On the expected discounted penalty function at ruin of a surplus process with interest.
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.
scientific article; zbMATH DE number 2004025