## Abstract

The insurance risk model in the presence of two horizontal absorbing barriers is considered. The lower barrier is the usual ruin barrier while the upper one corresponds to the dividend barrier. The distribution of two first-exit times of the risk process from the strip between the two horizontal lines is under study. The claim arrival process is governed by an Order Statistic Point Process (OSPP) which enables the derivation of formulas in terms of the joint distribution of the order statistics of a sample of uniform random variables.

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## References

Avanzi B (2009) Strategies for dividend distribution: a review. North Am Actuarial J 13(2):217–251

Borovkov KA, Dickson DCM (2008) On the ruin time distribution for a Sparre Andersen process with exponential claim sizes. Insurance: Math Econ 42(3):1104–1108

Crump KS (1975) On point processes having an order statistic structure. Sankhyā: The Indian J Stat Ser A (1961-2002) 37(3):396–404

Denuit M, Lefèvre C, Picard P (2003) Polynomial structures in order statistics distributions. J Stat Plan Inference 113(1):151–178

Dickson DCM, Gray JR (1984) Approximations to ruin probability in the presence of an upper absorbing barrier. Scand Actuar J 1984(2):105–115

Dimitrova DS, Ignatov ZG, Kaishev VK (2016) Ruin and deficit under claim arrivals with the order statistics property working paper or preprint

Dimitrova DS, Ignatov ZG, Kaishev VK (2017) On the first crossing of two boundaries by an order statistics risk process. Risks 5(3):43

Dimitrova DS, Kaishev VK, Zhao S (2016) On the evaluation of finite-time ruin probabilities in a dependent risk model. Appl Math Comput 275(Supplement C):268–286

Goffard P-O, Lefèvre C (2016) Duality in ruin problems for ordered risk models working paper or preprint

Goffard P-O, Lefèvre C (2017) Boundary crossing of order statistics point processes. J Math Anal Appl 447(2):890–907

Ignatov ZG, Kaishev VK (2016) First crossing time, overshoot and appell–hessenberg type functions. Stochastics 88(8):1240–1260

Lefèvre C, Picard P (2011) A new look at the homogeneous risk model. Insurance: Math Econ 49(3):512–519

Lefèvre C, Picard P (2014) Ruin probabilities for risk models with ordered claim arrivals. Methodol Comput Appl Probab 16(4):885–905

Lefèvre C, Picard P (2015) Risk models in insurance and epidemics: a bridge through randomized polynomials. Probab Eng Inf Sci 29(3):399–420

Lehmann A (1998) Boundary crossing probabilities of poisson counting processes with general boundaries. In: Kahle W, von Collani E, Franz J, Jensen U (eds) Advances in stochastic models for reliability, quality and safety. Statistics for Industry and Technology, Birkhäuser

Niederhausen H (1981) Sheffer polynomials for computing exact kolmogorov-Smirnov and rényi distributions. Ann Stat 9(5):923–944

Perry D, Stadje W, Zacks S (2005) A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodol Comput Appl Probab 7(1):51–62

Picard P, Lefèvre C (1994) On the first crossing of the surplus process with a given upper barrier. Insurance: Math Econ 14(2):163–179

Puri PS (1982) On the characterization of point processes with the order statistic property without the moment condition. J Appl Probab 19(1):39–51

Rullière D, Loisel S (2005) The win-first probability under interest force. Insurance: Math Econ 37(3):421–442

De Vylder FE, Goovaerts MJ (1999) Inequality extensions of Prabhu’s formula in ruin theory. Insurance: Math Econ 24(3):249–271

De Vylder FE, Goovaerts MJ (2000) Homogeneous risk models with equalized claim amounts. Insurance: Math Econ 26(2–3):223–238

Willmot GE (1989) The total claims distribution under inflationary conditions. Scand Actuar J 1989(1):1–12

Xu Y (2012) First exit times of compound Poisson processes with parallel boundaries. Seq Anal 31(2):135–144

## Acknowledgments

I am thankful to the anonymous referee for his useful comments and suggestions. I want to address special thanks to Mackenzie Wildman for helping me improving the presentation of my paper. My work was partially funded by a CAE grant of the Society of Actuaries.

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Goffard, PO. Two-Sided Exit Problems in the Ordered Risk Model.
*Methodol Comput Appl Probab* **21**, 539–549 (2019). https://doi.org/10.1007/s11009-017-9606-z

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DOI: https://doi.org/10.1007/s11009-017-9606-z

### Keywords

- Order statistic property
- Joint distribution of order statistics
- Finite-time ruin probabilities
- First-exit time
- Risk theory

### Mathematics Subject Classification (2010)

- 60G55
- 60G40
- 12E10