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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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