From repeated games to Brownian games

The subject of this paper is related to the analysis of the convergence rate of the value of the n-times repeated zero-sum game with one sided information and full monitoring. Particularly, the ultimate aim of this work is the proof of the existence of an asymptotic expansion for this value v(n): v(n) = v(infinity) + psi/root n + O (ln(n)/n). As suggested in the conclusion of [6], the function psi appearing in this expansion should be regarded as the value of a "continuously repeated" game, In this paper, we propose and analyze a game of this kind. In this game, the strategies are progressively measurable processes on the filtration generated by a Brownian motion and the payoff function is defined by use of the Ito-integral. Our main result is the proof of the existence of optimal strategies for both players in this game. (C) Elsevier, Paris.