### The Time at which a Lévy Process Creeps

**Philip S Griffin**

*(Syracuse University)*

**Ross A Maller**

*(Australian National University)*

#### Abstract

We show that if a Levy process creeps then the renewal function of the bivariate ascending ladder process satisfies certain continuity and differentiability properties. Then a left derivative of the renewal function is shown to be proportional to the distribution function of the time at which the process creeps over a given level, where the constant of proportionality is the reciprocal of the (positive) drift of the ascending ladder height process. This allows us to add the term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As an application, we derive a Laplace transform identity which generalises the second factorization identity. We also relate Doney and Kyprianou's extension of Vigon's equation amicale inversee to creeping. Some results concerning the ladder process, including the second factorization identity, continue to hold for a general bivariate subordinator, and are given in this generality.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2182-2202

Publication Date: November 13, 2011

DOI: 10.1214/EJP.v16-945

#### References

- Alili, L. and Chaumont, L. (2001) A new fluctuation identity for Levy
processes and some applications. Bernoulli 7, 557--569.
Math. Review 2002f:60090
**MR1836746** - Alili, L. and Kyprianou, A.E. (2005)
Some remarks on first passage of Levy processes, the American put and pasting principles,
Ann. Appl. Probab., 15, 2062--2080.
Math. Review 2006b:60078
**MR2152253** - Andrew, P. (2006)
A proof from "first principles" of Kesten's result for the probabilities with which a subordinator hits points,
Electron. Comm. Probab. 11, 58--63.
Math. Review 2007h:60031
**MR2219346** - Bertoin, J. (1996) Levy Processes. Cambridge Univ. Press.
Math. Review 98e:60117
**MR1406564** - Chaumont, L. and Doney, R.A. (2010)
Invariance principles for local times at the supremum of random walks and Levy processes.
Ann. Probab. 38, 1368--1389
Math. Review 2011j:60105
**MR2663630** - Doney, R.A. (2005) Fluctuation Theory for Levy Processes. Lecture Notes in Mathematics 1897, Ecole d'Ete de Probabilites de Saint-Flour XXXV, J. Picard, Ed. Math. Review number not available.
- Doney, R.A. and Kyprianou, A. (2006) Overshoots and undershoots of Levy processes.
Ann. Appl. Probab. 16, 91--106.
Math. Review 2007b:60117
**MR2209337** - Griffin, P.S. and Maller, R.A. (2011a) Path decomposition of ruinous behaviour for a general L evy insurance risk process. Ann. Appl. Probab., to appear. . Math. Review number not available.
- Griffin, P.S. and Maller, R.A. (2011b). Stability of the exit time for Levy processes. Adv. Appl. Probab., 43, 712--734. Math. Review number not available.
- Kesten, H. (1969) Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Math. Soc., 93. Math. Review number not available.
- Kyprianou, A. (2006).
Introductory Lectures on Fluctuations of Levy Processes with Applications.
Springer, Berlin Heidelberg New York.
Math. Review 2008a:60003
**MR2250061** - Percheskii, E.A. and Rogozin, B.A. (1969) On joint distributions of random variables associated with fluctuations of a process with independent increments, Theory Probab. Appl., 14, 410--423. Math. Review number not available.
- Sato, K. (1999). Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. Math. Review number not available.
- Savov, M. and Winkel, M. (2010)
Right inverses of Levy processes: the excursion measure in the general case,
Electron. Comm. Probab. 15, 572–584.
Math. Review 2746335
**MR2746335** - Vigon, V. (2002). Votre Levy rampe-t-il? J. London Math. Soc., 65, 243--256.
Math. Review 2002i:60101
**MR1875147** - Winkel, M. (2005) Electronic foreign exchange markets and passage events of independent subordinators.
J. Appl. Probab. 42, 138--152
Math. Review 2006b:60102
**MR2144899**

This work is licensed under a Creative Commons Attribution 3.0 License.