Abstract
So far we focused on the event of ruin, corresponding to the reserve process dropping below 0. This chapter studies various bankruptcy concepts, in which, besides the reserve level becoming negative, an additional condition has to be fulfilled. In the first variant, the net reserve process is inspected at Poisson instants, and bankruptcy occurs if the reserve level is below zero at such an inspection time. For this setting an adapted version of the Pollaczek-Khinchine theorem is derived, as well as an appealing decomposition. In the second variant there is bankruptcy if the reserve process is uninterruptedly below 0 for a sufficiently long time, whereas in the third variant the bankruptcy criterion is based on the total time with a negative surplus.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
I. Adan, O. Boxma, D. Perry, The G/M/1 queue revisited. Math. Methods Oper. Res. 62, 437–452 (2005)
H. Albrecher, J. Ivanovs, Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Stoch. Process. Their Appl. 127, 643–656 (2017)
H. Albrecher, V. Lautscham, From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bull. 43, 213–243 (2013)
H. Albrecher, E. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: dividends. ASTIN Bull. 41, 645–672 (2011)
H. Albrecher, H. Gerber, E. Shiu, The optimal dividend barrier in the Gamma-Omega model. Eur. Actuar. J. 1, 43–56 (2011)
H. Albrecher, E. Cheung, S. Thonhauser, Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scand. Actuar. J. 2013, 424–452 (2013)
H. Albrecher, J. Ivanovs, X. Zhou, Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382 (2016)
H. Albrecher, O. Boxma, R. Essifi, R. Kuijstermans, A queueing model with randomized depletion of inventory. Probab. Eng. Inf. Sci. 31, 43–59 (2017)
O. Boxma, M. Mandjes, A decomposition for Lévy processes inspected at Poisson moments. J. Appl. Probab. 60, 557–569 (2023)
O. Boxma, R. Essifi, A. Janssen, A queueing/inventory and an insurance risk model. Adv. Appl. Probab. 48, 1139–1160 (2016)
O. Boxma, O. Kella, M. Mandjes, On fluctuation-theoretic decompositions via Lindley-type recursions. Stoch. Process. Their Appl. 165, 316–336 (2023)
E. Cheung, W. Zhu, Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims. Insur. Math. Econ. 111, 84–101 (2023)
J. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1969)
A. Dassios, S. Wu, Parisian ruin with exponential claims. Unpublished manuscript (2008). http://stats.lse.ac.uk/angelos/docs/exponentialjump.pdf
A. Dassios, S. Wu, Ruin probabilities of the Parisian type for small claims. Unpublished manuscript (2008). http://stats.lse.ac.uk/angelos/docs/paper5a.pdf
H. Guérin, J.-F. Renaud, On the distribution of cumulative Parisian ruin. Insur. Math. Econ. 73, 116–123 (2017)
D. Landriault, J.-F. Renaud, X. Zhou, An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Probab. 16, 583–607 (2014)
R. Loeffen, I. Czarna, Z. Palmowski, Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609 (2013)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mandjes, M., Boxma, O. (2023). Advanced Bankruptcy Concepts. In: The Cramér–Lundberg Model and Its Variants. Springer Actuarial(). Springer, Cham. https://doi.org/10.1007/978-3-031-39105-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-39105-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-39104-0
Online ISBN: 978-3-031-39105-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)