On optimal stopping for time-dependent gain function

General questions of optimal stopping for an inhomogeneous Markov process for a time-dependent gain function are investigated. The connection between the optimal stopping problems for an inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. A detailed characterization of a value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous Markov process is given. The form of $\varepsilon$-optimal (optimal) stopping times is also found.