an:00167973
Zbl 0770.62090
Dickson, David C. M.
On the distribution of the surplus prior to ruin
EN
Insur. Math. Econ. 11, No. 3, 191-207 (1992).
00012159
1992
j
62P05
aggregate claim amount; severity of ruin; surplus prior to ruin; recursive calculation; classical risk model; initial surplus; premium rate; compound Poisson process; constant intensity; probability of ultimate ruin
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 \(G(u,y)\).
T.Mikosch (Z??rich)