## The Annals of Applied Probability

### Ruin probabilities and decompositions for general perturbed risk processes

#### Abstract

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Lévy process with zero expectation. We derive a Pollaczek–Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

#### Article information

Source
Ann. Appl. Probab., Volume 14, Number 3 (2004), 1378-1397.

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736289

Digital Object Identifier
doi:10.1214/105051604000000332

Mathematical Reviews number (MathSciNet)
MR2071427

Zentralblatt MATH identifier
1061.60075

#### Citation

Huzak, Miljenko; Perman, Mihael; Šikić, Hrvoje; Vondraček, Zoran. Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Probab. 14 (2004), no. 3, 1378--1397. doi:10.1214/105051604000000332. https://projecteuclid.org/euclid.aoap/1089736289

#### References

• Asmussen, S. (1996). Ruin Probabilities. World Scientific, Singapore.
• Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
• 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.
• Dufresne, F., Gerber, H. U. and Shiu, E. W. (1991). Risk theory with Gamma process. Astin Bull. 21 177–192.
• Furrer, H. (1998). Risk processes perturbed by $\alpha$-stable Lévy motion. Scand. Actuar. J. 59–74.
• Kingman, J. F. C. (1995). Poisson Processes. Oxford Univ. Press.
• Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2003). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Probab. 14(4).
• Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
• Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. Wiley, New York.
• Sato, K.-I. (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 Lévy motion. Insurance Math. Econom. 28 13–20.
• Yang, H. and Zhang, L. (2001). Spectrally negative Lévy processes with applications in risk theory. Adv. in Appl. Probab. 33 281–291.
• Zolotarev, V. M. (1964). The first passage time of a level and behavior at infinity for a class of processes with independent increments. Theory Probab. Appl. 9 653–661.