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Self-similar processes in collective risk theory

Michna, Zbigniew LU (1998) In Journal of Applied Mathematics and Stochastic Analysis 11(4). p.429-448
Abstract
A self-similar, continuous process with stationary increments is considered as an approximation to the surplus process in collective risk theory. This approximation can be seen as the weak limit of risk processes with linear premium income, where the claim sizes show a long-range dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where long-range dependence of claim sizes occurs is given in an example where the risk process evolves according to an environmental process with two states. If at least one of the distributions of the time between two changes of the state has a regularly varying tail, a long-range dependence is observed. By way of example,... (More)
A self-similar, continuous process with stationary increments is considered as an approximation to the surplus process in collective risk theory. This approximation can be seen as the weak limit of risk processes with linear premium income, where the claim sizes show a long-range dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where long-range dependence of claim sizes occurs is given in an example where the risk process evolves according to an environmental process with two states. If at least one of the distributions of the time between two changes of the state has a regularly varying tail, a long-range dependence is observed. By way of example, upper and lower bounds for the ruin probability of fractional Brownian motion with drift are obtained. Numerical calculations show, however, that these bounds are far from the true value. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Applied Mathematics and Stochastic Analysis
volume
11
issue
4
pages
429 - 448
publisher
Hindawi Publishing Corporation
external identifiers
  • scopus:77955730861
ISSN
1048-9533
language
English
LU publication?
yes
id
63d2107f-ba06-49df-a723-8ec57b8b7770 (old id 1273223)
date added to LUP
2008-12-09 14:08:01
date last changed
2017-01-01 07:53:28
@article{63d2107f-ba06-49df-a723-8ec57b8b7770,
  abstract     = {A self-similar, continuous process with stationary increments is considered as an approximation to the surplus process in collective risk theory. This approximation can be seen as the weak limit of risk processes with linear premium income, where the claim sizes show a long-range dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where long-range dependence of claim sizes occurs is given in an example where the risk process evolves according to an environmental process with two states. If at least one of the distributions of the time between two changes of the state has a regularly varying tail, a long-range dependence is observed. By way of example, upper and lower bounds for the ruin probability of fractional Brownian motion with drift are obtained. Numerical calculations show, however, that these bounds are far from the true value.},
  author       = {Michna, Zbigniew},
  issn         = {1048-9533},
  language     = {eng},
  number       = {4},
  pages        = {429--448},
  publisher    = {Hindawi Publishing Corporation},
  series       = {Journal of Applied Mathematics and Stochastic Analysis},
  title        = {Self-similar processes in collective risk theory},
  volume       = {11},
  year         = {1998},
}