On three methods for analytic Laplace inversion in the framework of Brownian motion and their excursions

Schröder, Michael

doi:10.1090/S0033-569X-2013-01324-5

Author: Michael Schröder
Journal: Quart. Appl. Math. 71 (2013), 549-572
MSC (2010): Primary 44A10, 41Axx, 33C15; Secondary 60J65, 91G20
DOI: https://doi.org/10.1090/S0033-569X-2013-01324-5
Published electronically: May 20, 2013
MathSciNet review: 3112828
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Working in a framework originating with Brownian motion and its excursions, this paper establishes a two-step Laplace inversion method for determining a function which is known through its transform after a convolution with another function with a known transform. The first step here has as its domain the class of parabolic cylinder functions, and it develops analytic Laplace inversion of their reciprocals. The second step pertains to convolutions on the positive reals with analytic factors where one of them is of exponential-order decay to zero at the origin; it develops two Laplace-inversion-based methods for handling these by asymptotic expansions. The results are shown to have applications to finance, yielding series representations and asymptotic expansions for the valuation and hedging of Parisian barrier options.

References

J. Azéma and M. Yor, Étude d’une martingale remarquable, Séminaire de Probabilités, XXIII, Lecture Notes in Math., vol. 1372, Springer, Berlin, 1989, pp. 88–130 (French). MR 1022900, DOI https://doi.org/10.1007/BFb0083962
J. Azéma and M. Yor, Sur les zéros des martingales continues, Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526, Springer, Berlin, 1992, pp. 248–306 (French). MR 1231999, DOI https://doi.org/10.1007/BFb0084326
Richard Beals, Advanced mathematical analysis. Periodic functions and distributions, complex analysis, Laplace transform and applications, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, No. 12. MR 0530403
Gustav Doetsch, Handbuch der Laplace-Transformation, Birkhäuser Verlag, Basel-Stuttgart, 1972 (German). Band II: Anwendungen der Laplace-Transformation. 1. Abteilung; Verbesserter Nachdruck der ersten Auflage 1955; Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, Band 15. MR 0344808
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1953 original. MR 698780

W. Gröbner und N. Hofreiter, Integraltafel. Teil I: Unbestimmte Integrale, 3. verb. Aufl., Springer, Wien, 1961.

N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075

Z. Palmowski, I. Czarna, R. Loeffen, Parisian ruin probabilities for spectrally negative Lévy processes,, arXiv:1102.4055v1 (2011).

Michael Schröder, Brownian excursions and Parisian barrier options: a note, J. Appl. Probab. 40 (2003), no. 4, 855–864. MR 2012672, DOI https://doi.org/10.1239/jap/1067436086

Marc Chesney, Monique Jeanblanc-Picqué, and Marc Yor, Brownian excursions and Parisian barrier options, Adv. in Appl. Probab. 29 (1997), no. 1, 165–184. MR 1432935, DOI https://doi.org/10.2307/1427865

A. Zygmund, Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Reprinting of the 1968 version of the second edition with Volumes I and II bound together. MR 0617944