On the expected discounted penalty function at ruin of a surplus process with interest.

*(English)*Zbl 1074.91027The 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.

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.

Reviewer: Emilia Di Lorenzo (Napoli)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

44A10 | Laplace transform |

45D05 | Volterra integral equations |

91B70 | Stochastic models in economics |

##### Keywords:

ruin penalty function; surplus prior to ruin; deficit at ruin; Laplace transform; Volterra equation
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\textit{J. Cai} and \textit{D. C. M. Dickson}, Insur. Math. Econ. 30, No. 3, 389--404 (2002; Zbl 1074.91027)

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##### References:

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