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

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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

Permanent link to this document
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

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60E07: Infinitely divisible distributions; stable distributions

Keywords
Two-sided exit time stable Ornstein-Uhlenbeck process Wright's generalized hypergeometric functions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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


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