On the probability of reaching a barrier in an Erlang(2) risk process
Abstract
HolaIn this paper the process of aggregated claims in a non-life insurance portfolio as defined in the classical model of risk theory is modified. The Compound Poisson process is replaced with a more general renewal risk process with interoccurrence times of Erlangian type. We focus our analysis on the probability that the process of surplus reaches a certain level before ruin occurs, χ(u,b). Our main contribution is the generalization obtained in the computation of χ(u,b) for the case of interoccurrence time between claims distributed as Erlang(2, β) and the individual claim amount as Erlang (n, γ).
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SORT, vol. 29 (2) July-December 2005, p. 235-248 On the probability of reaching a barrier in an Erlang(2) risk process M.M. Claramunt(1),M.M ´ armol(1) and R. Lacayo(2) (1)Universitat de Barcelona, (2)Universitat Aut` onoma de Barcelona Address for correspondence: M.M. Claramunt, M. M´ armol. Departament de Matem` atica Econ` omica, Financera i Actuarial. Universitat de Barcelona. Avda. Diagonal, 690. 08034 Barcelona. Tlf: 93.4035744. E-mail: [email protected], [email protected]. HolaIn this paper the process of aggregated claims in a non-life insurance portfolio as defined in the classical model of risk theory is modified. The Compound Poisson process is replaced with a more general renewal risk process with interoccurrence times of Erlangian type. We focus our analysis on the probability that the process of surplus reaches a certain level before ruin occurs, χ(u,b). Our main contribution is the generalization obtained in the computation of χ(u,b)for the case of interoccurrence time between claims distributed as Erlang(2, β) and the individual claim amount as Erlang (n,γ). MSC: 91B30, 62P05 Keywords: risk theory, Erlang distribution, upper barrier, ordinary differential equation, boundary conditions HolaEn aquest article es modifica el proc´ es de reclamacions agregades d’una cartera d’assegurances de no-vida en el model cl` assic de la teoria de risc. El proc´ es de Poisson compost ´ es reemplac¸at per un proc´ es de renaixement de risc m´ es general amb temps d’inter-ocurr` encia de tipus erlangi` a. Concentrem la nostra an` alisi en la probabilitat que el proc´ es de sobrepassar assoleixi cert nivell abans d’arribar a ru¨ ına, χ(u,b).La nostra principal contribuci´ o´ es la generalitzaci´ o obtinguda en el c` alcul d’χ(u,b)per al cas de temps d’inter-ocurr` encia entre reclamacions distribu¨ ıdes com Erlang (2, β)iles quantitats de reclamacions individuals com Erlang (n, γ). Paraules clau: teoria del risc, distribuci´ o d’Erlang, barreres superiors, equacions diferencials ordin` aries, condicions frontera