De Meyer and Moussa Saley  provide an endogenous justiﬁcation for the appearance of Brownian Motion in Finance by modeling the strategic interaction between two asymmetrically informed market makers with a zero-sum repeated game with one-sided information. The crucial point of this justiﬁcation is the appearance of the normal distribution in the asymptotic behavior of V n ( P )// n . In De Meyer and Moussa Saley’s model , agents can ﬁx a price in a continuous space. In the real world however, the market compels the agents to post prices in a discrete set. The previous remark raises the following question: Does the normal density still appear in the asymptotic of V n // n for the discrete market game? The main topic of this paper is to prove that for all discretization of the price set, V n ( P )// n converges uniformly to 0. Despite of this fact, we do not reject De Meyer, Moussa analysis: when the size of the discretization step is small as compared to n –1/2 , the continuous market game is a good approximation of the discrete one.
Marino, Alexandre and De Meyer, Bernard, "Continuous versus Discrete Market Games" (2005). Cowles Foundation Discussion Papers. 1822.