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

The First Passage Time of a Stable Process Conditioned to Not Overshoot Consider a stable Lévy process $$X=(X_t,t\ge 0)$$ X = ( X t , t ≥ 0 ) and let $$T_{x}$$ T x , for $$x>0$$ x > 0 , denote the first passage time of $$X$$ X above the level $$x$$ x . In this work, we give an alternative proof of the absolute continuity of the law of $$T_{x}$$ 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 $$T_{x}$$ T x . The random variable $$T_{x}^{0}$$ T x 0 , defined as the limit of $$T_{x}$$ T x when the corresponding overshoot tends to $$0$$ 0 , plays an important role in obtaining these results. Moreover, we establish a relation between the random variable $$T_{x}^{0}$$ T x 0 and the dual process conditioned to die at $$0$$ 0 . This relation allows us to link the expression of the density function of the law of $$T_{x}$$ T x presented in this paper to the already known results on this topic. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

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

Cordero, Fernando
Journal of Theoretical Probability , Volume 29 (3) – Jan 9, 2015
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References (31)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
DOI
10.1007/s10959-014-0592-6
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Journal of Theoretical Probability – Springer Journals

Published: Jan 9, 2015

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