# Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model

- 5 Downloads

## Abstract

In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.

### Keywords

Discounted aggregate claims until ruin Discrete Sparre Andersen renewal risk process Discounted moment-based joint distribution Higher moments Covariance Discrete Coxian (*K*

_{n}) distribution Claim causing ruin

### Mathematics Subject Classification (2010)

91B30 60K10## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

The authors are grateful to the anonymous referees and the Editor for their useful suggestions that have greatly improved the overall presentation and material of the paper.

### References

- Albrecher H, Boxma OJ (2004) A ruin model with dependence between claim sizes and claim intervals. Insur Math Econ 35(2):245–254MathSciNetCrossRefMATHGoogle Scholar
- Cai J, Feng R, Willmot GE (2009) On the total discounted operating costs up to default and its applications. Adv Appl Probab 41(2):495–522MathSciNetCrossRefMATHGoogle Scholar
- Cheng S, Gerber HU, Shiu ESW (2000) Discounted probabilities and ruin theory in the compound binomial model. Insur Math Econ 26(2):239–250MathSciNetCrossRefMATHGoogle Scholar
- Cheung ECK (2013) Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times. Insur Math Econ 53 (2):343–354MathSciNetCrossRefMATHGoogle Scholar
- Cheung ECK, Feng R (2013) A unified analysis of claim costs up to ruin in a Markovian arrival risk process. Insur Math Econ 53(1):98–109CrossRefMATHGoogle Scholar
- Cheung ECK, Landriault D, Willmot GE, Woo J-K (2010) Gerber-Shiu analysis with a generalized penalty function. Scand Actuar J 3:185–199MathSciNetCrossRefMATHGoogle Scholar
- Cheung ECK, Liu H, Woo J-K (2015) On the joint analysis of the total discounted payments to policyholders and shareholders: dividend barrier strategy. Risks 3(4):491–514CrossRefGoogle Scholar
- Cheung ECK, Woo J-K (2016) On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes. Scand Actuar J 1:63–91MathSciNetCrossRefGoogle Scholar
- De Vylder FE (1996) Advanced risk theory: a self-contained introduction. Editions de lUniversite de Bruxelles, BrusselsGoogle Scholar
- De Vylder FE, Marceau E (1996) Classical numerical ruin probabilities. Scand Actuar J, 109–123Google Scholar
- Dickson DCM (1994) Some comments on the compound binomial model. ASTIN Bull 24:33–45CrossRefGoogle Scholar
- Dickson DCM, Hipp C (2001) On the time to ruin for Erlang(2) risk processes. Insur Math Econ 29(3):333–344MathSciNetCrossRefMATHGoogle Scholar
- Feng R (2009a) On the total operating costs up to default in a renewal risk model. Insur Math Econ 45(2):305–314Google Scholar
- Feng R (2009b) A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model. Bull Swiss Assoc Actuar 1&2:71–87Google Scholar
- Genest C, Nes̆lehová J (2007) A primer on copulas for count data. ASTIN Bull 37:475–515MathSciNetCrossRefMATHGoogle Scholar
- Gerber HU (1988) Mathematical fun with compound binomial process. ASTIN Bull 18(2):161–168CrossRefGoogle Scholar
- Gerber HU, Shiu ESW (1998) On the time value of ruin. North Amer Actuar J 2(1):48–78MathSciNetCrossRefMATHGoogle Scholar
- Klugman SA, Panjer HH, Willmot GE (2008) Loss models: from data to decisions, 3rd edn. Wiley, New YorkCrossRefMATHGoogle Scholar
- Krishna H, Pundir PS (2009) A bivariate geometric distribution with applications to reliability. J Commun Stat Theory Methods 38(7):1079–1093MathSciNetCrossRefMATHGoogle Scholar
- Landriault D, Lee WY, Willmot GE, Woo J-K (2014) A note on deficit analysis in dependency models involving Coxian claim amounts. Scand Actuar J 5:405–423MathSciNetCrossRefGoogle Scholar
- Léveillé G, Garrido J (2001a) Moments of compound renewal sums with discounted claims. Insur Math Econ 28(2):217–231Google Scholar
- Léveillé G, Garrido J (2001b) Recursive moments of compound renewal sums with discounted claims. Scand Actuar J 2:98–110Google Scholar
- Li S (2005a) On a class of discrete time renewal risk models. Scand Actuar J 4:241–260Google Scholar
- Li S (2005b) Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time renewal risk models. Scand Actuar J 4:271–284Google Scholar
- Lindsay BG, Pilla RS, Basak P (2000) Moment-based approximations of distributions using mixtures: theory and applications. Ann Institut Statist Math 52 (2):215–230MathSciNetCrossRefMATHGoogle Scholar
- Marceau E (2009) On the discrete-time compound renewal risk model with dependence. Insur Math Econ 44(2):245–259MathSciNetCrossRefMATHGoogle Scholar
- Nikoloulopoulos AK, Karlis D (2008) Fitting copulas to bivariate earthquake data: the seismic gap hypothesis revisited. Environmetrics 19:251269MathSciNetCrossRefGoogle Scholar
- Panjer HH (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull 12(1):22–26MathSciNetCrossRefGoogle Scholar
- Shiu ESW (1989) The probability of eventual ruin in the compound binomial model. ASTIN Bull 19(2):179–190CrossRefGoogle Scholar
- Sparre Andersen E (1957) On the collective theory of risk in the case of contagion between claims. In: Proceedings of the Transactions of the XVth international congress on actuaries, vol II. New York, pp 219–229Google Scholar
- Willmot GE (1993) Ruin probabilities in the compound binomial model. Insur Math Econ 12(2):133–142MathSciNetCrossRefMATHGoogle Scholar
- Willmot GE (2007) On the discounted penalty function in the renewal risk model with general interclaim times. Insur Math Econ 41(1):17–31MathSciNetCrossRefMATHGoogle Scholar
- Willmot GE, Woo J-K (2012) On the analysis of a general class of dependent risk processes. Insur Math Econ 51(1):134–141MathSciNetCrossRefMATHGoogle Scholar
- Woo J-K (2012) A generalized penalty function for a class of discrete renewal processes. Scand Actuar J 2:130–152MathSciNetCrossRefMATHGoogle Scholar
- Wu X, Li S (2008) On the discounted penalty function in a discrete time renewal risk model with general interclaim times. Scand Actuar J 1:1–14Google Scholar