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On the distribution of surplus immediately after ruin under interest force. (English) Zbl 1012.91027
A compound Poisson model is considrered \(dU_\delta(t)= pdt+ U_\delta(t) \delta dt-dX(t)\). Here \(U_\delta(t)\) is the value of the reserve at time \(t\), \(p\) is the premium rate that the insurance company receives, \(\delta\) is the interest force, and \(X(t)= \sum^{N(t)}_{j=1} Y_j\), where \(N(t)\) is a homogeneous Poisson process counting the number of claims in the time interval \((0,t]\), and \(Y_j\) is the amount of the \(j\)th claim. Using the techniques of B. Sundt and J. L. Tengels [Isur. Math. Econ. 16, 7-22 (1995; Zbl 0838.62098)], the equations are obtained for the probability of ruin \(G_\delta (u,y)\) beginning with initial reserve \(u\) and the deficit at most \(y> 0\) immediately after the claim causing ruin. The bounds for \(G_\delta (0,y)\) are derived. In the case \(\delta=0\), the asymptotic result for \(G_\delta (u,y)\), as \(u\to +\infty\), is obtained, which generalizes the result in J. Grandell [Aspects of Risk Theory. Springer, Berlin (1991; Zbl 0717.62100)] to the case where the severity of ruin is taken into account.

91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
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