Perturbation and truncation bounds for one class of Markov processes of birth-and-death type with catastrophes

A class of inhomogeneous continuous-time Markov chains of birth-and-death type with a countable state space is considered. Two types of additional transitions are allowed in the chain, which bring it either to the boundary state or to the state adjacent to it. It is assumed that with an increase in the state number, the birth (death) intensities monotonically decrease (increase). Perturbation bounds are obtained using special weighted norms associated with the total variation. An estimate of the approximation error, when one replaces the original chain by a process with a finite number of states, is constructed. For the case when all the intensities are state-dependent, conditions are provided (using the logarithmic norm method), which guarantee (weak) ergodicity in the norm of total variation. Results are accompanied by illustrative examples.