×

zbMATH — the first resource for mathematics

Exponential behavior in the presence of dependence in risk theory. (English) Zbl 1097.62110
Summary: We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G50 Sums of independent random variables; random walks
91B30 Risk theory, insurance (MSC2010)
60K05 Renewal theory
PDF BibTeX XML Cite
Full Text: DOI Euclid
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. Appl. Prob. 14 , 143–170. JSTOR: · Zbl 0501.60076 · doi:10.2307/1426737 · links.jstor.org
[2] Asmussen, S. (2000). Ruin Probabilities . World Scientific, Singapore. · Zbl 0960.60003
[3] Baltrūnas, A. (2001). Some asymptotic results for transient random walks with applications to insurance risk. J. Appl. Prob. 38 , 108–121. · Zbl 0983.60042 · doi:10.1239/jap/996986647
[4] Breiman, L. (1968). Probability . Addison-Wesley, Reading, MA. · Zbl 0174.48801
[5] Feller, W. (1966). An Introduction to Probability Theory and its Applications , Vol. 2. John Wiley, New York. · Zbl 0138.10207
[6] Genest, C. and Rivest, L. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88 , 1034–1043. JSTOR: · Zbl 0785.62032 · doi:10.2307/2290796 · links.jstor.org
[7] Hürlimann, W. (2000). A Spearman multivariate distribution with fixed margins – theory and applications.
[8] Joe, H. (1997). Multivariate Models and Dependence Concepts . Chapman and Hall, London. · Zbl 0990.62517
[9] Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions , Vol. 1, Models and Applications . John Wiley, New York. · Zbl 0946.62001
[10] Nelsen, R. (1999). An Introduction to Copulas . Springer, Berlin. · Zbl 0909.62052
[11] Prabhu, N. U. (1965). Stochastic Processes. Basic Theory and its Applications . Macmillan, New York. · Zbl 0138.10403
[12] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1999). Stochastic Processes for Insurance and Finance . John Wiley, Chichester. · Zbl 0940.60005
[13] Veraverbeke, N. and Teugels, J. L. (1975). The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 12 , 279–288. JSTOR: · Zbl 0307.60061 · doi:10.2307/3212441 · links.jstor.org
[14] Veraverbeke, N. and Teugels, J. L. (1976). The exponential rate of convergence of the distribution of the maximum of a random walk. II. J. Appl. Prob. 13 , 733–740. JSTOR: · Zbl 0353.60072 · doi:10.2307/3212528 · links.jstor.org
[15] Whittaker, E. and Watson, G. (1963). A Course of Modern Analysis , 4th edn. Cambridge University Press. · Zbl 0951.30002
[16] Widder, D. (1942). The Laplace Transform . Princeton University Press. · Zbl 0061.23304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.