an:02004025
Zbl 1074.91027
Cai, Jun; Dickson, David C. M.
On the expected discounted penalty function at ruin of a surplus process with interest
EN
Insur. Math. Econ. 30, No. 3, 389-404 (2002).
00086329
2002
j
91B30 44A10 45D05 91B70
ruin penalty function; surplus prior to ruin; deficit at ruin; Laplace transform; Volterra equation
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.
Emilia Di Lorenzo (Napoli)