Mean field game of controls and an application to trade crowding

Abstract

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price.In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically.Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.

Some remarks on mean field games

“…Each solution A of this equation produces a solution for the system (7). Hence if φ and m 0 are such that the previous equation admits a unique solution then we have proved uniqueness.…”

“…Thus terms of the form G(∇u(·), m(·)) = T d g(∇u(y))m(y)dy seem to be quite general for applications. See the example studied in [7] for example. Moreover such a term satisfies the assumption of theorem 1.…”

“…Even though this section is not particularly concerned with the question of existence, let us mention that a solution of such a system exists as soon as φ is a Lipschitz function. The following lemma gives a general property satisfied by the 6 solutions of (7). Proposition 1 details a precise example of a function φ which yields uniqueness even though it is not constant : Lemma 2.…”

“…where G # m stands for the image measure of m by the measurable application G. Existence of solutions of such a system was proven for particular Hamiltonians in [20] ; we refer to [7] for a more general result of existence for (2) and to [15,12] for the study of such a coupling in a probabilistic setup. In general, uniqueness of such solutions is not known.…”

Cemracs 2017: numerical probabilistic approach to MFG

“…In many applications, individuals may interact through their controls, instead of their states. One example is an application of trade crowding which was tackled with a mean field game approach in the paper of Cardaliaguet and Lehalle [7]. There is also a model for price impact in the book of Carmona and Delarue which we take as one of our example problems in Section 4.4.2.…”