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Contents of PMS, Vol. 38, Fasc. 1,
pages 27 - 37
 

THE AREA OF A SPECTRALLY POSITIVE STABLE PROCESS STOPPED AT ZERO

Julien Letemplier
Thomas Simon

Abstract: A multiplicative identity in law for the area of a spectrally positive Lévy α -stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.

2010 AMS Mathematics Subject Classification: Primary: 60G52; Secondary: 60E07, 60G51.

Keywords and phrases: Hitting time, integrated process, stable Lévy process, tail asymptotics.

References

[1]   G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge 1999.

[2]   N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge 1987.

[3]   M. Csörgö, Z. Shi, and M. Yor, Some asymptotic properties of the local time of the uniform empirical process, Bernoulli 5 (6) (1999), pp. 1035–1058.

[4]   D. Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. 1 (1990), pp. 39–79.

[5]   P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoret. Comput. Sci. 144 (1995), pp. 3–58.

[6]   S. Janson, Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas, Probab. Surv. 4 (2007), pp. 80–145.

[7]   M. Kanter, Stable densities under change of scale and total variation inequalities, Ann. Probab. 3 (1975), pp. 697–707.

[8]   A. Kuznetsov and J.-C. Pardo, Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes, Acta Appl. Math. 123 (2013), pp. 113–139.

[9]   A. E. Kyprianou and J.-C. Pardo, Continuous state branching processes and self-similarity, J. Appl. Probab. 45 (4) (2008), pp. 1140–1160.

[10]   A. Lachal, Sur le premier instant de passage de l’intégrale du mouvement brownien, Ann. Inst. Henri Poincaré Probab. Stat. 27 (3) (1991), pp. 385–405.

[11]   M. Lefebvre, First-passage densities of a two-dimensional process, SIAM J. Appl. Math. 49 (5) (1989), pp. 1514–1523.

[12]   J. Letemplier and T. Simon, On the law of homogeneous stable functionals, arXiv: 1510.07441.

[13]   J. Pitman and M. Z. Rácz, Beta-gamma tail asymptotics, Electron. Commun. Probab. 20 (2015), paper no. 84.

[14]   C. Profeta and T. Simon, Persistence of integrated stable processes, Probab. Theory Related. Fields 162 (2015), pp. 463–485.

[15]   K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge 1999.

[16]   T. Simon, Hitting densities for spectrally positive stable processes, Stochastics 83 (2) (2011), pp. 203–214.

[17]   T. Simon, Comparing Fréchet and positive stable laws, Electron. J. Probab. 19 (16) (2014), pp. 1–25.

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