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Journal of Theoretical Probability

, Volume 29, Issue 3, pp 776–796 | Cite as

The First Passage Time of a Stable Process Conditioned to Not Overshoot

  • Fernando Cordero
Article

Abstract

Consider a stable Lévy process \(X=(X_t,t\ge 0)\) and let \(T_{x}\), for \(x>0\), denote the first passage time of \(X\) above the level \(x\). In this work, we give an alternative proof of the absolute continuity of the law of \(T_{x}\) and we obtain a new expression for its density function. Our constructive approach provides a new insight into the study of the law of \(T_{x}\). The random variable \(T_{x}^{0}\), defined as the limit of \(T_{x}\) when the corresponding overshoot tends to \(0\), plays an important role in obtaining these results. Moreover, we establish a relation between the random variable \(T_{x}^{0}\) and the dual process conditioned to die at \(0\). This relation allows us to link the expression of the density function of the law of \(T_{x}\) presented in this paper to the already known results on this topic.

Keywords

Lévy processes Stable processes First passage times Absolute continuity 

Mathematics Subject Classification (2010)

Primary 60G52 Secondary 60G51 60G40 

Notes

Acknowledgments

I would like to thank the anonymous referee whose valuable suggestions and comments contributed to the quality of this version of the paper. The author gratefully acknowledges financial support from the European Research Council (ERC) under Grant Agreement No. 247033.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of TechnologyUniversity of BielefeldBielefeldGermany

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