Publicação
Further developments in the Erlang(n) risk process
| Resumo: | For actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustration |
|---|---|
| Autores principais: | Bergel, Agnieszka I. |
| Outros Autores: | Reis, Alfredo D. Egídio dos |
| Assunto: | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| Ano: | 2015 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Lisboa |
| Idioma: | inglês |
| Origem: | Repositório da Universidade de Lisboa |
| _version_ | 1866810984614592512 |
|---|---|
| author | Bergel, Agnieszka I. |
| author2 | Reis, Alfredo D. Egídio dos |
| author2_role | author |
| author_facet | Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
| author_role | author |
| contributor_name_str_mv | Repositório Científico de Acesso Aberto da ULisboa |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Bergel, Agnieszka I.\"},{\"Person.name\":\"Reis, Alfredo D. Egídio dos\"}] |
| datacite.contributors.contributor.contributorName.fl_str_mv | Repositório Científico de Acesso Aberto da ULisboa |
| datacite.creators.creator.creatorName.fl_str_mv | Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
| datacite.date.Accepted.fl_str_mv | 2015-01-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2022-06-01T10:17:23Z |
| datacite.date.embargoed.fl_str_mv | 2022-06-01T10:17:23Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| datacite.titles.title.fl_str_mv | Further developments in the Erlang(n) risk process |
| dc.contributor.none.fl_str_mv | Repositório Científico de Acesso Aberto da ULisboa |
| dc.creator.none.fl_str_mv | Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
| dc.date.Accepted.fl_str_mv | 2015-01-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2022-06-01T10:17:23Z |
| dc.date.embargoed.fl_str_mv | 2022-06-01T10:17:23Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | http://hdl.handle.net/10400.5/24449 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Taylor & Francis |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| dc.title.fl_str_mv | Further developments in the Erlang(n) risk process |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | For actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustration |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| fulltext.url.fl_str_mv | https://repositorio.ulisboa.pt/bitstreams/08f85988-4f33-4bf5-82ce-41746ea3fcbe/download |
| id | ul_8eddaa33a2cebb3eef448a9a71a63b6d |
| identifier.url.fl_str_mv | http://hdl.handle.net/10400.5/24449 |
| instacron_str | ul |
| institution | Universidade de Lisboa |
| instname_str | Universidade de Lisboa |
| language | eng |
| network_acronym_str | ul |
| network_name_str | Repositório da Universidade de Lisboa |
| oai_identifier_str | oai:repositorio.ulisboa.pt:10400.5/24449 |
| organization_str_mv | urn:organizationAcronym:ul |
| person_str_mv | Bergel, Agnieszka I. Reis, Alfredo D. Egídio dos |
| publishDate | 2015 |
| publisher.none.fl_str_mv | Taylor & Francis |
| reponame_str | Repositório da Universidade de Lisboa |
| repository_id_str | urn:repositoryAcronym:ul |
| service_str_mv | urn:repositoryAcronym:ul |
| spelling | engTaylor & Francispt_PTFor actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustrationapplication/pdfpt_PTFurther developments in the Erlang(n) risk processBergel, Agnieszka I.Reis, Alfredo D. Egídio dosHostingInstitutionOrganizationalRepositório Científico de Acesso Aberto da ULisboae-mailmailto:repositorio@reitoria.ulisboa.ptrepositorio@reitoria.ulisboa.pt2022-06-01T10:17:23Z20152015-01-01T00:00:00ZHandlehttp://hdl.handle.net/10400.5/24449http://purl.org/coar/access_right/c_abf2open accessSparre–Andersen Risk ModelGeneralized Erlang(n) Interclaim TimesFundamental Lundberg’s EquationProbability of Reaching an Upper BarrierMaximum Severity of RuinExpected Discounted Dividends Prior to Ruin525929 bytesliteraturehttp://purl.org/coar/resource_type/c_6501journal articlehttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://repositorio.ulisboa.pt/bitstreams/08f85988-4f33-4bf5-82ce-41746ea3fcbe/download |
| spellingShingle | Further developments in the Erlang(n) risk process Bergel, Agnieszka I. Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| status | SINGLETON |
| subject.fl_str_mv | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| title | Further developments in the Erlang(n) risk process |
| title_full | Further developments in the Erlang(n) risk process |
| title_fullStr | Further developments in the Erlang(n) risk process |
| title_full_unstemmed | Further developments in the Erlang(n) risk process |
| title_short | Further developments in the Erlang(n) risk process |
| title_sort | Further developments in the Erlang(n) risk process |
| topic | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| topic_facet | Sparre–Andersen Risk Model Generalized Erlang(n) Interclaim Times Fundamental Lundberg’s Equation Probability of Reaching an Upper Barrier Maximum Severity of Ruin Expected Discounted Dividends Prior to Ruin |
| url | http://hdl.handle.net/10400.5/24449 |
| visible | 1 |