Selfsimilar processes in collective risk theory
(1998) In Journal of Applied Mathematics and Stochastic Analysis 11(4). p.429448 Abstract
 A selfsimilar, 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 longrange dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where longrange 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 longrange dependence is observed. By way of example,... (More)
 A selfsimilar, 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 longrange dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where longrange 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 longrange 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1273223
 author
 Michna, Zbigniew ^{LU}
 organization
 publishing date
 1998
 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 Limited
 external identifiers

 scopus:77955730861
 ISSN
 10489533
 language
 English
 LU publication?
 yes
 id
 63d2107fba0649dfa7238ec57b8b7770 (old id 1273223)
 date added to LUP
 20160404 09:41:03
 date last changed
 20220129 19:04:59
@article{63d2107fba0649dfa7238ec57b8b7770, abstract = {{A selfsimilar, 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 longrange dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where longrange 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 longrange 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 = {{10489533}}, language = {{eng}}, number = {{4}}, pages = {{429448}}, publisher = {{Hindawi Limited}}, series = {{Journal of Applied Mathematics and Stochastic Analysis}}, title = {{Selfsimilar processes in collective risk theory}}, volume = {{11}}, year = {{1998}}, }