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

Published online by Cambridge University Press:  08 November 2017

Eric C. K. Cheung ,
Suhang Dai  and
Weihong Ni
Eric C. K. Cheung
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, NSW 2052, Australia
Suhang Dai
Affiliation:
Institute for Financial and Actuarial Mathematics, University of Liverpool, Liverpool L69 7ZL, UK
Weihong Ni*
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong
*
*Correspondence to: Weihong Ni, Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Tel: +852 39178155; E-mail: wni.grace@gmail.com
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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.

Keywords

Ruin probabilityMarkov additive processMarkov fluid flowErlangian approximation
Type
Paper
Information
Annals of Actuarial Science , Volume 12 , Issue 2 , September 2018 , pp. 269 - 295
Copyright
© Institute and Faculty of Actuaries 2017 

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