FOR TIHE OTHER

I Abstract. In the paper a general zero-sum game with a, I stopping strategy for the fist player and a continuous one fur the second player is considered. The author proves the existence of a value of the game and an optimal strategy for the first player under fairly general assumptions. 1. IntrodwOioo. There is a considerable number of papers dealing with general zero-sum stochastic games with optimal stopping [I, 2, 6, 8, 91. A good survey on these results is given by Zabczyk in 1101. From another point of view Davis-Elliott [3] have studied a zero-sum game with continuous strategies. In this paper we consider the so-calfed mixed zero-sum game, where the first (resp. the second) player chooses a stopping time S (resp. a continuous strategy u) and looks for maximize (resp. minimize) a payoff E,(C", YT). Section 2 gives a precise model of the game. In Section 3 we prove that the upper value function of the game ~ ( r) is "aggregable " under right continuity assumptions on the processes, i.e. there exists a right continuous process w such that @(r) = WT P a.e. for every stopping time 7: The method is based on the results of Dellacherie-Lenglart 141. The last section contains essential results of this paper: the value of the game and, with additional assumption on the left regularity of the processes, the existence of an optimal strategy for the first player