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Joint distributions of some actuarial random vectors containing the time of ruin. (English) Zbl 1024.62045
Summary: We introduce the renewal measure of the defective renewal sequence constituted by the zero points of the classical risk model \(U(t)\), \(t\geq 0\). The density function of this renewal measure is derived. By this density function together with the strong Markov property of the surplus process, we obtain explicit expressions for the ruin probability and the joint distributions of the actuarial random vectors \((T,U(T^{-}), |U(T)|)\) and \[ (T,U(T^{-}),|U(T)|,\;\sup_{0\leq t<T} U(t),\;\sup_{T\leq t<L} U(t),\;\inf_{0\leq t<L} U(t)), \] where \(T\) represents the time of ruin and \(L\) the time of the surplus process leaving zero ultimately. Finally, a special case with the claim amount being exponentially distributed is considered.

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