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On the distribution of the surplus prior to ruin. (English) Zbl 0770.62090
Consider the classical risk model $$Z_ t=u+ct-X_ t$$, where $$u$$ is the initial surplus, $$c$$ is the premium rate with a positive loading and $$X_ t$$ are aggregate claims up to time $$t$$. It is supposed that $$X_ t$$ satisfies the standard assumption of a compound Poisson process with constant intensity $$\lambda$$. Let $$\psi(u)$$ denote the probability of ultimate ruin starting with the initial capital $$u$$ and let $$T$$ denote the time of ruin. Then $$\psi(u)=P(T<\infty|\;Z_ 0=u)$$. The quantity $$G(u,y)=P(T<\infty, Z_ T>-y| Z_ 0=u)$$ denotes the probability that ruin occurs from initial surplus $$u$$ and that the deficit at the time of ruin is less than $$y$$. Let $$Z_{\widetilde T}$$ denote the surplus immediately prior to ruin (given that ruin occurs) and $$F(u,x)=P(T<\infty, Z_{\widetilde T}<x|\;Z_ 0=u)$$. The results derive $$F(u,x)$$ as a function of $$\psi(u)$$ and $$G(u,y)$$.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics
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##### References:
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