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On the expected discounted penalty function at ruin of a surplus process with interest. (English) Zbl 1074.91027
The paper deals with the ruin problem for an insurer, who receives interest on its surplus at time \(t\), \(U_{\delta}(t)\), at the constant force \(\delta\) per unit time. In particular the expected value of a discounted penalty function at ruin is investigated.
Denoted by \(T_{\delta}\) the time of ruin, \(u\) the inizial surplus and \(\alpha\) a non-negative parameter, the expected value of a discounted function of the surplus immediately prior to ruin and the deficit at ruin is given by \[ \Phi_{\delta,\alpha}(u)=E(w(U(T_{\delta}^-),| U(T_{\delta})| )e^{-\alpha T_{\delta}} I(T_{\delta}<\infty)), \] where \(I(A)\) is the indicator function of a set \(A\) and \(w\) is a non-negative function.
The authors provide an integral equation involving \(\Phi_{\delta,\alpha}\) and obtain the exact solution for \(\Phi_{\delta,0}(0)\).
Finally some classical formulae concerning the distribution of the surplus immediately prior to ruin are generalized to the surplus process with interest.

MSC:
91B30 Risk theory, insurance (MSC2010)
44A10 Laplace transform
45D05 Volterra integral equations
91B70 Stochastic models in economics
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References:
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