Appl. Math. 0ptim. 9:1-24 (1982) Applied Mathematics and Optimization Zero-sum Markov Games with Stopping and Impulsive Strategies Lukasz Stettner Institute of Mathematics, Polish Academyof Sciences,Sniadeckich8, 00-950 Warsaw, Poland Communicated by A. V. Balakrishnan Abstract. Three kinds of zero-sum Markov games with stopping and impulsive strategies are considered. For these games we find the saddle point strategies and prove that, the value of the game depends continuously on the initial state. In the paper the following three kinds of zero-sum, two-person Markov games are considered: games with stopping, games with impulsive controls and games with impulsive control and stopping. These games do not exhaust the variety of the zero-sum games with stopping and impulsive strategies but are typical in this theory. The form of the associated cost functional depends on the kind of game. For instance, for the game of the first type the functional is of the form: Jx(%8)---- Ex{ fo~Aa e '~Sf(xs)ds+ x~<ae-~'~-Xt'l(X~)+ Xa<.~e-'~axP2(xa) } If T and 8 are stopping times chosen by the first and second player respectively, then the first player pays to the second one in average the total amount equal to Jx(~-, 6). A similar, but a more complicated functional is in the case of games with impulsive controls, and games with impulsive control and stopping. Under assumptions introduced by M. Robin in [8], we prove the existence of the saddle point strategies for these games. We show also that the values of the games depend continuously on the initial state. This way we generalize Bismut results contained in [2] and solve a problem posed by M. Robin in [8]. An analogous game to the one with impulsive controls was considered independently and Under stronger assumptions by J. P. Lepeltier in [6]. 0095-4616/82/0009-0001 $04.80 ©1982 Springer-Verlag New York Inc.