The Annals of Applied Probability

Path decomposition of ruinous behavior for a general Lévy insurance risk process

Philip S. Griffin and Ross A. Maller

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Abstract

We analyze the general Lévy insurance risk process for Lévy measures in the convolution equivalence class $\mathcal{S}^{(\alpha)}$, $\alpha>0$, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level $u\to\infty$, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as $u\to\infty$, conditional on ruin occurring under our assumptions. Existing asymptotic results under the $\mathcal{S}^{(\alpha)}$ assumption are synthesized and extended, and proofs are much simplified, by comparison with previous methods specific to the convolution equivalence analyses. Additionally, limiting expressions for penalty functions of the type introduced into actuarial mathematics by Gerber and Shiu are derived as straightforward applications of our main results.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 4 (2012), 1411-1449.

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614199

Digital Object Identifier
doi:10.1214/11-AAP797

Mathematical Reviews number (MathSciNet)
MR2985165

Zentralblatt MATH identifier
1259.60051

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60F17: Functional limit theorems; invariance principles
Secondary: 91B30: Risk theory, insurance 62P05: Applications to actuarial sciences and financial mathematics

Keywords
Lévy insurance risk process convolution equivalence time to ruin overshoot expected discounted penalty function

Citation

Griffin, Philip S.; Maller, Ross A. Path decomposition of ruinous behavior for a general Lévy insurance risk process. Ann. Appl. Probab. 22 (2012), no. 4, 1411--1449. doi:10.1214/11-AAP797. https://projecteuclid.org/euclid.aoap/1344614199


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References

  • [1] 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.
  • [2] Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl. 64 103–125.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [4] Bertoin, J. and Doney, R. A. (1994). Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 363–365.
  • [5] Biffis, E. and Morales, M. (2010). On a generalization of the Gerber–Shiu function to path-dependent penalties. Insurance Math. Econom. 46 92–97.
  • [6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [7] Braverman, M. (1997). Suprema and sojourn times of Lévy processes with exponential tails. Stochastic Process. Appl. 68 265–283.
  • [8] Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Probab. Theory Related Fields 72 529–557.
  • [9] Doney, R. A. (2007). Fluctuation Theory for Lévy Processes. Lecture Notes in Math. 1897. Springer, Berlin.
  • [10] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91–106.
  • [11] Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stochastic Process. Appl. 13 263–278.
  • [12] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [13] Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Am. Actuar. J. 2 48–78.
  • [14] Griffin, P. S. and Maller, R. A. (2011). The time at which a Lévy processes creeps. Electron. J. Probab. 16 2182–2202.
  • [15] Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Probab. Theory Related Fields 82 259–269.
  • [16] Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14 1766–1801.
  • [17] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes With Applications. Springer, Berlin.
  • [18] Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 407–424.
  • [19] Pakes, A. G. (2007). Convolution equivalence and infinite divisibility: Corrections and corollaries. J. Appl. Probab. 44 295–305.
  • [20] Park, H. S. and Maller, R. (2008). Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes. Adv. in Appl. Probab. 40 716–733.
  • [21] Tang, Q. and Wei, L. (2010). Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalence. Insurance Math. Econom. 46 19–31.
  • [22] Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.