Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

Author: Michael Schröder
Journal: Quart. Appl. Math. 71 (2013), 549-572
MSC (2010): Primary 44A10, 41Axx, 33C15; Secondary 60J65, 91G20
Published electronically: May 20, 2013
MathSciNet review: 3112828
Full-text PDF

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 [Enhancements On Off] (What's this?)

  • 1. 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, 10.1007/BFb0083962
  • 2. 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, 10.1007/BFb0084326
  • 3. 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
  • 4. 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
  • 5. 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
  • 6. 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
  • 7. W. Gröbner und N. Hofreiter, Integraltafel. Teil I: Unbestimmte Integrale, 3. verb. Aufl., Springer, Wien, 1961.
  • 8. 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
  • 9. Z. Palmowski, I. Czarna, R. Loeffen, Parisian ruin probabilities for spectrally negative Lévy processes,, arXiv:1102.4055v1 (2011).
  • 10. Michael Schröder, Brownian excursions and Parisian barrier options: a note, J. Appl. Probab. 40 (2003), no. 4, 855–864. MR 2012672
  • 11. 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, 10.2307/1427865
  • 12. 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

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 44A10, 41Axx, 33C15, 60J65, 91G20

Retrieve articles in all journals with MSC (2010): 44A10, 41Axx, 33C15, 60J65, 91G20

Additional Information

Michael Schröder
Affiliation: Keplerstraße 30, D-69469 Weinheim (Bergstraße), Germany

DOI: https://doi.org/10.1090/S0033-569X-2013-01324-5
Keywords: Brownian motion subject to excursion conditions, constructive methods for handling convolutions, analytic Laplace inversion, Parisian barrier options
Received by editor(s): October 4, 2011
Published electronically: May 20, 2013
Article copyright: © Copyright 2013 Brown University
The copyright for this article reverts to public domain 28 years after publication.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website