## Electronic Communications in Probability

### Two-sided exit problem for a Spectrally Negative $\alpha$-Stable Ornstein-Uhlenbeck Process and the Wright's generalized hypergeometric functions

Pierre Patie

#### Abstract

The Laplace transform of the first exit time from a finite interval by a regular spectrally negative $\alpha$-stable Ornstein-Uhlenbeck process is provided in terms of the Wright's generalized hypergeometric function. The Laplace transform of first passage times is also derived for some related processes such as the process killed when it enters the negative half line and the process conditioned to stay positive. The law of the maximum of the associated bridges is computed in terms of the $q$-resolvent density. As a byproduct, we deduce some interesting analytical properties for some Wright's generalized hypergeometric functions.

#### Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 16, 146-160.

Dates
Accepted: 8 May 2007
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465224959

Digital Object Identifier
doi:10.1214/ECP.v12-1265

Mathematical Reviews number (MathSciNet)
MR2318162

Zentralblatt MATH identifier
1128.60031

Rights

#### Citation

Patie, Pierre. Two-sided exit problem for a Spectrally Negative $\alpha$-Stable Ornstein-Uhlenbeck Process and the Wright's generalized hypergeometric functions. Electron. Commun. Probab. 12 (2007), paper no. 16, 146--160. doi:10.1214/ECP.v12-1265. https://projecteuclid.org/euclid.ecp/1465224959

#### References

• Bally, Vlad; Stoica, Lucretiu. A class of Markov processes which admit local times. Ann. Probab. 15 (1987), no. 1, 241–262.
• Bertoin, Jean. An extension of Pitman's theorem for spectrally positive Lévy processes. Ann. Probab. 20 (1992), no. 3, 1464–1483.
• Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0
• Bertoin, Jean. On the first exit time of a completely asymmetric stable process from a finite interval. Bull. London Math. Soc. 28 (1996), no. 5, 514–520.
• Blumenthal, R. M.; Getoor, R. K. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29 Academic Press, New York-London 1968 x+313 pp. (41 #9348)
• Borodin, Andrei N.; Salminen, Paavo. Handbook of Brownian motion—facts and formulae. Second edition. Probability and its Applications. Birkhäuser Verlag, Basel, 2002. xvi+672 pp. ISBN: 3-7643-6705-9
• Doney, R. A. Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. (2) 44 (1991), no. 3, 566–576.
• Doob, J. L. The Brownian movement and stochastic equations. Ann. of Math. (2) 43, (1942). 351–369.
• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. Higher transcendental functions. Vol. III. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. xvii+292 pp.
• W.E. Feller. An Introduction to Probability Theory and its Applications, volume 2. Wiley, New York, 2nd edition, 1971.
• Fitzsimmons, Pat; Pitman, Jim; Yor, Marc. Markovian bridges: construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 101–134, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993.
• Gorenflo, Rudolf; Mainardi, Francesco; Srivastava, Hari M. Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena. Proceedings of the Eighth International Colloquium on Differential Equations (Plovdiv, 1997), 195–202, VSP, Utrecht, 1998.
• Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, 2000. xlvii+1163 pp. ISBN: 0-12-294757-6
• Hadjiev, Dimitar I. The first passage problem for generalized Ornstein-Uhlenbeck processes with nonpositive jumps. Séminaire de probabilités, XIX, 1983/84, 80–90, Lecture Notes in Math., 1123, Springer, Berlin, 1985.
• Kent, John. Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978), no. 5, 760–770.
• Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7
• Lebedev, N. N. Special functions and their applications. Revised edition, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication. Dover Publications, Inc., New York, 1972. xii+308 pp.
• Novikov, A. A. The martingale approach in problems on the time of the first crossing of nonlinear boundaries. (Russian) Analytic number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 158 (1981), 130–152, 230.
• Patie, Pierre. On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance. Stochastic Process. Appl. 115 (2005), no. 4, 593–607.
• Pitman, Jim; Yor, Marc. Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. Itô's stochastic calculus and probability theory, 293–310, Springer, Tokyo, 1996.
• Pitman, Jim; Yor, Marc. On the lengths of excursions of some Markov processes. Séminaire de Probabilités, XXXI, 272–286, Lecture Notes in Math., 1655, Springer, Berlin, 1997.
• Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
• Schneider, W. R. Stable distributions: Fox functions representation and generalization. Stochastic processes in classical and quantum systems (Ascona, 1985), 497–511, Lecture Notes in Phys., 262, Springer, Berlin, 1986.
• Sharpe, Michael J. Supports of convolution semigroups and densities. Probability measures on groups and related structures, XI (Oberwolfach, 1994), 364–369, World Sci. Publ., River Edge, NJ, 1995.
• Shepp, L. A. A first passage problem for the Wiener process. Ann. Math. Statist. 38 1967 1912–1914.
• Shiga, Tokuzo. A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type. Probab. Theory Related Fields 85 (1990), no. 4, 425–447.
• Srivastava, H. M.; Karlsson, Per W. Multiple Gaussian hypergeometric series. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985. 425 pp. ISBN: 0-85312-602-X
• Uchaikin, Vladimir V.; Zolotarev, Vladimir M. Chance and stability. Stable distributions and their applications. With a foreword by V. Yu. Korolev and Zolotarev. Modern Probability and Statistics. VSP, Utrecht, 1999. xxii+570 pp. ISBN: 90-6764-301-7
• Zolotarev, V. M. On the representation of the densities of stable laws by special functions. (Russian) Teor. Veroyatnost. i Primenen. 39 (1994), no. 2, 429–437; translation in Theory Probab. Appl. 39 (1994), no. 2, 354–362 (1995)