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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: [email protected]
∗∗ Postal address: Department of Risk Management and Insurance, Nankai University, Tianjin 300071, China.
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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 

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