Factorization identity for semicontinuous process defined on a Markov chain. (English. Ukrainian original) Zbl 0998.60043

Gusak, D. V.

Theory Probab. Math. Stat. 64, 37-50 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 35-47 (2001).

The paper deals with the 2-dimensional Markov process \(Z(t)=\{\xi(t)\); \(x(t)\), \(t\geq 0\}\), where \(\{x(t)\), \(t\geq 0\}\) is a Markov chain with a finite state space and \(\xi(t)\) is a process with conditionally independent increments, constructed on the base of increments of the Poisson processes with negative jumps and positive drifts. Using more precise form for matrix components in factorization identity the author finds rather simple relations for distributions of \(\sup_{0\leq u\leq t} \xi(u)\) and \(\inf_{0\leq u\leq t} \xi(u)\) as well as their asymptotic behaviour as \(t\to\infty\). These results are successfully used for the investigation of the ruin probabilities since the risk processes in Markovian environment often can be considered as semicontinuous Poisson processes on a Markov chain.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60G50	Sums of independent random variables; random walks
62P05	Applications of statistics to actuarial sciences and financial mathematics
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

stochastic process; independent increments; Poisson process; negative jumps; extreme values; characteristic function; factorization identity; Markov chain; risk process; ruin probability