Optimal stopped games

Let ($\Omega$, $\mathscr F$, $\mathbf P$) be the probability space, $\mathscr F_0\subseteq\mathscr F_1\subseteq\dots\subseteq\mathscr F_n\subseteq\dots\subseteq\mathscr F$ a nondecreasing sequence of $\sigma$-algebras. Let random variables $x_n$, $\varphi_n$ be $\mathscr F_n$-measurable ($n=0,1,\dots$).
The process may be stopped by the 1st player at the $n$th step if $\varphi_n>0$, and by the 2nd player if $\varphi_n<0$. The 2nd player gets from the 1st one the sum $x_n$ provided the process is stopped on the $n$th step. The process where the role of $\varphi_n$ plays
$$ \varphi_n^L= \begin{cases} \varphi_n,&\varphi_n>0, \\ 0,&\varphi_\le0, \end{cases} $$
is called the minorizing process and the process where the role of $\varphi_n$ plays
$$ \varphi_n^M= \begin{cases} 0,&\varphi_\ge0, \\ \varphi_n,&\varphi_n<0, \end{cases} $$
is called the majorizing process. We suppose that $\mathbf M(\sup\limits_n|x_n|)<\infty$.
We prove that if there exists an optimal policy in the minorizing (majorizing) process, starting at the $n$th step, then the policy
\begin{gather*} \sigma^k=\inf\{t\colon t\ge k,\quad\varphi_t>0,\quad x_t\le w_t\}\quad(\tau^k=\inf\{t\colon t\ge k,\quad\varphi_t<0,\quad x_t\ge w_t\}) \\ (k=0,\dots,n) \end{gather*}
is optimal for the first (second) player in the initial game starting at the $k$th step. (Here $w_t$ is the value of the initial game starting at the $t$th step. The existence of $w_t$ is proved in [1].)