Abstract
In this chapter we focus on a multivariate variant of the conventional Cramér-Lundberg model. Imposing an ordering condition on the individual net cumulative claim processes, it turns out that the distribution of the joint running maximum can be derived, which can be used to evaluate ruin probabilities in a multivariate context. We start by analyzing the bivariate case, to then extend the reasoning to the higher-dimensional setting. The method relies upon the Kolmogorov forward equations underlying the associated queueing process. The solution reveals a so-called quasi-product form structure. We also point out how the results from this section can be translated into corresponding results for tandem queueing networks. We conclude the chapter by deriving the corresponding multivariate Gerber-Shiu metrics (including ruin times, undershoots, and overshoots).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H. Albrecher, E. Cheung, H. Liu, J.-K. Woo, A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process. Insur. Math. Econ. 103, 96–118 (2022)
F. Avram, Z. Palmowski, M. Pistorius, A two-dimensional ruin problem on the positive quadrant. Insur. Math. Econ. 42, 227–234 (2008)
F. Avram, Z. Palmowski, M. Pistorius, Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. Ann. Appl. Probab. 18, 2421–2449 (2008)
F. Baccelli, Two parallel queues created by arrivals with two demands: the M/G/2 symmetrical case. Research Report 426, INRIA-Rocquencourt, 1985
A. Badescu, E. Cheung, L. Rabehasaina, A two-dimensional risk model with proportional reinsurance. J. Appl. Probab. 48, 749–765 (2011)
E. Badila, O. Boxma, J. Resing, E. Winands, Queues and risk models with simultaneous arrivals. Adv. Appl. Probab. 46, 812–831 (2014)
J. Cohen, Analysis of Random Walks (IOS Press, Amsterdam, 1992)
S. de Klein, Fredholm Integral Equations in Queueing Analysis. Ph.D. Thesis, University of Utrecht, 1988
K. Dȩbicki, T. Dieker, T. Rolski, Quasi-product forms for Lévy-driven fluid networks. Math. Oper. Res. 32, 629–647 (2007)
K. Dȩbicki, M. Mandjes, M. van Uitert, A tandem queue with Lévy input: a new representation of the downstream queue length. Probab. Eng. Inf. Sci. 21, 83–107 (2007)
M. Evgrafov, Analytic Functions (Dover Publications, New York, 1978)
L. Gong, A. Badescu, E. Cheung, Recursive methods for a multi-dimensional risk process with common shocks. Insur. Math. Econ. 50, 109–120 (2012)
O. Kella, Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Probab. 3, 682–695 (1993)
O. Kella, Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Probab. 6, 186–199 (1996)
O. Kella, Stochastic storage networks: stationarity and the feedforward case. J. Appl. Probab. 34, 498–507 (1997)
O. Kella, Non-product form of two-dimensional fluid networks with dependent Lévy inputs. J. Appl. Probab. 37, 1117–1122 (1997)
O. Kella, W. Whitt, A tandem fluid network with Lévy input, in Queueing and Related Models, ed. by U. Bhat, I. Basawa (Oxford University Press, Oxford, 1992), pp. 112–128
M. Mandjes, M. van Uitert, Sample-path large deviations for tandem and priority queues with Gaussian inputs. Ann. Appl. Probab. 15, 1193–1226 (2005)
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). Multivariate Ruin. In: The Cramér–Lundberg Model and Its Variants. Springer Actuarial(). Springer, Cham. https://doi.org/10.1007/978-3-031-39105-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-39105-7_7
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)