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.

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.

Reviewer: A.V.Swishchuk (Kyïv)