ZERO-SUM GAMES WITH STOPPING TIMES AND VARIATIONAL INEQUALITIES

This paper deals with a zero-sum game, whose state changes correspondingly with a kind of Markov process. In the game the two players\u27 strategies are when to stop the game to maximize their own rewards. In this paper we investigate optimal stopping times, optimal rewards and the conditions which should be satisfied. To describe more precisely, it is as follows. Let $ M $ be an $ m $-symmetric Hunt process and let $ (mathfrak{F}, mathfrak{E}) $ be the corresponding Dirichlet space. For $ alpha > 0 $ and $ W in mathfrak{F} $ we consider a variational inequality : (1.1) $ U^alpha in Re $, $ mathfrak{E}_alpha(U^alpha - W, V - U^alpha) geqq 0 $ for all $ V in Re $, where $ Re $ is a certain subset of $ mathfrak{F} $. In this paper we investigate properties of solutions $ U^alpha $ of (1.1) and we show that solutions $ U^alpha $ has a quasi-continuous version which becomes to be a value of a certain zero-sum game associated with regions $ X - F_i (i=1,2) $ not to permit stopping the game. The aim of this paper is to discuss these in the case of $ m $-symmetric Hunt processes, which have $ m $-symmetric diffusion processes and $ m $-symmetric jump processes as examples. When the processes are transient, we can treat the case of # alpha = 0 $ and we discuss it similarly