Continuous versus Discrete Market Games
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.