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Ruin problem for an inhomogeneous semicontinuous integer-valued process. (English. Ukrainian original) Zbl 0972.60034
Ukr. Math. J. 52, No. 2, 234-248 (2000); translation from Ukr. Mat. Zh. 52, No. 2, 208-219 (2000).
Various boundary problems for homogeneous semi-continuous Poisson processes were studied by V. S. Korolyuk, M. S. Bratijchuk and B. Pirdzhanov (1975-1990) [see, for example, V. S. Korolyuk, “Limit problems for complicated Poisson processes” (1975; Zbl 0329.60040); V. S. Korolyuk, M. S. Bratijchuk and B. Pirdzhanov, “Boundary problems for random walks” (Ashkhabad, 1990); M. S. Bratijchuk and D. V. Gusak, “Boundary problems for processes with independent increments” (1990; Zbl 0758.60074)]; and for semi-continuous homogeneous Poisson processes with integer-valued jumps and respective boundary problems by D. V. Gusak [Theory Probab. Math. Stat. 59, 41-46 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 41-46 (1998; Zbl 0958.60076)]. Inhomogeneous processes $$\xi(t)$$, given on superposition of two renewal processes, and boundary problems for them were investigated by V. S. Korolyuk, B. Pirliev, M. Bratijchuk and B. Pirdzhanov (1984-1990) [see M. S. Bratijchuk and B. Pirliev, Ukr. Math. J. 37, 560-566 (1985), translation from Ukr. Mat. Zh. 37, No. 6, 689-695 (1985; Zbl 0598.60077); V. S. Korolyuk and B. Pirliev, ibid. 36, 349-352 (1984), resp. ibid. 36, No. 4, 433-436 (1984; Zbl 0549.60063); M. Bratijchuk and B. Pirdzhanov, “Theory of random processes and its applications” (Kyïv, 1990)].
In the book by D. V. Gusak [“Boundary problems for processes with independent increments on Markov chains and for semi-Markov processes” (1998; Zbl 0909.60051)] for such processes a factorization approach were developed for the study of boundary problems, connected with distributions of extremes of these processes, moments of first achievement of the extremes or fixed level, excess-process or jump, that covers fixed level. The main attention was paid to the case when jumps of the process $$\xi(t)$$ are continuously distributed.
In the given paper, the case, when jumps are integer-valued is considered, and the ruin probability is studied, connected with a random walk in a bounded interval. The notion of resolvent, introduced by V. S. Korolyuk (loc. cit.) for homogeneous semi-continuous Poisson processes, is generalized. For the process $$\xi(t)=\xi_{1}(t)+\psi(t),$$ $$t\geq 0,$$ $$\xi(0)=0,$$ which is inhomogeneous in time, a ruin probability associated with corresponding random walk in a finite interval, where $$\xi_{1}(t)$$ is a homogeneous Poisson process with positive integer-valued jumps, $$\psi(t)$$ is an inhomogeneous lower semi-continuous process with integer-valued jumps $$\xi_{n}\geq 1$$, is studied.
##### MSC:
 60G50 Sums of independent random variables; random walks 60J50 Boundary theory for Markov processes 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010) 60G51 Processes with independent increments; Lévy processes
##### Keywords:
ruin problem; random walk; Poisson process
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