Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T11:09:04.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluctuation identities for lévy processes and splitting at the maximum

Published online by Cambridge University Press:  01 July 2016

Priscilla Greenwood  and
Jim Pitman
Priscilla Greenwood*
Affiliation:
University of British Columbia
Jim Pitman*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada.
∗∗Postal address: Department of Statistics, University of California, Berkeley, CA 94720, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.

Keywords

LEVY PROCESSSPLITTING TIMEFLUCTUATION THEORYPOINT PROCESS OF EXCURSIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 893 - 902
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This author's research was partly supported by NSF Grant No. MCS 78-25301.

References

Bingham, N. H. (1975) Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Freedman, D. (1971) Markov Chains. Holden-Day, San Francisco.Google Scholar
Fristedt, B. (1974) Sample functions of stochastic processes with stationary independent increments. Advances in Probability 3, 241–396.Google Scholar
Greenwood, P. and Pitman, J. W. (1979) Construction of local time and Poisson point processes from nested arrays. J. London Math. Soc. 22, 182–192.Google Scholar
Huff, B. W. (1969) The strict subordination of differential processes. Sankya A , 31, 403–412.Google Scholar
Itô, K. (1970) Poisson point processes attached to Markov processes. Proc. 6th Berk. Symp. Math. Statist. Prob. , 225–239.Google Scholar
Maisonneuve, B. (1975) Exit systems. Ann. Prob. 3, 399–411.Google Scholar
Meyer, P. A. (1971a) Processus de Poisson ponctuels, d'apres K. Itô. In Seminaire de Probabilités V , Lecture Notes in Mathematics 191, Springer-Verlag, Berlin, 177–190.Google Scholar
Meyer, P. A. (1971b) Un théorème de répartition des temps locaux. In Seminaire de Probabilités V, Lecture Notes in Mathematics 191, Springer-Verlag, Berlin, 209–210.Google Scholar
Millar, P. W. (1977a) Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc. 226, 365–391.Google Scholar
Millar, P. W. (1977b) Random times and decomposition theorems. Proc. Symp. Pure Math. 31, 91–103.Google Scholar
Pecherskii, E. A. and Rogozin, B. A. (1969) On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. XIV, 410–423.Google Scholar
Rogozin, B. A. (1966) On the distribution of functionals related to boundary problems for processes with independent increments. Theory Prob. Appl. 11, 580–591.Google Scholar
Silverstein, M. L. (1979) Classification of coharmonic and co-invariant functions for a Lévy process. (to appear).Google Scholar
Williams, D. (1974) Path decomposition and continuity of local time for one-dimensional diffusions. Proc. London Math. Soc. 28, 738–768.Google Scholar