Construction of the cost and optimal policies in a game problem of stopping of a Markov process

A minimax version of optimal stopping of a Markov process $\{x_n,\mathscr F_n,\mathbf P_x\}$, $n\ge 0$, with a phase space $(E,\mathscr B)$ (a game of two persons with opposite interests) is considered. The process $x_n$ can be stopped at any moment $n\ge 0$. If it is stopped by the first, second or both of the players, the first one gets, correspondingly, a reward $g(x_n)$, $G(x_n)$ or $e(x_n)$. If the process is not stopped, the first player gets reward $\displaystyle\varliminf_{n\to\infty}C(x_n)$. A recurrent procedure of constructing the cost and structure of optimal and $\varepsilon$-optimal stopping times is investigated.