zbMATH — the first resource for mathematics

The deficit at ruin in the stationary renewal risk model. (English) Zbl 1092.62115
The paper deals with a stationary renewal risk model, or, equivalently, a stationary Sparre Andersen model. Let \(U(t)\) be the risk process. Denote by \(T\) the relevant time of ruin. Then \(| U(T)| \) is the deficit at ruin (if ruin occurs). The authors present a number of results concerning the distribution of \(| U(T)| \). A mixture representation for the conditional distribution of the deficit of ruin (given that ruin occurs) is derived, as well as a stochastic decomposition involving the residual lifetime associated with the maximum aggregate loss. Special cases where the claim amounts follow a distribution belonging to a family of distributions with decreasing failure rate, or from a phase-type family of distributions, are studied. For instance, it is proved that when individual claims have a phase-type distribution, then the deficit at ruin is also of phase-type.

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
[1] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[2] Barlow R, Statistical Theory of Reliability and Life Testing: Probability Models (1975)
[3] Bowers N, Actuarial Mathematics (1997)
[4] Cai J, Proceedings of Statistics 2001 Canada pp pp. 114–131– (2002)
[5] DOI: 10.1016/0167-6687(89)90028-0 · Zbl 0682.62083 · doi:10.1016/0167-6687(89)90028-0
[6] Drekic S, Scandinavian Actuarial Journal pp 105– (2004)
[7] Embrechts P, Modelling Extremal Events (1997) · doi:10.1007/978-3-642-33483-2
[8] DOI: 10.2307/3215038 · Zbl 0806.60075 · doi:10.2307/3215038
[9] Feller W, An Introduction to Probability Theory and Its Applications 2 (1971) · Zbl 0219.60003
[10] Grandell J, Aspects of Risk Theory (1991) · doi:10.1007/978-1-4613-9058-9
[11] DOI: 10.1137/1.9780898719734 · Zbl 0922.60001 · doi:10.1137/1.9780898719734
[12] Neuts M, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach (1981) · Zbl 0469.60002
[13] DOI: 10.1002/9780470317044 · doi:10.1002/9780470317044
[14] DOI: 10.2307/3214200 · Zbl 0599.60085 · doi:10.2307/3214200
[15] Wikstad N, ASTIN Bulletin 6 pp 147– (1971) · doi:10.1017/S0515036100010874
[16] DOI: 10.1016/S0167-6687(02)00122-1 · Zbl 1039.62097 · doi:10.1016/S0167-6687(02)00122-1
[17] Willmot G, Journal of Applied Probability (2004)
[18] Willmot G, Lundberg Approximations for Compound Distributions with Insurance Applications (2001) · doi:10.1007/978-1-4613-0111-0
[19] DOI: 10.1239/aap/1005091359 · Zbl 1003.60081 · doi:10.1239/aap/1005091359
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.