Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 4, Pages 429-448

Self-similar processes in collective risk theory

Zbigniew Michna

University of Lund, Department of Mathematical Statistics, Solvegatan 18, Box 118, Lund 221 00, Sweden

Received 1 March 1997; Revised 1 March 1998

Copyright © 1998 Zbigniew Michna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Collective risk theory is concerned with random fluctuations of the total assets and the risk reserve of an insurance company. In this paper we consider self-similar, continuous processes with stationary increments for the renewal model in risk theory. We construct a risk model which shows a mechanism of long range dependence of claims. We approximate the risk process by a self similar process with drift. The ruin probability within finite time is estimated for fractional Brownian motion with drift. A similar model is applicable in queueing systems, describing long range dependence in on/off processes and associated fluid models. The obtained results are useful in communication network models, as well as storage and inventory models.