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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.

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