Transform approach for discounted aggregate claims in a risk model with descendant claims
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Transform approach for discounted aggregate claims in a risk model with descendant claims Hyunjoo Yoo1 · Bara Kim1 · Jeongsim Kim2 · Jiwook Jang3 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider a risk model with three types of claims: ordinary, leading, and descendant claims. We derive an expression for the Laplace–Stieltjes transform of the distribution of the discounted aggregate claims. By using this expression, we can then obtain the mean and variance of the discounted aggregate claims. For actuarial applications, the VaR and CTE are computed by numerical inversion of the Laplace transforms for the tail probability and the conditional tail expectation of the discounted aggregate claims. The net premium for stop-loss reinsurance contract is also computed. Keywords Laplace–Stieltjes transform · Risk model · Descendant claims · Discounted aggregate claims
1 Introduction In risk models, there are several quantities of interest, such as the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin, and distribution of aggregate claim amount. These quantities have been investigated in various kinds of risk models. For details in various risk models, refer to Asmussen (2000).
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Jeongsim Kim [email protected] Hyunjoo Yoo [email protected] Bara Kim [email protected] Jiwook Jang [email protected]
1
Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea
2
Department of Mathematics Education, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju, Chungbuk 28644, Korea
3
Department of Actuarial Studies & Business Analytics, Macquarie Business School, Macquarie University, Sydney, NSW 2109, Australia
123
Annals of Operations Research
Dickson et al. (1995) studied the ruin probability in the Cramér-Lundberg risk model (or classical compound Poisson risk model). Willmot et al. (2004) studied the distribution of deficit at ruin in the Sparre Andersen model (or renewal risk model). Lu and Li (2005) studied the ruin probability in a Markov-modulated risk model (or regime switching model) where inter-claim times, claim sizes, and premiums vary according to a Markovian environment. Ng and Yang (2006) obtained the joint distribution of surplus before and after ruin in a Markovian regime switching risk model where the claim sizes are phase-type distributed. Breuer (2010) applied the Markov additive processes with phase-type jumps to an insurance risk model. For an insurance risk model where claims occur according to a Markovian arrival process, refer to Cheung and Feng (2013), and Ren (2016). In recent years, some special risk models have been suggested to account for the possibility of clustering of insurance events. Albrecher and Asmussen (2006) studied the ruin probability and aggregate claim distributions for shot noise Cox processes. Stabile and Torrisi (2010) and Zhu (2013) used Hawkes processes in risk models. Markov Poisson cluster processes have also been of interest in in
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