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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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