Abstract. Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location and the distribution of the remaining agents. Here, we consider stationary MFGs with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFGs with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.

Mean-field games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in high-density areas.Here, we study stationary MFGs with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing density. Our main contribution is a new variational formulation for MFGs with congestion. This formulation was not previously known, and, thanks to it, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.

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We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), high-frequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order book, similar to the one considered in the Queue-Reactive models [12,18,19], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy. . D. Evangelista was partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452. ¶ CMAP, École Polytechnique. othmane.mounjid@polytechnique.edu. arXiv:1802.08135v2 [q-fin.TR] 9 Nov 2018 v(t, z) := sup φ∈C(t,z)Remark 3.3. Note that v is bounded from above by 0 by definition. On the other hand, for all E[U (P t,z,0 T , 0, 0, g, i, 0, 0, 0, 0, j)] = e −η(g− j) min i∈[−I * ,I * ] E[U (P t,z,0 T , 0, 0, 0, i, 0, 0, 0, 0, 0)], where P t,z,0 corresponds to the dynamics in the case that the MM does not act on the order book up to T . Moreover, it follows from (3.3) that E[U (P t,z,0 T , 0, 0, 0, i, 0, 0, 0, 0, 0)] ≥ −e ηI * |p b | E[e ηI * (|P t,z,0,b T −p b |+2d+κ) ]where supby Remark 2.1 and the fact that the price can jump only by d when a market event occurs. Thus, v belongs to the class L exp ∞ of functions ϕ such that ϕ/L is bounded, in which The dynamic programming equationThe derivation of the dynamic programming equation is standard, and is based on the dynamic programming principle. We state below the weak version of Bouchard and Touzi [10], we let v * and v * denote the lower-and upper-semicontinuous envelopes of v. Proposition 3.1. Fix (t, z) ∈ [0, T ] × D Z and a family {θ φ , φ ∈ C(t, z)} such that each θ φ is a [t, T ]-valued F t,z,φ -stopping time and Z t,z,φ θ φ L∞ < ∞. Then, sup φ∈C(t,z) E v * (θ φ , Z t,z,φ θ φ ) ≤ v(t, z) ≤ sup φ∈C(t,z)E v * (θ φ , Z t,z,φ θ φ ) .