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Ruin probabilities for competing claim processes

Part of: Markov processes Special processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Miljenko Huzak ,
Mihael Perman ,
Hrvoje Šikić  and
Zoran Vondraček
Miljenko Huzak*
Affiliation:
University of Zagreb
Mihael Perman*
Affiliation:
University of Ljubljana
Hrvoje Šikić*
Affiliation:
University of Zagreb
Zoran Vondraček*
Affiliation:
University of Zagreb
*
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
∗∗ Postal address: Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
∗ Postal address: Department of Mathematics, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
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Abstract

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Keywords

Risk theoryruin probabilityPollaczek–Khinchin formulasubordinatorspectrally negative Lévy process

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; L'evy processes 60J75: Jump processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 679 - 690
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Doney, R. A., and Maller, R. M. (2002). Stability and attraction to normality for Lévy processes at zero and infinity. J. Theoret. Prob. 15, 751–792.10.1023/A:1016228101053Google Scholar
Dufresne, F., and Gerber, H. U. (1991). Risk theory for a compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.10.1016/0167-6687(91)90023-QGoogle Scholar
Dufresne, F., Gerber, H. U., and Shiu, E. W. (1991). Risk theory with gamma process. Astin Bull. 21, 177–192.10.2143/AST.21.2.2005362Google Scholar
Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1998, 59–74.10.1080/03461238.1998.10413992Google Scholar
Huzak, M., Perman, M., Sikić, H. and Vondraček, Z. (2004). Ruin probabilities for general perturbed risk processes. To appear in Ann. Appl. Prob.Google Scholar
Revuz, D., and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.10.1007/978-3-662-21726-9Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.10.1002/9780470317044Google Scholar
Schmidli, H. (2001). Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion. Insurance Math. Econom. 28, 13–20.10.1016/S0167-6687(00)00062-7Google Scholar
Takács, L. (1977). Combinatorial Methods in the Theory of Stochastic Processes. Krieger, Huntington, NY.Google Scholar
Winkel, M. (2001). Electronic foreign exchange markets and the level passage event for multivariate subordinators. Preprint. Available at http://www.stats.ox.ac.uk/∼winkel/.Google Scholar
Yang, H., and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. Appl. Prob. 33, 281–291.10.1017/S0001867800010740Google Scholar