Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T10:53:50.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the absolute maximum of a Lévy process

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Mykola Bratiychuk
Mykola Bratiychuk*
Affiliation:
Silesian Technical University
*
∗ Postal address: Instytut Matematyki, Kaszubska st. 23, 44-100 Gliwice, Poland. Email address: bratiich@zeus.polsl.gliwice.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

Keywords

Lévy processslowly oscillating functionrisk process

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 2 , June 2002 , pp. 282 - 295
Copyright
Copyright © Applied Probability Trust 2002 

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.)

References

[1] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[2] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, Berlin.Google Scholar
[3] Bratiychuk, M. S. (1990). The size of the overshot and behaviour of the absolute maximum for process with independent increments. Ukrainian Math. J. 42, 397–403.Google Scholar
[4] Dufresne, F., and Gerber, H. U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.Google Scholar
[5] Embrechts, P., and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
[6] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.Google Scholar
[7] Feller, F. (1971). An Introduction to Probability Theory and Its Application. John Wiley, New York.Google Scholar
[8] Gerber, H. U. (1970). An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuar Tidskr. 53, 205–210.Google Scholar
[9] Gusak, D. V., and Koroljuk, V. S. (1970). Distribution of the functionals on the homogeneous process with independent increments. Theory Prob. Math. Statist. 1, 55–73.Google Scholar
[10] Koroljuk, V. S., Suprun, V. N., and Surenkov, V. M. (1976). Potential method in the boundary problems for processes with independent increments and jumps of the same sign. Theory Prob. Appl. 22, 419–525.Google Scholar
[11] Rogers, L. C. G. (1984). A new identity for real Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 20, 21–34.Google Scholar
[12] 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
[13] Rogozin, B. A. (1969). Distribution of the maximum of the process with independent increments. Siberian Math. J. 10, 1334–1363.Google Scholar
[14] Rogozin, B. A. (1976). The asymptotic of the renewal function. Theory Prob. Appl. 21, 689–706.Google Scholar
[15] Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
[16] Veraverbeke, N. (1993). Asymptotic estimates for the probability of ruin in a Poisson model with diffusion. Insurance Math. Econom. 13, 57–62.Google Scholar
[17] Yakymiv, A. L. (1987). Asymptotical behavior of a class of infinitely divisible distributions. Theory Prob. Appl. 32, 628–639.Google Scholar