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# Lévy Processes

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## 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.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.D. Applebaum,
*Lévy Processes and Stochastic Calculus*, 2nd edn. (Cambridge Univ. Press, Cambridge, 2009)CrossRefGoogle Scholar - 9.O. Barndorff-Nielsen, Processes of normal inverse Gaussian type. Finance Stochast.
**2**(1), 41–68 (1998)MathSciNetCrossRefGoogle Scholar - 23.J. Bertoin,
*Lévy Processes*(Cambridge Univ. Press, Cambridge, 1996)zbMATHGoogle Scholar - 38.S. Boyarchenko, S. Levendorskiı̆,
*Non-Gaussian Merton-Black-Scholes Theory*(World Scientific, River Edge, 2002)Google Scholar - 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 - 60.R. Cont, P. Tankov,
*Financial Modelling with Jump Processes*(Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar - 75.C. Dellacherie, P.-A. Meyer,
*Probabilities and Potential*(North-Holland, Amsterdam, 1978)zbMATHGoogle Scholar - 76.C. Dellacherie, P.-A. Meyer,
*Probabilities and Potential B: Theory of Martingales*(North-Holland, Amsterdam, 1982)zbMATHGoogle Scholar - 82.E. Eberlein, U. Keller, Hyperbolic distributions in finance. Bernoulli
**1**(3), 281–299 (1995)CrossRefGoogle Scholar - 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 - 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 - 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 - 152.J. Jacod,
*Calcul Stochastique et Problèmes de Martingales*, volume 714 of*Lecture Notes in Math*(Springer, Berlin, 1979)CrossRefGoogle Scholar - 154.J. Jacod, A. Shiryaev,
*Limit Theorems for Stochastic Processes*, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar - 182.J. Kallsen, A. Shiryaev, The cumulant process and Esscher’s change of measure. Finance Stochast.
**6**(4), 397–428 (2002)MathSciNetCrossRefGoogle Scholar - 186.I. Karatzas, S. Shreve,
*Brownian Motion and Stochastic Calculus*, 2nd edn. (Springer, New York, 1991)zbMATHGoogle Scholar - 195.S. Kou, A jump-diffusion model for option pricing. Manag. Sci.
**48**(8), 1086–1101 (2002)CrossRefGoogle Scholar - 200.U. Küchler, S. Tappe, Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl.
**118**(2), 261–283 (2008)MathSciNetCrossRefGoogle Scholar - 202.A. Kyprianou,
*Fluctuations of Lévy Processes with Applications*, 2nd edn. (Springer, Heidelberg, 2014)CrossRefGoogle Scholar - 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 - 210.D. Madan, E. Seneta, The variance gamma (V.G.) model for share market returns. J. Bus.
**63**(4), 511–524 (1990)CrossRefGoogle Scholar - 220.R. Merton, Option pricing when underlying stock returns are discontinuous. J. Financ. Econ.
**3**(1–2), 125–144 (1976)CrossRefGoogle Scholar - 238.P. Protter,
*Stochastic Integration and Differential Equations*, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar - 241.D. Revuz, M. Yor,
*Continuous Martingales and Brownian Motion*, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar - 256.G. Samorodnitsky, M. Taqqu,
*Stable Non-Gaussian Random Processes*(Chapman & Hall, New York, 1994)zbMATHGoogle Scholar - 259.K.-I. Sato,
*Lévy Processes and Infinitely Divisible Distributions*(Cambridge Univ. Press, Cambridge, 1999)zbMATHGoogle Scholar - 263.W. Schoutens,
*Lévy Processes in Finance*(Wiley, New York, 2003)CrossRefGoogle Scholar - 288.E. von Hammerstein,
*Generalized Hyperbolic Distributions: Theory and Applications to CDO Pricing*. PhD thesis, University of Freiburg, 2010Google Scholar - 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

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