Neveu's martingale conditions and closedness in Dynkin stopping problem with a finite constraint (Q1819816)

Consider three random sequences \(\{X_ n\), \(n\in N\}\), \(\{Y_ n\), \(n\in N\}\) and \(\{W_ n\), \(n\in N\}\) such that \(X_ n\leq W_ n\leq Y_ n\) a.s. The following modified version of Dynkin's game is considered: 2 players play a game in which, if \(\tau\) is the stopping time selected by player 1 and \(\sigma\) is the stopping time selected by player 2, then the gain obtained by player 1 is \[ g(\tau,\sigma)=X_{\tau}I_{\{\tau <\sigma \}}+W_{\tau}I_{\{\tau =\sigma \}}+Y_{\tau}I_{\{\tau >\sigma \}}, \] where \(I_ A\) denotes the indicator function of the set A. The aim of player 1 is to maximize E[g(\(\tau\),\(\sigma)\)], and that of the second is to minimize this expectation. The game is said to be closed if its lower value equals its higher value. Under the assumption that the game terminates in a finite number of steps, necessary and sufficient conditions are given for the game to be closed. Also, the existence of saddle points is proved when the game is closed.