Abstract
In this chapter we discuss the conventional Cramér-Lundberg ruin model. The focus lies on evaluating transforms related to the all-time and time-dependent ruin probabilities (where the latter, in our setting, concerns ruin before an exponentially distributed amount of time). An important role is played by a duality with the M/G/1 queueing model. We present four independent analysis techniques; they differ in the sense whether the ruin model or the corresponding queueing model (or a mixture of both) has been used.
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References
J. Abate, W. Whitt, Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43 (1995)
N. Asghari, P. den Iseger, M. Mandjes, Numerical techniques in Lévy fluctuation theory. Methodol. Comput. Appl. Probab. 16, 31–52 (2014)
S. Asmussen, Applied Probability and Queues, 2nd edn. (Springer, New York, 2003)
S. Asmussen, H. Albrecher, Ruin Probabilities (World Scientific, Singapore, 2010)
J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1998)
P. Brill, Level Crossing Methods in Stochastic Models, 2nd edn. (Springer, New York, 2017)
J. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1969)
H. Cramér, On the mathematical theory of risk. Skandia Jubilee4 (1930)
K. Dȩbicki, M. Mandjes, Queues and Lévy Fluctuation Theory (Springer, New York, 2015)
P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events for Insurance and Finance. Applications of Mathematics, vol. 33 (Springer, Berlin, 1997)
H. Gerber, E. Shiu, The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insur. Math. Econ. 21, 129–137 (1997)
H. Gerber, E. Shiu, On the time value of ruin. N. Am. Actuar. J. 2, 48–78 (1998)
G. Grimmett, D. Stirzaker, Probability and Random Processes (Oxford University Press, Oxford, 2001)
P. Gruntjes, P. den Iseger, M. Mandjes, A Wiener-Hopf based approach to numerical computations in fluctuation theory for Lévy processes. Math. Methods Oper. Res. 78, 101–118 (2013)
A. Hasofer, On the integrability, continuity and differentiability of a family of functions introduced by L. Takács. Ann. Math. Stat. 34, 1045–1049 (1963)
P. den Iseger, Numerical transform inversion using Gaussian quadrature. Probab. Eng. Inf. Sci. 20, 1–44 (2006)
O. Kella, W. Whitt, Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probab. 33, 1169–1180 (1992)
A. Kyprianou, Gerber-Shiu Risk Theory (Springer, New York, 2013)
A. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, 2nd edn. (Springer, New York, 2014)
F. Lundberg, Approximerad Framställning af Sannolikhetsfunktionen: Aterförsäkering af Kollektivrisker. Ph.D Thesis, Almqvist & Wiksell, 1903
F. Lundberg, Försäkringsteknisk Riskutjämning: Teori (F. Englunds Boktryckeri A.B., Stockholm, 1926)
N. Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication, 2nd edn. (Springer, New York, 1998)
T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic Processes for Insurance and Finance (Wiley, Chichester, 2009)
K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, revised version (Cambridge University Press, Cambridge, 2013)
L. Takács, Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101–129 (1955)
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Mandjes, M., Boxma, O. (2023). Cramér-Lundberg Model. In: The Cramér–Lundberg Model and Its Variants. Springer Actuarial(). Springer, Cham. https://doi.org/10.1007/978-3-031-39105-7_1
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DOI: https://doi.org/10.1007/978-3-031-39105-7_1
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