Ruin probabilities in a Sparre Andersen model with dependency structure based on a threshold window

Abstract

We analyse ruin probabilities for an insurance risk process with a more generalised dependence structure compared to the one introduced in Constantinescu et al. (2016). In this paper, we assume that a random threshold window is generated every time after a claim occurs. By comparing the previous inter-claim time with the threshold window, the distributions of the current threshold window and the inter-arrival time are determined. Furthermore, the statuses for the previous and current inter-arrival times give rise to the current claim size distribution as well. Like Constantinescu et al. (2016), we first identify the embedded Markov additive process where all the randomness takes a general form. Inspired by the Erlangisation technique, the key message of this paper is to analyse such risk process using a Markov fluid flow model where the underlying random variables follow phase-type distributions. This would further allow us to approximate the fixed observation windows by Erlang random variables. Then ruin probabilities under the process with Erlang(n) observation windows are proved to be Erlangian approximations for those related to the process with fixed threshold windows at the limit. An exact form of the limit can be obtained whose application will be illustrated further by a numerical example.

Reinsurance and Solvency Capital: Mitigating Insurance Companies’ Ruin Probability

“…Interestingly, despite the complex approach, the results are similar to the Markovian approach. Some authors argue that the independence hypothesis between the severity and claims occurrences may not be realistic for certain portfolios (Cheung, Dai, & Ni, 2018;Eryilmaz & Gebizlioglu, 2017). In a similar line, but imposing a dependence structure between the claims occurrences in different lines of business, Dong and Wang (2018) use renovation processes with stochastic financial returns to derive asymptotic formulas for the ruin probability in finite and infinite time.…”

Resseguro e Capital de Solvência: Atenuantes da Probabilidade de Ruína de Seguradoras

“…Alguns autores argumentam que a hipótese de independência entre as ocorrências de sinistros pode não ser realista para certos portfólios (Cheung, Dai, & Ni, 2018;Eryilmaz & Gebizlioglu, 2017). Em linha similar, mas impondo estrutura de dependência entre as ocorrências de sinistros de diferentes linhas de negócios, Dong e Wang (2018) usam processos de renovação com retornos financeiros estocásticos para derivar fórmulas assintóticas para a probabilidade de ruína em tempo finito e infinito.…”