×

On three methods for analytic Laplace inversion in the framework of Brownian motion and their excursions. (English) Zbl 1278.44001

The framework, in which the paper (under review) analyzes the problem undertaken, originates with Brownian motion and its properties. It describes three mutually connected issues. The first pertains to an analytic approach, which involves isolation of a function out of a convolution for a known Laplace transform of the convolution and of the complementary factor of the function present. The second that is considered involves the means for the valuation and creates a fence of a class of barrier options (the Parisian barrier options); whereas the third issue is to study the explicit structure of minimal-length excursion of the stochastic process. The functions, denoted by a certain symbol, studied here are defined on the positive real and depend on two complex parameters there in the symbol. The origin of the Laplace transform equation is due to Azémas martingales (see [J. Azéma and M. Yor, Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 88–130 (1989; Zbl 0743.60045), ibid. XXVI, Lect. Notes Math. 1526, 248–306 (1992; Zbl 0765.60038)]) by virtue of Brownian excursions and Parisian options. In Section 3, the endeavour is to connect with the second issue (mentioned above). Complete proofs of the results and applications is given through Sections 4–6.

MSC:

44A10 Laplace transform
60J65 Brownian motion
91G20 Derivative securities (option pricing, hedging, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[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). · Zbl 0743.60045 · doi: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). · Zbl 0765.60038 · doi: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. · Zbl 0294.46029
[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. · Zbl 0065.34001
[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. · Zbl 0055.36401
[6] Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. 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. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1955 original. · Zbl 0058.34103
[7] W. Gröbner und N. Hofreiter, Integraltafel. Teil I: Unbestimmte Integrale, 3. verb. Aufl., Springer, Wien, 1961. · Zbl 0126.33107
[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. · Zbl 0271.33001
[9] Z. Palmowski, I. Czarna, R. Loeffen, Parisian ruin probabilities for spectrally negative Lévy processes,, arXiv:1102.4055v1 (2011). · Zbl 1267.60054
[10] Michael Schröder, Brownian excursions and Parisian barrier options: a note, J. Appl. Probab. 40 (2003), no. 4, 855 – 864. · Zbl 1056.60040
[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. · Zbl 0882.60042 · doi: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. · Zbl 0367.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.