Schauder Estimates for a Class of Potential Mean Field Games of Controls
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Abstract
An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray–Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.
Keywords
Mean field games of controls Extended mean field games Strongly coupled mean field games Potential formulation Hölder estimatesMathematics Subject Classification
91A13 49N701 Introduction
An interpretation of the system (MFGC) is as follows. Consider a stock trading market. A typical trader, with an initial level of stock \(X_0=x\), controls its level of stock \((X_t)_{ t \in [0,T]}\) through the purchasing rate \(\alpha _t\) with stochastic dynamic \(\mathrm {d} X_t = \alpha _t \mathrm {d} t+ \sqrt{2 \sigma } \mathrm {d} B_t\). The agent aims at minimizing the expected cost (2) where \(P(t)\) is the price of the stock at time t. The agent is considered to be infinitesimal and has no impact on \(P(t)\), so it assumes the price as given in its optimization problem. On the other hand, in the equilibrium configuration, the price \(P(t) \; (t \in [0,T])\) becomes endogenous and indeed, is a function of the optimal behaviour of the whole population of agents as formulated in (iii) (MFGC). The expression \(D(t) := \int _{ \mathbb {T}^d} \phi (x,t) v(x,t) m(x,t) \; \mathrm {d} x\) can be considered as a weighted net demand formulation and the relation \(P = \varPsi (D)\) is the result of supplydemand relation which determines the price of the good at the market. Concerning the role of the mapping \(\phi \), one can think for example to the case of two exchangeable goods, i.e. \(x \in \mathbb {R}^2\), with a price given by \(P(t)= \varPsi ( \int _{\mathbb {T}^d} (\phi _1(x,t)v_1(x,t)+\phi _2(x,t) v_2(x,t)) m(x,t) \, \text {d} x )\), where \(\varPsi :\mathbb {R}\rightarrow \mathbb {R}\). The use of a mapping \(\phi \), which is valued in \(\mathbb {R}^{1 \times 2}\) and whose values depend on the scale choosed for the goods, is in such a situation necessary. Thus, the system (MFGC) captures an equilibrium configuration. Similar models have been proposed in the electrical engineering literature, see for example [2, 10, 11] and the references therein.
In most mean field game models, the individual players interact through their position only, that is, via the variable m. The problem that we consider belongs to the more general class of problems, called extended mean field games, for which the players interact through the joint probability distribution \(\mu \) of states and controls. Several existence results have been obtained for such models: in [13] for stationary mean field games, in [15] for deterministic mean field games. In [6, Section 5], a class of problems where \(\mu \) enters in the drift and the integral cost of the agents is considered. We adopt the terminology mean field games of controls employed by the authors of the latter reference. Let us mention that our existence proof is different from the one of [6], which includes control bounds. In [3, Section 1], a model where the drift of the players depends on \(\mu \) is analyzed. In [14], a mean field game model is considered where at all time t, the average control (with respect to all players) is prescribed. We finally mention that extended mean field games have been studied with a probabilistic approach in [1, 8] and in [7, Section 4.6], and that a class of linearquadratic extended mean field games has been analyzed in [20].
The existence of a classical solution of (MFGC) is established with the Leray–Schauder theorem and classical estimates for parabolic equations. A similar approach has been employed in [16, 17], and [18] for the analysis of a mean field game problem proposed by Chan and Sircar [9]. In this model, each agent exploits an exhaustible resource and fixes its price. The evolution of the capacity of a given producer depends on the price set by the producer, but also on the average price (with respect to all producers).
The application of the Leray–Schauder theorem relies on a priori bounds for fixed points. These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are also solutions to an optimal control problem of the Fokker–Planck equation. We are not aware of any other publication making use of such a potential formulation for a mean field game of controls, with the exception of [17] for the Chan and Sircar model. Let us mention that besides the derivation of a priori bounds, the potential formulation of the problem can be very helpful for the numerical resolution of the problem and the analysis of learning procedures (which are out of the scope of the present work).
The article is structured as follows. We list in Sect. 2 the assumptions employed all along. The main result (Theorem 1) is stated in Sect. 3. We provide in Sect. 4 a first incomplete potential formulation of the problem, incomplete in so far as the term f(m) is not integrated. We also introduce some auxiliary mappings, which allow to express P and v as functions of m and u. We give some regularity properties for these mappings in Sect. 5. In Sect. 6 we establish some a priori bounds for solutions to the coupled system. We prove our main result in Sect. 7. In Sec. 8, we give a full potential formulation of the problem, prove the uniqueness of the solution to (MFGC) and prove that (u, P, f(m)) is the solution to an optimal control problem of the HJB equation, under an additional monotonicity condition on f. Some parabolic estimates, used all along the article, are provided and proved in the appendix.
2 Assumptions on Data
The gradient of the data functions with respect to some variable is denoted with an index, for example, \(H_p\) denotes the gradient of H with respect to p. The same notation is used for the Hessian matrix. The gradient of u with respect to x is denoted by \(\nabla u\). Let us mention that very often, the variables x and t are omitted, to alleviate the calculations. We also denote by \(\int \phi v m\) the integral \(\int _{\mathbb {T}^d} \phi v m \; \mathrm {d} x\) when used as a second argument of \(\varPsi \). For a given normed space X, the ball of center 0 and radius R is denoted B(X, R).
2.1 Convexity Assumptions
2.2 Regularity Assumptions
We assume that \(L_v\) is differentiable with respect to x and v and that \(\phi \) is differentiable with respect to x. All along the article, we make use of the following assumptions.
2.3 Growth Assumptions
2.4 Hölder Continuity Assumptions
 For all \(R>0\), there exists \(\alpha \in (0,1)\) such thatwhere \(B_R= Q \times B(\mathbb {R}^d,R)\) and \(B_R'= [0,T] \times B(\mathbb {R}^k,R)\).$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{l} L \in \mathcal {C}^{\alpha }(B_R), \\ L_v \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^d), \\ L_{vx} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d}), \\ L_{vv} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d}), \end{array} \end{array}\right. } \quad {\left\{ \begin{array}{ll} \begin{array}{l} \varPsi \in \mathcal {C}^{\alpha }(B_R',\mathbb {R}^d), \\ \phi \in \mathcal {C}^{\alpha }(Q,\mathbb {R}^{k \times d}), \\ D_x \phi \in \mathcal {C}^{\alpha }(Q,\mathbb {R}^{k \times d \times d}), \end{array} \end{array}\right. } \qquad \end{aligned}$$(A6)
 There exists \(\alpha \in (0,1)\) and \(C>0\) such thatfor all \((x_1,t_1)\) and \((x_2,t_2) \in Q\) and for all \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\).$$\begin{aligned} \begin{aligned}&\qquad \qquad  f(x_2,t_2,m_2)  f(x_1,t_1,m_1)  \\&\qquad \qquad \qquad \le C \big ( x_2  x_1 + t_2  t_1^\alpha + \Vert m_2  m_1 \Vert _{L^{\infty }(\mathbb {T}^d)}^\alpha \big ), \end{aligned} \end{aligned}$$(A7)
 $$\begin{aligned} \text {There exists }\alpha \in (0,1)\text { such that }m_0 \in \mathcal {C}^{2+ \alpha }(\mathbb {T}^d)\text {, }g \in \mathcal {C}^{2 +\alpha }(\mathbb {T}^d). \end{aligned}$$(A8)
Some regularity properties for the Hamiltonian can be deduced from the convexity assumption (A1) and the Hölder continuity of L and its derivatives (assumption (A6)). They are collected in the following lemma.
Lemma 1
The Hamiltonian H is differentiable with respect to p and \(H_p\) is differentiable with respect to x and p. Moreover, for all \(R>0\), there exists \(\alpha \in (0,1)\) such that \(H \in \mathcal {C}^{\alpha }(B_R)\), \(H_p \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^d)\), \(H_{px} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d})\), and \(H_{pp} \in \mathcal {C}^{\alpha }(B_R, \mathbb {R}^{d \times d})\)
Proof
 1.
The mapping \(x \in \mathbb {T}^d\mapsto K(x,0,0)\) lies in \(L^1(\mathbb {T}^d)\).
 2.There exist a mapping \(C_3 \in L^1(\mathbb {T}^d)\) and \(\alpha \in (0,1)\) such that for a.e. \(x \in \mathbb {T}^d\), for all \(t_1\) and \(t_2 \in [0,T]\) and for all \(w_1\) and \(w_2 \in [C_2,C_2]\),$$\begin{aligned} K(x,t_2,w_2)K(x,t_1,w_1) \le C_3(x) \big ( t_2t_1^\alpha + w_2  w_1^\alpha \big ). \end{aligned}$$
Proof
3 Main Result and General Approach
Theorem 1
The result is obtained with the Leray–Schauder theorem, recalled below.
Theorem 2
(Leray–Schauder) Let X be a Banach space and let \(\mathcal {T} :X \times [0,1] \rightarrow X\) be a continuous and compact mapping. Let \(x_0 \in X\). Assume that \(\mathcal {T}(x,0)=x_0\) for all \(x \in X\) and assume there exists \(C>0\) such that \(\Vert x \Vert _X < C\) for all \((x,\tau ) \in X \times [0,1]\) such that \(\mathcal {T} (x,\tau ) = x\). Then, there exists \(x \in X\) such that \(\mathcal {T}(x,1) = x\).
Lemma 3
Let \(\tau \in [0,1]\), let \((m,v) \in W^{2,1,p}(Q) \times L^\infty (Q,\mathbb {R}^d)\) be a weak solution to the Fokker–Planck equation \(\partial _t m  \sigma \varDelta m + \tau \text {div }(vm)= 0\), \(m(\cdot ,0)= m_0(\cdot )\). Then \(m \ge 0\) and for all \(t \in [0,T]\), \(\int _{\mathbb {T}^d} m(x,t) \; \mathrm {d} x = 1\).
Proof
4 Potential Formulation
In this section, we first establish a potential formulation of the mean field game problem (MFGC\(_\tau \)), that is to say, we prove that for \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)), \((m_\tau ,v_\tau )\) is a solution to an optimal control problem. We prove then that for all t, \(v_\tau (\cdot ,t)\) is the unique solution of some optimization problem, which will enable us to construct the announced auxiliary mappings.
Lemma 4
Remark 1
Let us emphasize that the above optimal control problem is only an incomplete potential formulation, since the term \(\tilde{f}_\tau \) still depends on \(m_\tau \).
Proof
We have proved that the pair \((m_\tau ,v_\tau )\) is the solution to an optimal control problem. Therefore, for all t, \(v_\tau (\cdot ,t)\) minimizes the Hamiltonian associated with problem (17). Let us introduce some notation, in order to exploit this property. For \(m \in \mathcal {D}_1(\mathbb {T}^d)\), we denote by \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\) the Hilbert space of measurable mappings \(v :\mathbb {T}^d\rightarrow \mathbb {R}^d\) such that \(\int _{\mathbb {T}^d} v^2 m < \infty \), equipped with the scalar product \(\int _{\mathbb {T}^d} \langle v_1,v_2 \rangle m\). An element of \(L_m^2(\mathbb {T}^d)\) is an equivalent class of functions equal malmost everywhere. Note that \(L^\infty (\mathbb {T}^d) \subset L_m^2(\mathbb {T}^d)\).
Lemma 5
Proof
Step 2 existence of \(\mathbf {v}(t,m,w)\) and a priori bound.
Step 3 uniqueness of \(\mathbf {v}(t,m,w)\).
Let \(v_1\) and \(v_2 \in L^\infty (\mathbb {T}^d,\mathbb {R}^d)\) satisfy (22). Then \(DJ(v_i;t,m,w)= 0\), proving that \(v_1\) and \(v_2\) are minimizers of \(J(\cdot ;t,m,w)\) and thus are equal in \(L_m^2(\mathbb {T}^d,\mathbb {R}^d)\). Therefore \({\textstyle \int }\phi (x',t) v_1(x') m(x') \; \mathrm {d} x' = {\textstyle \int }\phi (x',t) v_2(x') m(x') \; \mathrm {d} x'\) and finally that \(v_1= v_2\), by (22). \(\square \)
5 Regularity Results for the Auxiliary Mappings
We provide in this section some regularity results for the mappings \(\mathbf {v}\) and \(\mathbf {P}\). We begin by proving that \(\mathbf {P}(\cdot ,\cdot ,\cdot )\) is locally Hölder continuous. For this purpose, we perform a stability analysis of the optimality condition (23).
Lemma 6
Proof
Lemma 7
Proof
Remark 2
As a consequence of Lemma 7, the images of the mappings given by (31) and (32) are bounded. This fact will be used in the steps 3 and 5 of the proof of Proposition 1.
Lemma 8
Let \(R>0\) and \(\beta \in (0,1)\). Then, there exists \(\alpha \in (0,1)\) and \(C>0\) such that for all \(u \in B(W^{2,1,p}(Q),R)\) and for all \(m \in B(\mathcal {C}^{\beta }(Q),R) \cap L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), \(\Vert \mathbf {P}(m,\nabla u) \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).
Proof
We recall that by Lemma 12, \(\Vert \nabla u \Vert _{\mathcal {C}^{\alpha }(Q,\mathbb {R}^d)} \le C \Vert u \Vert _{W^{2,1,p}(Q)}\). We obtain then the bound on \(\Vert \mathbf {P}(m,\nabla u) \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)}\) with Lemma 6. \(\square \)
Lemma 9
6 A Priori Estimates for Fixed Points
Proposition 1
Proof
Let us fix \(\tau \in [0,1]\) and \((u_\tau ,m_\tau ,v_\tau ,P_\tau ) \in X'\) satisfying (MFGC\(_\tau \)). All constants C and all exponents \(\alpha \in (0,1)\) involved below are independent of \((u_\tau ,m_\tau ,v_\tau ,P_\tau )\) and \(\tau \). Let us recall that \(\tilde{f}_\tau \in L^\infty (Q)\) has been defined in Lemma 4 by \(\tilde{f}_\tau (x,t)= f(x,t,m_\tau (t))\).
Step 1\(\Vert P_\tau \Vert _{L^2(0,T;\mathbb {R}^k)} \le C\).
Step 3\(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)} \le C\).
By Lemma 3, \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). We have that \(\Vert \nabla u_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\) and \(P_\tau = \mathbf {P}(m_\tau ,\nabla u_\tau )\). The bound on \(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)}\) follows then from Lemma 7 and Remark 2.
Step 4\(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\).
By assumption (A6), \(\phi \) is bounded. We have proved that \(\Vert P_\tau \Vert _{L^\infty (0,T;\mathbb {R}^k)} \le C\) and by Lemma 1, H is continuous. Thus, \(\Vert H(\nabla u_\tau + \phi ^\intercal P_\tau ) \Vert _{L^\infty (Q)} \le C\). By assumption (A5), \(\Vert \tau \tilde{f}_\tau \Vert _{L^\infty (Q)} \le C\). It follows that \(u_\tau \), as the solution to the HJB equation (i) (MFGC\(_\tau \)), is the solution to a parabolic equation with bounded coefficients. Thus, by Theorem 6, \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\). We also obtain with Lemma 12 that \(\Vert u_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\) and \(\Vert \nabla u_\tau \Vert _{\mathcal {C}^\alpha (Q,\mathbb {R}^d)} \le C\).
Step 5\(\Vert v_\tau \Vert _{L^\infty (Q,\mathbb {R}^d)} \le C\), \(\Vert D_x v_\tau \Vert _{L^p(Q,\mathbb {R}^{d \times d})} \le C\).
We have proved that \(v_\tau = \mathbf {v}(m_\tau ,\nabla u_\tau )\) and \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\). The estimate follows directly with Lemma 7 and Remark 2.
Step 6\(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\).
The Fokker–Planck equation can be written in the form of a parabolic equation with coefficients in \(L^p\): \(\partial _t m_{\tau }  \sigma \varDelta m_\tau + \tau \langle v_\tau , \nabla m_\tau \rangle + \tau m_\tau \text {div}(v_\tau )= 0\), since \(\Vert D_x v_\tau \Vert _{L^p(Q,\mathbb {R}^{d \times d})} \le C\). Combining Theorem 4 and Lemma 12, we get that \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\).
Step 7\(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).
We already know that \(\Vert u_\tau \Vert _{W^{2,1,p}(Q)} \le C\), that \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\), and that \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). Thus Lemma 8 applies and yields that \(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\).
Step 8\(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C\).
We have proved that \(\Vert \nabla u_\tau \Vert _{\mathcal {C}^\alpha (Q,\mathbb {R}^d)} \le C\) and \(\Vert P_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^k)} \le C\). Moreover, we have assumed that \(\phi \) is Hölder continuous and know that H is Hölder continuous on bounded sets. It follows that \(\Vert H(\nabla u_\tau + \phi ^\intercal P_\tau ) \Vert _{\mathcal {C}^\alpha (Q)} \le C\). It follows from assumption (A7) that \(\tau \tilde{f}_\tau \) is Hölder continuous. Since \(g \in \mathcal {C}^{2+\alpha }(\mathbb {T}^d)\), we finally obtain that \(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+ \alpha /2}(Q)} \le C\), by Theorem 7.
Step 9\(\Vert v_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^{d})} \le C\) and \(\Vert D_x v_\tau \Vert _{\mathcal {C}^\alpha (0,T;\mathbb {R}^{d \times d})} \le C\).
We have \(\Vert u_\tau \Vert _{\mathcal {C}^{2+\alpha ,1+\alpha /2}(Q)} \le C\) and \(\Vert m_\tau \Vert _{\mathcal {C}^\alpha (Q)} \le C\). Thus Lemma 9 applies and the announced estimates hold true.
Step 10\(\Vert m_\tau \Vert _{\mathcal {C}^{2+ \alpha , 1 + \alpha /2}(Q)} \le C\).
A direct consequence of Step 9 is that \(m_\tau \) is the solution to a parabolic equation with Hölder continuous coefficients. Therefore \(\Vert m_\tau \Vert _{\mathcal {C}^{2+\alpha ,1 + \alpha /2}(Q)} \le C\), by Theorem 7, which concludes the proof of the proposition. \(\square \)
7 Application of the Leray–Schauder Theorem
Proof
(Theorem 1) Step 1 construction of \(\mathcal {T}\).

For all \(m \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\), \(\rho (m)=m\).

The mapping \(\rho \) is locally Lipschitz continuous, from \(L^\infty (Q)\) to \(L^\infty (Q)\).

For all \(\alpha \in (0,1)\), there exists a constant \(C>0\) such that if \(m \in \mathcal {C}^\alpha (Q)\), then \(\rho (m) \in \mathcal {C}^\alpha (Q)\) and \(\Vert \rho (m) \Vert _{\mathcal {C}^{\alpha }(Q)} \le C \Vert m \Vert _{\mathcal {C}^{\alpha }(Q)}\).
Step 2 a priori bound.
Let \(\tau \in [0,1]\) and let \((u_\tau ,m_\tau )\) be such that \((u_\tau ,m_\tau )= \mathcal {T}(u_\tau ,m_\tau ,\tau )\). Then, by Lemma 3, \(m_\tau \in L^\infty (0,T;\mathcal {D}_1(\mathbb {T}^d))\). Thus, \(m_\tau = \rho (m_\tau )\) and finally, by Lemma 5, the quadruplet \((u_\tau ,m_\tau ,P_\tau ,v_\tau )\), with \(P_\tau = \mathbf {P}(m_\tau ,\nabla u_\tau )\) and \(v_\tau = \mathbf {v}(m_\tau ,\nabla u_\tau )\), is a solution to (MFGC\(_\tau \)). We directly conclude with Proposition 1 that \(\Vert (u_\tau ,m_\tau ) \Vert _X \le C\), where C is independent of \(\tau \).
Step 3 continuity of \(\mathcal {T}\).
Step 4 compactness of \(\mathcal {T}\).
Step 5 Conclusion.
The existence of a fixed point (u, m) to \(\mathcal {T}(\cdot ,\cdot ,1)\) follows. With the same arguments as those of Step 2, we obtain that \((u,m,\mathbf {P}(m,\nabla u),\mathbf {v}(m,\nabla u))\) is a solution to (MFGC\(_\tau \)) with \(\tau =1\) and that (15) holds, by Proposition 1. \(\square \)
8 Uniqueness and Duality
Remark 3
 1.It follows from (40) that f is monotone:for all \(m_1\) and \(m_2 \in \mathcal {D}_1(\mathbb {T}^d)\) and for a.e. t. Conversely, (40) holds true if (41) is satisfied and if F is a primitive of f(., t, .) in the sense that$$\begin{aligned} \int _{ \mathbb {T}^d} (f(x,t,m_2) f(x,t,m_2)) (m_2(x)  m_1(x)) \; \mathrm {d} x \ge 0, \end{aligned}$$(41)We refer to [5, Proposition 1.2] for a further characterization of functions f deriving from a potential.$$\begin{aligned} F(t,m_2)  F(t,m_1) = \int _{0}^{1} \int _{\mathbb {T}^d} f(x,t, sm_2 + (1s)m_1) (m_2(x)  m_1(x)) \; \mathrm {d} s. \end{aligned}$$
 2.Consider the mapping \(f_K\) proposed in Lemma 2. Assume that for all \((x,t) \in Q\), \(K(x,t,\cdot )\) is nondecreasing and consider the function \(\mathcal {K}\) defined by \(\mathcal {K}(x,t,w) := \int _0^w K(x,t,w') \; \mathrm {d} w'\), for \((x,t,w) \in Q \times [C_2,C_2]\). Then inequality (40) holds true with \(F_K\) defined byIndeed, since \(\mathcal {K}\) is convex in its third argument, we have$$\begin{aligned} F_K(t,m)= \int _{\mathbb {T}^d} \mathcal {K} (x,t,m * \varphi (x)) \; \mathrm {d} x. \end{aligned}$$as was to be proved.$$\begin{aligned}&F_K(t,m_2)F_K(t,m_1) = \int _{\mathbb {T}^d} \mathcal {K}(x,t,m_2 * \varphi (x))  \mathcal {K}(x,t,m_1 * \varphi (x)) \; \mathrm {d} x \\&\quad \ge \int _{\mathbb {T}^d} K(x,t,m_1* \varphi (x))((m_2m_1)*\varphi )(x) \; \mathrm {d} x \\&\quad = \int _{\mathbb {T}^d} (K(\cdot ,t,m_1*\varphi (\cdot ))* \tilde{\varphi })(x) (m_2(x)  m_1(x)) \; \mathrm {d} x \\&\quad = \int _{\mathbb {T}^d} f_K(x,t,m)(m_2(x)m_1(x)) \; \mathrm {d} x, \end{aligned}$$
Proposition 2
Proof
Lemma 10
Proof
9 Conclusion
The existence and uniqueness of a classical solution to a mean field game of controls have been demonstrated. A particularly important aspect of the analysis is the fact that the equations (iii) and (iv) (MFGC), encoding the coupling of the agents through the controls, are equivalent to the optimality system of a ‘static’ convex problem. This observation enabled us to eliminate the variables v and P from the coupled system.
Future research will aim at exploiting the potential structure of the problem, which can be used to solve it numerically and to prove the convergence of learning procedures, as was done in [5].
Notes
Acknowledgements
Open access funding provided by University of Graz. The authors want to thank an anonymous referee for his useful remarks. The first author acknowledges support from the FiME Lab (Institut Europlace de Finance). The first two authors acknowledges support from the PGMO project “Optimal control of conservation equations”, itself supported by iCODE (IDEX ParisSaclay) and the Hadamard Mathematics LabEx.
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest
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