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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.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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