Factorization identity for semicontinuous process defined on a Markov chain (Q2771520)

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.