Ruin problems in the generalized Erlang (n) risk model
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.5/24447 |
Resumo: | For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (2005), by considering a general claim amount distribution |
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Ruin problems in the generalized Erlang (n) risk modelSparre-Andersen Risk ModeGeneralized Erlang(n) Interclaim TimesGeneralized Lundberg’s EquationProbability of Reaching an Upper BarrierMaximum Severity of RuinExpected Discounted Dividends Prior to RuinFor actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (2005), by considering a general claim amount distributionSpringerRepositório da Universidade de LisboaBergel, Agnieszka I.Reis, Alfredo D. Egídio dos2022-06-01T09:17:33Z20162016-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.5/24447engBergel, Agnieszka I. and Alfredo D. Egídio dos Reis. (2016). "Ruin problems in the generalized Erlang (n) risk model." European Actuarial Journal. Vol. 6, No.1: pp. 257-275.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-03-06T14:54:05Zoai:www.repository.utl.pt:10400.5/24447Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T17:08:29.551962Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Ruin problems in the generalized Erlang (n) risk model |
title |
Ruin problems in the generalized Erlang (n) risk model |
spellingShingle |
Ruin problems in the generalized Erlang (n) risk model Bergel, Agnieszka I. Sparre-Andersen Risk Mode Generalized Erlang(n) Interclaim Times Generalized Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
title_short |
Ruin problems in the generalized Erlang (n) risk model |
title_full |
Ruin problems in the generalized Erlang (n) risk model |
title_fullStr |
Ruin problems in the generalized Erlang (n) risk model |
title_full_unstemmed |
Ruin problems in the generalized Erlang (n) risk model |
title_sort |
Ruin problems in the generalized Erlang (n) risk model |
author |
Bergel, Agnieszka I. |
author_facet |
Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
author_role |
author |
author2 |
Reis, Alfredo D. Egídio dos |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Repositório da Universidade de Lisboa |
dc.contributor.author.fl_str_mv |
Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
dc.subject.por.fl_str_mv |
Sparre-Andersen Risk Mode Generalized Erlang(n) Interclaim Times Generalized Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
topic |
Sparre-Andersen Risk Mode Generalized Erlang(n) Interclaim Times Generalized Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
description |
For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (2005), by considering a general claim amount distribution |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016 2016-01-01T00:00:00Z 2022-06-01T09:17:33Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10400.5/24447 |
url |
http://hdl.handle.net/10400.5/24447 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Bergel, Agnieszka I. and Alfredo D. Egídio dos Reis. (2016). "Ruin problems in the generalized Erlang (n) risk model." European Actuarial Journal. Vol. 6, No.1: pp. 257-275. |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Springer |
publisher.none.fl_str_mv |
Springer |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799131178638245888 |