Optimal stopping in games with continuous time

Let ($\Omega$, $\mathscr F$, $\mathbf P$) be a probability space, $T$ a subset of $[0,\infty)$ such that there exists a countable set $R$, $R\subset T$, and the union of $R$ and the set of all limits from the right over $R$ coincides with $T$. Let $\{\mathscr F_t,\ t\in T\}$ be a non-decreasing and right-continuous in $t$ family of $\sigma$-subalgebras of $\mathscr F$ and $x_t$, $\varphi_t$, $\psi_t$ right-continuous in $t$ $\mathscr F_t$-measurable functions. The process $x_t$ may be stopped by the first player at a moment $t$ if $\varphi_t=1$ and by the second one if $\psi_t=1$. The second player gets from the first one the sum $x_t$ if the process is stopped at time $t$.
Suppose that $\mathbf M(\sup\limits_t|x_t|)<\infty$; then we prove that there exists a $w_t$ such that
(a) $w_t=\overline w_t=\underline w_t$ a.e., $\overline w_t$ and $\underline w_t$ being defined by (1) and (2);
(b) the policies $\eta_\varepsilon^s=\inf\{t\colon t\ge s,\ \varphi_t=1,\ x_t<w_t+\varepsilon\}$ and $\theta_\varepsilon^s=\inf\{t\colon t\ge s,\ \psi_t=1,\ x_t>w_t-\varepsilon\}$ are $\varepsilon$-optimal.