an:01516530
Zbl 0955.60072
Kartashov, N. V.
On ruin probability for a risk process with bounded reserves
EN
Theory Probab. Math. Stat. 60, 53-65 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 46-58 (1999).
00070564
1999
j
60J05 60J45
ruin probability; storage process; risk theory; Poisson process; absorbing upper barrier
Let \(U(t)\) be a classical risk process
\[
U(t)=u+ct-X(t), \tag{1}
\]
where \(u\) and \(c\) are constants and \(X(t)=\sum_{k=1}^{N(t)}X_k\), where \(N(t)\) is a Poisson process with intensity \(\lambda\), and \(X_{k}\), \(k\geq 1,\) are independent identically distributed random variables. The random variable \(X_1\) has density \(f(x),\) finite mean \(\mu=E X_1\) and \(E X_1^2 <\infty.\) The author considers the following limit modification of the process (1),
\[
dU_{b}(t)=D U(t)-\mathbf{1}_{U_{b}(t)=b} c dt,\tag{2}
\]
where \(b\geq U\) is a limit level. Process (2) coincides with process (1) until the process \(X(t)\) achieves the level \(b\) and stay on this level until the first jump of the process \(X(t).\) The process \(U_{b}\) may be interpreted as a risk process with bounded reserves. The Laplace transform of the distribution of the ruin time, probability of ruin on certain interval and other characteristics of the process \(U_{b}\) are found.
Yu.V.Kozachenko (Ky??v)