Overshoot functionals for almost semicontinuous processes defined on a Markov chain. (Ukrainian, English) Zbl 1199.60163

Karnaukh, E. V.

Teor. Jmovirn. Mat. Stat. 76, 45-52 (2007); translation in Theory Probab. Math. Stat. 76, 49-57 (2008).

The distributions of extremal values and over-jump functionals for semicontinuous processes (that is, for those processes that cross either a positive or a negative barrier in a continuous way) defined on a Markov chain are considered by many authors (see, for example, S. Asmussen [Ruin probabilities, Singapore: World Scientific (1997; Zbl 0960.60003)], D.V. Gusak [Boundary value problems for processes with independent increments in risk theory, Kyïv: Instytut Matematyky NAN Ukraïny (2007; Zbl 1199.60001)] and D. Gusak [Theory Stoch. Process. 7(23), No. 1–2, 109–120 (2001; Zbl 0977.60059)]). The distributions of extremal values are considered in the paper [D. V. Gusak and E. V. Karnaukh, Theory Stoch. Process. 11, No. 27, Part 1–2, 40–47 (2005; Zbl 1142.60343)] for almost semicontinuous processes (that is, for those processes that cross either a positive or a negative barrier by means of exponential jumps only). Under some assumptions these processes can be considered as surplus risk processes with random premiums in a Markov environment. The distributions of some over-jump functionals are studied in this paper for lower almost semicontinuous processes defined on a Markov chain.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G50	Sums of independent random variables; random walks
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60K10	Applications of renewal theory (reliability, demand theory, etc.)
60K15	Markov renewal processes, semi-Markov processes

Keywords:

semicontinuous process; over-jump functional; Markov chain

Citations:

Zbl 0960.60003; Zbl 0977.60059; Zbl 1142.60343; Zbl 1199.60001