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On ruin probability for a risk process with bounded reserves. (English. Ukrainian original) Zbl 0955.60072
Theory Probab. Math. Stat. 60, 53-65 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 46-58 (1999).
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.

60J05 Discrete-time Markov processes on general state spaces
60J45 Probabilistic potential theory