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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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##### References:
 [1] Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. The S.S. Huebner Foundation for Educational University of Pennsylvania. [2] Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047 [3] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, Astin bulletin, 17, 151-163, (1987) [4] Grandell, J., 1991. Aspects of Risk Theory. Springer, New York. · Zbl 0717.62100 [5] Picard, R., On some measures of the severity of ruin in the classical Poisson model, Insurance: mathematics and economics, 14, 107-117, (1994) · Zbl 0813.62093 [6] Revuz, D., Yor, M., 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002 [7] Vylder, F.E.D.; Goovaerts, M.J., Explicit finite-time and infinite-time ruin probability in the continuous case, Insurance: mathematics and economics, 24, 155-172, (1999) · Zbl 0963.91062 [8] Wei, L.; Wu, R., The joint distributions of several important actuarial diagnostics in the classical risk model, Insurance: mathematics and economics, 30, 451-462, (2002) · Zbl 1071.91027 [9] Wu, R.; Zhang, C.; Wang, G., About a joint distribution in classical risk model, Acta mathematicae applicatae sinicas, 25, 554-560, (2002), (in Chinese) · Zbl 1022.62104
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