Stochastic Control and Mean Field Games with an application to Optimal Trading

Cardaliaguet and Lehalle (in their paper "Mean Field Game of Controls and An Application To Trade Crowding", 2016) formulate the problem of optimal trading within a mean field game: a set of investors (viewed as players) have to buy or sell in a given time interval some shares of the same stock, whose public price is influenced by their average trading speed. They consider a simple model for the permanent market impact and prove that, under suitable conditions, this game admits a unique Nash equilibrium. Moreover, they explicitly describe it under the additional assumption that all the investors have the same risk aversion. Our main purpose is to extend these results to a more general model for the market impact, from which many other models widely used in the literature can be derived as special cases. More precisely, we prove the existence and uniqueness of the Nash equilibrium according to this general model; then, we address the case of the same risk aversion for (almost) all the investors, where our general framework lead to equations for the Nash Equilibrium analogous to those formulated by Cartaliaguet and Lehalle. We perform also a few numerical experiments. For simplicity, we focus on two particular cases of our general model: the one used by Cardaliaguet and Lehalle and another one, by Obizhaeva and Wang. Finally, we reformulate the whole problem by expressing the risk aversion of the players through suitable utility functions, and we analyze how this formulation changes the results previously presented