  This paper deals with a stochastic order-driven market model with waiting
costs, for order books with heterogenous traders. Offer and demand of liquidity
drives price formation and traders anticipate future evolutions of the order
book. The natural framework we use is mean field game theory, a class of
stochastic differential games with a continuum of anonymous players. Several
sources of heterogeneity are considered including the mean size of orders. Thus
we are able to consider the coexistence of Institutional Investors and High
Frequency Traders (HFT). We provide both analytical solutions and numerical
experiments. Implications on classical quantities are explored: order book
size, prices, and effective bid/ask spread. According to the model, in markets
with Institutional Investors only we show the existence of inefficient
liquidity imbalances in equilibrium, with two symmetrical situations
corresponding to what we call liquidity calls for liquidity. During these
situations the transaction price significantly moves away from the fair price.
However this macro phenomenon disappears in markets with both Institutional
Investors and HFT, although a more precise study shows that the benefits of the
new situation go to HFT only, leaving Institutional Investors even with higher
trading costs.
