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A note on a class of delayed renewal risk processes. (English) Zbl 1114.60068
A delayed or modified renewal process is considered where the time until the first renewal (claim) is a positive random variable with density $k_1(t) = q \frac{e^{-\alpha t} \int_t^\infty e^{\alpha y} k(y) dy} {\int_0^\infty e^{\alpha y} \bar{K}(y) dy} +(1-q) \alpha e^{-\alpha t} .$ $$K(t)$$ is the distribution function of the random time between the $$(i-1)$$th and $$i$$th claim, $$i=2,3,\dots$$, $$k$$ is its probability density, and $$\bar{K}(t) = 1- K(t)$$. This choice of $$k_1$$ includes two important cases, an equilibrium distribution and the exponential distribution. The Gerber-Shiu discounted penalty function, which is a generalization of the ruin probability, is considered. A mathematically tractable formula is derived for the Gerber-Shiu function (Theorem 2.1). Further, ruin probabilities and those related to the surplus immediately prior to ruin or to the deficit at ruin can be evaluated for specified parameters. An example for the ruin probability is given and it is shown, that it can be expressed in terms of the ruin probability in the ordinary renewal risk model.

##### MSC:
 60K05 Renewal theory 60K10 Applications of renewal theory (reliability, demand theory, etc.) 91B30 Risk theory, insurance (MSC2010)
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