Repeated market games with lack of information on both sides.

De Meyer and Moussa Saley explains endogenously the appearance of Brownian Motion in finance by modelling the strategic interaction between two asymmetrically informed market makers with a zero-sum repeated game with One-sided information. In this paper, we generalize this model to a setting of a bilateral asymmetry of information. This new model leads us to the analyze of a repeated zero sum game with lack of information on both sides. In De Meyer and Moussa Saley's analysis, the appearance of the normal distribution in the asymptotic behaviour of Vn(P)/ûn is the crucial point of the appearance of the B.M. In the context of bilateral asymmetry of information, the same analysis provides naturally the B.M as a limit of random walks. This allows us to describe the limit of Vn(P,Q)/ûn as the value of a associated «Brownian game», similar to those introduced by De Meyer. Furthermore, the value of this «Brownian game» allows us to consider the limit of ûn(P,Q)ûn as the solution of a heuristic partial differential equation.