Abstract
This chapter provides asymptotic expressions for the ruin probability in the regime that the initial reserve level u grows large. One needs to distinguish between the case that the claim sizes are light-tailed, in which the ruin probability decays essentially exponentially, and the case that the claim sizes are heavy-tailed, in which the ruin probability decays as the complementary distribution function of a residual claim size. While the focus in this chapter is on the asymptotics of the all-time ruin probability \(p(u)\), we also briefly discuss the asymptotics of its finite-time counterpart \(p(u,t).\) We conclude this chapter by some comments on another limiting regime, corresponding to the heavy-traffic scaling in queueing theory.
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References
G. Arfwedson, Research in collective risk theory: the case of equal risk sums. Scand. Actuar. J. 1955, 53–100 (1955)
S. Asmussen, Applied Probability and Queues, 2nd edn. (Springer, New York, 2003)
S. Asmussen, H. Albrecher, Ruin Probabilities (World Scientific, Singapore, 2010)
S. Asmussen, J. Teugels, Convergence rates for M/G/1 queues and ruin problems with heavy tails. J. Appl. Probab. 33, 1181–1190 (1996)
B. von Bahr, Asymptotic ruin probabilities when exponential moments do not exist. Scand. Actuar. J. 1975, 6–10 (1975)
J. Bertoin, R. Doney, Cramér’s estimate for Lévy processes. Stat. Probab. Lett. 21, 363–365 (1994)
N. Bingham, C. Goldie, J. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987)
A. Borovkov, Asymptotic Methods in Queueing Theory (Springer, Berlin, 1976)
O. Boxma, J. Cohen, Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed service time distributions. Queueing Syst. 33, 177–204 (1999)
H. Cramér, On the Mathematical Theory of Risk. Skandia Jubilee 4 (1930)
K. Dȩbicki, M. Mandjes, Queues and Lévy Fluctuation Theory (Springer, New York, 2015)
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications (Springer, New York, 1998)
T. Dieker, Applications of factorization embeddings for Lévy processes. Adv. Appl. Probab. 38, 768–791 (2006)
N. Duffield, N. O’Connell, Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Philos. Soc. 118, 363–374 (1995)
H. Hadwiger, Über die Wahrscheinlichkeit des Ruins bei einer grossen Zahl von Geschäften. Archiv für Mathematische Wirtschafts- und Sozialforschung 6, 131–135 (1940)
J. Kingman, On queues in heavy traffic. J. R. Stat. Soc. B24, 383–392 (1962)
D. Konstantinides, Risk Theory: a Heavy-tail Approach (World Scientific, Singapore, 2018)
F. Lundberg, Försäkringsteknisk Rriskutjämning: Teori (F. Englunds Boktryckeri A.B., Stockholm, 1926)
A. Pakes, On the tail of waiting time distributions. J. Appl. Probab. 12, 555–564 (1975)
E. Pitman, Subexponential distribution functions. J. Aust. Math. Soc. 29A, 337–347 (1980)
C. Segerdahl, When does ruin occur in the collective theory of risk? Scand. Actuar. J. 1955, 22–36 (1955)
D. Siegmund, Sequential Analysis (Springer, New York, 1985)
J. Teugels, Estimation of ruin probabilities. Insur. Math. Econ. 1, 163–175 (1982)
J. Teugels, N. Veraverbeke, Cramér-type estimates for the probability of ruin. CORE Discussion Paper, No 7316 (1973)
W. Whitt, Stochastic Process Limits (Springer, New York, 2002)
B. Zwart, Queueing Systems with Heavy Tails. Ph.D. Thesis, Eindhoven University of Technology, 2001. http://alexandria.tue.nl/extra2/200112999.pdf
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Mandjes, M., Boxma, O. (2023). Asymptotics. In: The Cramér–Lundberg Model and Its Variants. Springer Actuarial(). Springer, Cham. https://doi.org/10.1007/978-3-031-39105-7_2
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DOI: https://doi.org/10.1007/978-3-031-39105-7_2
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