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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.

##### MSC:
 91B30 Risk theory, insurance (MSC2010)
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##### References:
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