Printing Office, Washington, D.C., 1964)zbMATHGoogle Scholar
  2. 2.
    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge Univ. Press, Cambridge, 2009)CrossRefGoogle Scholar
  3. 9.
    O. Barndorff-Nielsen, Processes of normal inverse Gaussian type. Finance Stochast. 2(1), 41–68 (1998)MathSciNetCrossRefGoogle Scholar
  4. 23.
    J. Bertoin, Lévy Processes (Cambridge Univ. Press, Cambridge, 1996)zbMATHGoogle Scholar
  5. 38.
    S. Boyarchenko, S. Levendorskiı̆, Non-Gaussian Merton-Black-Scholes Theory (World Scientific, River Edge, 2002)Google Scholar
  6. 51.
    P. Carr, H. Geman, D. Madan, M. Yor, The fine structure of asset returns: An empirical investigation. J. Bus. 75(2), 305–332 (2002)CrossRefGoogle Scholar
  7. 60.
    R. Cont, P. Tankov, Financial Modelling with Jump Processes (Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar
  8. 75.
    C. Dellacherie, P.-A. Meyer, Probabilities and Potential (North-Holland, Amsterdam, 1978)zbMATHGoogle Scholar
  9. 76.
    C. Dellacherie, P.-A. Meyer, Probabilities and Potential B: Theory of Martingales (North-Holland, Amsterdam, 1982)zbMATHGoogle Scholar
  10. 82.
    E. Eberlein, U. Keller, Hyperbolic distributions in finance. Bernoulli 1(3), 281–299 (1995)CrossRefGoogle Scholar
  11. 88.
    E. Eberlein, K. Prause, The generalized hyperbolic model: financial derivatives and risk measures, in Mathematical Finance—Bachelier Congress, 2000 (Paris) (Springer, Berlin, 2002), pp. 245–267zbMATHGoogle Scholar
  12. 90.
    E. Eberlein, S. Raible, Some analytic facts on the generalized hyperbolic model, in European Congress of Mathematics, Vol. II (Barcelona, 2000), volume 202 of Progr. Math. (Birkhäuser, Basel, 2001), pp. 367–378CrossRefGoogle Scholar
  13. 91.
    E. Eberlein, E. von Hammerstein, Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes, in Seminar on Stochastic Analysis, Random Fields and Applications IV, vol. 58 (Birkhäuser, Basel, 2004), pp. 221–264zbMATHGoogle Scholar
  14. 152.
    J. Jacod, Calcul Stochastique et Problèmes de Martingales, volume 714 of Lecture Notes in Math (Springer, Berlin, 1979)CrossRefGoogle Scholar
  15. 154.
    J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar
  16. 182.
    J. Kallsen, A. Shiryaev, The cumulant process and Esscher’s change of measure. Finance Stochast. 6(4), 397–428 (2002)MathSciNetCrossRefGoogle Scholar
  17. 186.
    I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)zbMATHGoogle Scholar
  18. 195.
    S. Kou, A jump-diffusion model for option pricing. Manag. Sci. 48(8), 1086–1101 (2002)CrossRefGoogle Scholar
  19. 200.
    U. Küchler, S. Tappe, Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl. 118(2), 261–283 (2008)MathSciNetCrossRefGoogle Scholar
  20. 202.
    A. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd edn. (Springer, Heidelberg, 2014)CrossRefGoogle Scholar
  21. 209.
    E. Luciano, P. Semeraro, A generalized normal mean-variance mixture for return processes in finance. Int. J. Theor. Appl. Finance 13(3), 415–440 (2010)MathSciNetCrossRefGoogle Scholar
  22. 210.
    D. Madan, E. Seneta, The variance gamma (V.G.) model for share market returns. J. Bus. 63(4), 511–524 (1990)CrossRefGoogle Scholar
  23. 220.
    R. Merton, Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1–2), 125–144 (1976)CrossRefGoogle Scholar
  24. 238.
    P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar
  25. 241.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
  26. 256.
    G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, New York, 1994)zbMATHGoogle Scholar
  27. 259.
    K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions (Cambridge Univ. Press, Cambridge, 1999)zbMATHGoogle Scholar
  28. 263.
    W. Schoutens, Lévy Processes in Finance (Wiley, New York, 2003)CrossRefGoogle Scholar
  29. 288.
    E. von Hammerstein, Generalized Hyperbolic Distributions: Theory and Applications to CDO Pricing. PhD thesis, University of Freiburg, 2010Google Scholar
  30. 289.
    E. von Hammerstein, Tail behaviour and tail dependence of generalized hyperbolic distributions, in Advanced Modelling in Mathematical Finance: In Honour of Ernst Eberlein, vol. 189, ed. by J. Kallsen, A. Papapantoleon (Springer, 2016), pp. 3–40Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ernst Eberlein
    • 1
  • Jan Kallsen
    • 2
  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsKiel UniversityKielGermany

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