{"id":{"repo_id":"vilnius","oai_identifier":"oai:vu.lt:elaba:210642105"},"canonical_url":"https://search.dev.ndltd.org/etd/vilnius/oai:vu.lt:elaba:210642105","repository":{"repo_id":"vilnius","name":"Vilnius University","base_url":"https://epublications.vu.lt/oai"},"display":{"title":"Atstatymo lygties sprendinio aproksimavimas klasikiniame rizikos modelyje /","abstract":"The main objective of this work is to derive numerical methods from the renewal equation of Cramér – Lundberg model that would approximate the probability of ultimate ruin for any loss distribution, along with empirical method analysis. By employing the recovery equation, three expressions for ψ(u) are derived analytically for testing when claims are distributed according to: exponential distribution, mixture of two exponential distributions, and the second Erlang distribution. Subsequently, utilizing classical numerical integration schemes such as left and right Riemann sums, trapezoidal rule, and Simpson’s rule, four methods suitable for the recovery equation are derived. We empirically assess the convergence rate of the methods, determine the step sizes h for which the methods are stable, and propose possible implementations to reduce time complexity: online fast Fourier transform and solved recursion. Additionally, we provide the R program code, naive implementation of all four methods, and one of solved recursion.","abstract_html":"The main objective of this work is to derive numerical methods from the renewal equation of Cramér – Lundberg model that would approximate the probability of ultimate ruin for any loss distribution, along with empirical method analysis. By employing the recovery equation, three expressions for ψ(u) are derived analytically for testing when claims are distributed according to: exponential distribution, mixture of two exponential distributions, and the second Erlang distribution. Subsequently, utilizing classical numerical integration schemes such as left and right Riemann sums, trapezoidal rule, and Simpson’s rule, four methods suitable for the recovery equation are derived. We empirically assess the convergence rate of the methods, determine the step sizes h for which the methods are stable, and propose possible implementations to reduce time complexity: online fast Fourier transform and solved recursion. Additionally, we provide the R program code, naive implementation of all four methods, and one of solved recursion.","abstract_has_math":false,"creators":["Lukoševičius, Gustas,"],"institution":"Institutional Repository of Vilnius University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024","date_published":"2024","updated_at":"2026-07-24T05:55:52Z","subjects":[],"languages":["lit"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.vu.lt/VU:ELABAETD210642105&prefLang=en_US","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lukoševičius, Gustas,"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024"]},{"key":"dc:publisher","label":"Institution","values":["Institutional Repository of Vilnius University"]},{"key":"dc:relation","label":"Dc Relation","values":["https://epublications.vu.lt/object/elaba:210642105/210642105.pdf"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/bachelorThesis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["lit"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://repository.vu.lt/VU:ELABAETD210642105&prefLang=en_US"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The main objective of this work is to derive numerical methods from the renewal equation of Cramér – Lundberg model that would approximate the probability of ultimate ruin for any loss distribution, along with empirical method analysis. By employing the recovery equation, three expressions for ψ(u) are derived analytically for testing when claims are distributed according to: exponential distribution, mixture of two exponential distributions, and the second Erlang distribution. Subsequently, utilizing classical numerical integration schemes such as left and right Riemann sums, trapezoidal rule, and Simpson’s rule, four methods suitable for the recovery equation are derived. We empirically assess the convergence rate of the methods, determine the step sizes h for which the methods are stable, and propose possible implementations to reduce time complexity: online fast Fourier transform and solved recursion. Additionally, we provide the R program code, naive implementation of all four methods, and one of solved recursion."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Atstatymo lygties sprendinio aproksimavimas klasikiniame rizikos modelyje /","Approximating solutions to the renewal equation within the classical compound - poisson risk model."]}]}],"canonical_facts":{"dc:creator":["Lukoševičius, Gustas,"],"dc:date":["2024"],"dc:description":["The main objective of this work is to derive numerical methods from the renewal equation of Cramér – Lundberg model that would approximate the probability of ultimate ruin for any loss distribution, along with empirical method analysis. By employing the recovery equation, three expressions for ψ(u) are derived analytically for testing when claims are distributed according to: exponential distribution, mixture of two exponential distributions, and the second Erlang distribution. Subsequently, utilizing classical numerical integration schemes such as left and right Riemann sums, trapezoidal rule, and Simpson’s rule, four methods suitable for the recovery equation are derived. We empirically assess the convergence rate of the methods, determine the step sizes h for which the methods are stable, and propose possible implementations to reduce time complexity: online fast Fourier transform and solved recursion. Additionally, we provide the R program code, naive implementation of all four methods, and one of solved recursion."],"dc:format":["application/pdf"],"dc:identifier":["https://repository.vu.lt/VU:ELABAETD210642105&prefLang=en_US"],"dc:language":["lit"],"dc:publisher":["Institutional Repository of Vilnius University"],"dc:relation":["https://epublications.vu.lt/object/elaba:210642105/210642105.pdf"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:title":["Atstatymo lygties sprendinio aproksimavimas klasikiniame rizikos modelyje /","Approximating solutions to the renewal equation within the classical compound - poisson risk model."],"dc:type":["info:eu-repo/semantics/bachelorThesis"]},"updated_at":"2026-07-24T05:55:52Z"}