# A Ciesielski–Taylor type identity for positive self-similar Markov processes

@article{Kyprianou2011ACT,
title={A Ciesielski–Taylor type identity for positive self-similar Markov processes},
author={Andreas E. Kyprianou and P. Patie},
journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
year={2011},
volume={47}
}
• Published 8 August 2009
• Mathematics
• Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Levy processes into itself. Secondly, some classical…
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## References

SHOWING 1-10 OF 24 REFERENCES

### Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

• Mathematics
• 2006
Using Lamperti’s relationship between Levy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law ℙx of a pssMp starting at x>0, in the Skorohod space

### Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of L\'evy processes

We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new

### Recurrent extensions of self-similar Markov processes and Cramér’s condition II

Let P = (Fx, x > 0) be a family of probability measures on Skohorod's space D+, the space of cadlag paths defined on [0, oof with values in R+. The space D+ is endowed with the Skohorod topology and

### Smoothness of scale functions for spectrally negative Lévy processes

• Mathematics
• 2009
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require

### Old and New Examples of Scale Functions for Spectrally Negative Levy Processes

• Mathematics
• 2011
We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of

### Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Nous commencons par caracteriser les fonctions propres croissantes, au sens strict, de la famille d'operateurs integro-differentiels (0.1), pour tout α > 0, γ ≥0, f une function definie sur R + et

### Introductory Lectures on Fluctuations of Lévy Processes with Applications

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance

### First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

• Mathematics
• 1962
also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the