The Annals of Applied Probability

Ruin probabilities and overshoots for general Lévy insurance risk processes

Claudia Klüppelberg, Andreas E. Kyprianou, and Ross A. Maller

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We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Lévy processes.

Article information

Ann. Appl. Probab., Volume 14, Number 4 (2004), 1766-1801.

First available in Project Euclid: 5 November 2004

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Zentralblatt MATH identifier

Primary: 60J30 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes 90A46
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G17: Sample path properties 60J15

Insurance risk process Lévy process conditional limit theorem first passage time overshoot ladder process ruin probability subexponential distributions convolution equivalent distributions heavy tails


Klüppelberg, Claudia; Kyprianou, Andreas E.; Maller, Ross A. Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 (2004), no. 4, 1766--1801. doi:10.1214/105051604000000927.

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  • Asmussen, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the $GI/G/1$ queue. Adv. in Appl. Probab. 14 143–170.
  • Asmussen, S. (2001). Ruin Probabilities. World Scientific, Singapore.
  • Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl. 64 103–125.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7 156–169.
  • Bertoin, J. and Doney, R. (1994). Cramér's estimate for \LL processes. Statist. Probab. Lett. 21 363–365.
  • Bertoin, J. and Doney, R. (1996). Some asymptotic results for transient random walks. Adv. in Appl. Probab. 28 207–226.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Chover, J., Ney, P. and Wainger, S. (1973). Degeneracy properties of subcritical branching processes. Ann. Probab. 1 663–673.
  • Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43 347–365.
  • Doney, R. A. and Maller, R. A. (2002). Stability and attraction to normality for Lévy processes at zero and infinity. J. Theoret. Probab. 15 751–792.
  • Dufresne, F. and Gerber, H. U. (1998). On the discounted penalty at ruin in a jump-diffusion and perpetual put option. Insurance Math. Econom. 22 263–276.
  • Embrechts, P. (1983). A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Probab. 20 537–544.
  • Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stochastic Process. Appl. 13 263–278.
  • Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335–347.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Embrechts, P. and Samorodnitsky, G. (2003). Ruin problem, operational risk and how fast stochastic processes mix. Ann. Appl. Probab. 13 1–36.
  • Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55–72.
  • Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185 371–381.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications II. Wiley, New York.
  • Furrer, H. (1998). Risk processes perturbed by $\al$-stable Lévy motion. Scand. Actuar. J. 59–74.
  • Furrer, H. and Schmidli, H. (1994). Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance Math. Econom. 15 23–36.
  • Gerber, H. U. and Shiu, E. S. W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math. Econom. 21 129–137.
  • Goldie, C. M. and Klüppelberg, C. (1998). Subexponential distributions. In A Practical Guide to Heavy Tails (R. Adler, R. Feldman and M. S. Taqqu, eds.) 435–459. Birkhäuser, Boston.
  • Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.
  • Huzak, M., Perman, M., Hrvoje, S. and Vondracek, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14 1378–1397.
  • Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Probab. Theory Related Fields 82 259–269.
  • Klüppelberg, C. (1989). Estimation of ruin probabilities by means of hazard rates. Insurance Math. Econom. 8 279–285.
  • Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy tailed steps. Ann. Appl. Probab. 10 1025–1064.
  • Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 407–424.
  • Rogozin, B. A. (2000). On the constant in the definition of subexponential distributions. Theory Probab. Appl. 44 409–412.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Schmidli, H. (2001). Distribution of the first ladder height of a stationary risk process perturbed by $\alpha$-stable \LL motion. Insurance Math. Econom. 28 13–20.
  • Teugels, J. (1968). Renewal theorems when the first or second moment is infinite. Ann. Math. Statist. 39 1210–1219.
  • Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stochastic Process. Appl. 5 27–37.
  • Vigon, V. (2002). Votre Lévy ramp-t-il? J. London Math. Soc. 65 243–256.
  • Winkel, M. (2002). Electronic foreign exchange markets and the level passage event for multivariate subordinators. MaPhySto Research Report 41, Univ. Aarhus.
  • Zolotarev, M. V. (1986). One Dimensional Stable Distributions. Amer. Math. Soc., Providence, RI.