Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-17T08:42:23.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectrally negative Lévy processes with applications in risk theory

Part of: Special processes Applications

Published online by Cambridge University Press:  01 July 2016

Hailiang Yang  and
Lianzeng Zhang
Hailiang Yang*
Affiliation:
The University of Hong Kong
Lianzeng Zhang*
Affiliation:
Nankai University
*
∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: hlyang@hkusua.hku.hk
∗∗ Postal address: Department of Risk Management and Insurance, Nankai University, Tianjin 300071, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, results on spectrally negative Lévy processes are used to study the ruin probability under some risk processes. These processes include the compound Poisson process and the gamma process, both perturbed by diffusion. In addition, the first time the risk process hits a given level is also studied. In the case of classical risk process, the joint distribution of the ruin time and the first recovery time is obtained. Some results in this paper have appeared before (e.g., Dufresne and Gerber (1991), Gerber (1990), dos Reis (1993)). We revisit them from the Lévy process theory's point of view and in a unified and simple way.

Keywords

spectrally negative Levy processessubordinatorrisk processes perturbed by diffusiongamma processruin probabilityfirst hitting timesexponential integralfirst recovery time

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 281 - 291
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertoin, J. (1998). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. (1999a). Some elements on Lévy processes. To appear in Stochastic Processes: Modelling and Simulation. Theory and Methods (Handbook of Statistics 20). North-Holland, Amsterdam.Google Scholar
Bertoin, J. (1999b). Subordinators: examples and applications. In Lectures on Probability Theory and Statistics (Lecture Notes in Math. 1717), ed. Bernard, P.. Springer, New York.CrossRefGoogle Scholar
Dos Reis, A. E. (1993). How long is the surplus below zero? Insurance: Math. Econ. 12, 23–38.Google Scholar
Dufresne, F. and Gerber, H. U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance: Math. Econ. 10, 51–59.Google Scholar
Dufresne, F. and Gerber, H. U. (1993). The probability of ruin for the inverse Gaussian and related processes. Insurance: Math. Econ. 12, 9–22.Google Scholar
Dufresne, F., Gerber, H. U. and Shiu, E. S. W. (1991). Risk theory with the gamma process. Astin Bull. 21, 177–192.Google Scholar
Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scand. Actuarial J. 1998, 59–74.CrossRefGoogle Scholar
Gerber, H. U. (1990). When does the surplus reach a given target? Insurance: Math. Econ. 9, 115–119.Google Scholar
Gerber, H. U. (1992). On the probability of ruin for infinitely divisible claim amount distributions. Insurance: Math. Econ. 11, 163–166.Google Scholar
Gerber, H. U. and Landry, B. (1998). On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263–276.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1994). Martingale approach to pricing perpetual American options. Astin Bull. 24, 195–220.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.Google Scholar
Lundberg, F. (1903a). Approximerad Framställning av Sannolikhetsfunktionen. Almqvist and Wiksell, Uppsala.Google Scholar
Lundberg, F. (1903b). Återförsäkring av Kollektivrisker. Almqvist and Wiksell, Uppsala.Google Scholar
Lundberg, F. (1926). Försäkringsteknisk Riskutjämning. F. Englunds, Stockholm.Google Scholar
Marchal, P. (1998). Distribution of the occupation time for a Lévy process at passage times at 0. Stoch. Proc. Appl. 74, 123–131.Google Scholar
Prabhu, N. U. (1970). Ladder variables for a continuous stochastic process. Z. Wahrscheinlichkeitsth. 16, 157–164.Google Scholar
Prabhu, N. U. (1998). Stochastic Storage Process, Queues, Insurance Risk, Dams, and Data Communication, 2nd edn. Springer, New York.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.CrossRefGoogle Scholar
Spiegel, M. R. (1965). Laplace Transforms. McGraw-Hill, New York.Google Scholar
Zolotarev, V. M. (1964). The first passage time of a level and the behavior at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–661.Google Scholar