## Electronic Communications in Probability

### A convergent series representation for the density of the supremum of a stable process

#### Abstract

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic series which were obtained in Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.

#### Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 8, 84-95.

Dates
Accepted: 23 January 2011
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465261964

Digital Object Identifier
doi:10.1214/ECP.v16-1601

Mathematical Reviews number (MathSciNet)
MR2763530

Zentralblatt MATH identifier
1231.60040

Subjects
Primary: 60G52: Stable processes

Rights

#### Citation

Hubalek, Friedrich; Kuznetsov, Alexey. A convergent series representation for the density of the supremum of a stable process. Electron. Commun. Probab. 16 (2011), paper no. 8, 84--95. doi:10.1214/ECP.v16-1601. https://projecteuclid.org/euclid.ecp/1465261964

#### References

• E.W. Barnes. The genesis of the double gamma function. Proc. London Math. Soc., 31:358–381., 1899.
• E.W. Barnes. The theory of the double gamma function. Phil. Trans. Royal Soc. London (A), 196:265–387., 1901.
• C. Baxa and J. Schoißengeier. Calculation of improper integrals using $(n\alpha)$-sequences. Monatsh. Math., 135(4):265–277, 2002.
• V. Bernyk, R.C. Dalang, and G. Peskir. The law of the supremum of a stable Levy process with no negative jumps. Ann. Probab., 36(5):1777–1789, 2008.
• J. Bertoin. Levy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.
• N.H. Bingham. Fluctuation theory in continuous time. Advances in Appl. Probability, 7(4):705–766, 1975.
• R.A. Doney. On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab., 15(4):1352–1362, 1987.
• R.A. Doney. A note on the supremum of a stable process. Stochastics, 80(2-3):151–155, 2008.
• R.A. Doney and M.S. Savov. The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab., 38(1):316–326, 2010.
• I.S. Gradshteyn and I.M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007.
• G.H. Hardy and J.E. Littlewood. Notes on the theory of series. XXIV. A curious power-series. Proc. Cambridge Philos. Soc., 42:85–90, 1946.
• A.Y. Khinchin. Continued fractions. The University of Chicago Press, Chicago, Ill.-London, 1964.
• A. Kuznetsov. On extrema of stable processes. to appear in Ann. Probab., 2010.
• A.E. Kyprianou. Introductory lectures on fluctuations of Levy processes with applications. Universitext. Springer-Verlag, Berlin, 2006.
• J.C. Oxtoby. Measure and category. A survey of the analogies between topological and measure spaces. Volume 2 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1980.
• P.Patie. A few remarks on the supremum of stable processes. Statist. Probab. Lett., 79(8):1125–1128, 2009.
• V.M. Zolotarev. Mellin-Stieltjes transformations in probability theory. Teor. Veroyatnost. i Primenen., 2:444–469, 1957.
• V.M. Zolotarev. One-dimensional stable distributions. Volume 65 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1986.