# A Variational Approach For Price Formation Models In One Dimension

Yuri Ashrafyan  ,  Tigran Bakaryan  ,  Diogo Gomes  and  Julian Gutierrez

###### Abstract.

In this paper, we study a class of first-order mean-field games (MFGs) that
model price formation. Using Poincaré Lemma, we eliminate one of the equations
and obtain a variational problem for a single function. This variational
problem offers an alternative approach for the numerical solution of the
original MFGs system. We show a correspondence between solutions of the MFGs
system and the variational problem. Moreover, we address the existence of
solutions for the variational problem using the direct method in the calculus
of variations. We end the paper with numerical results for a linear-quadratic
model.

###### Key words and phrases:

Mean Field Games; Price formation; Potential Function, Lagrange multiplier

King Abdullah University of Science and Technology (KAUST), CEMSE Division,
Thuwal 23955-6900. Saudi Arabia. e-mail<EMAIL_ADDRESS>
King Abdullah University of Science and Technology (KAUST), CEMSE Division,
Thuwal 23955-6900. Saudi Arabia. e-mail<EMAIL_ADDRESS>
King Abdullah University of Science and Technology (KAUST), CEMSE Division,
Thuwal 23955-6900. Saudi Arabia. e-mail<EMAIL_ADDRESS>
King Abdullah University of Science and Technology (KAUST), CEMSE Division,
Thuwal 23955-6900. Saudi Arabia. e-mail<EMAIL_ADDRESS>
The authors were partially supported by King Abdullah University of Science
and Technology (KAUST) baseline funds and KAUST OSR-CRG2021-4674.

##  1\. Introduction

Here, we consider the numerical solution of the first-order mean-field games
(MFGs) system introduced in [31] to model price formation. The solution to
this system determines the price $\varpi$ of a commodity with supply $Q$ when
a large group of rational agents trades that commodity. The original price
problem reads as follows:

###### Problem 1.

Suppose that $m_{0}\in\mathcal{P}({\mathbb{R}})$, $H\in C^{1}({\mathbb{R}})$,
and $Q$, $V$, and $u_{T}$ are continuous. Assume further that $H$ is uniformly
convex. Find $u,m:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ and
$\varpi:[0,T]\to{\mathbb{R}}$ satisfying $m\geqslant 0$,

$\begin{cases}-u_{t}+H(\varpi+u_{x})+V(x)=0&[0,T]\times{\mathbb{R}},\\\
m_{t}-\left(H^{\prime}(\varpi+u_{x})m\right)_{x}=0&[0,T]\times{\mathbb{R}},\\\
-\int_{{\mathbb{R}}}H^{\prime}(\varpi+u_{x})m{\rm d}x=Q(t)&[0,T],\end{cases}$
(1.1)

and

$\begin{cases}m(0,x)=m_{0}(x)&\\\
u(T,x)=u_{T}(x)&\end{cases}x\in{\mathbb{R}}.$ (1.2)

The existence of solutions $(u,m,\varpi)$ to the previous problem was proved
in [31]. The first equation is solved in the viscosity sense by the value
function of a typical player $u\in C([0,T]\times{\mathbb{R}})$. The second
equation is solved in the distributional sense by the probability distribution
of the agents, $m\in C([0,T],{\mathcal{P}}({\mathbb{R}}))$. The price $\varpi$
is a continuous function on $[0,T]$.

Price formation models offer a load-adaptive pricing strategy relevant in
energy markets. For instance, [6] and [7] modeled intraday electricity
markets, obtaining a price from the solution of forward-backward equations. In
[24] and [28] authors studied the effects of a major player in the market. The
latest paper considered $N$-agent setting. A deterministic $N$-agent price
model was studied in [8]. A MFG model of homogeneous agents for the
electricity markets was considered in [25]. In [4], the price equilibrium is
obtained for a finite number of agents who optimally control their production
and trading rates in order to satisfy a demand subjected to common noise.
Stackelberg games for price formation under revenue optimization were proposed
in [13] and [40], and Cournot models in [21]. A MFG of optimal switching was
presented in [5] to model the transition to renewable energies. Other works
incorporating market-clearing conditions are [41] and [27], the former
specialized to Solar Renewable Energy Certificate Markets and the latter in
exchange markets. The stochastic supply case was studied in [29], where
authors obtained a price from a Lagrange multiplier rule for the balance
constraint.

The standard MFG system exhibits a coupling of two partial differential
equations with initial and terminal conditions (see for example [16]). Several
numerical methods have been proposed to solve these MFG systems. Finite
differences schemes and Newton-based methods were introduced in [1] and [2]. A
recent survey can be found in [3]. Optimization methods and Fourier series
approximations were proposed in [35]. Machine learning methods have been
studied in [17], [18], [39], and [34]. However, the MFG system (1.1)-(1.2) not
only couples a forward equation for $m$ with a backward equation for $u$ but
also determines the coupling term $\varpi$ through an integral constraint,
which is the third equation in (1.1). Therefore, the numerical approximation
of the solution $(u,m,\varpi)$ of Problem 1 is challenging, and the main
application of our methods is a novel numerical scheme for Problem 1.

The word Potential in MFGs is used in two unrelated contexts. Potential MFGs
([33], [16], [36]) are MFG systems given by the first-order optimality
conditions of a minimization problem. Previously, standard optimization
techniques were used for its numerical solution ([14]). In contrast, our
potential approach relies on the structure of the continuity equation and
Poincaré lemma ([19], Theorem 1.22). We introduce a potential functional that
integrates the transport equation in (1.1).

Poincaré lemma was used for the continuity equation in [12] for the MFG
planning problem. The authors obtained a variational problem for a potential
function by eliminating one of the equations in the MFG system. Moreover, the
solution $(u,m)$ of the planning MFG can be recovered using only the solution
of the variational problem. The structure of the MFG planning problem differs
from that in Problem 1 in two critical aspects: the initial-terminal
conditions and the way the constraint couples the equations.

In Section 2, we use the existence result for Problem 1 provided in [31] to
formally obtain a potential function, $\varphi$. We show that (1.1)
corresponds to the Euler-Lagrange equation of a constrained variational
problem depending on $\varphi$. To introduce this problem, let $F$ be the
Legendre transform of $H$; that is,

$F(y)=\sup_{p\in{\mathbb{R}}}\left[py-H(p)\right],\quad y\in{\mathbb{R}},$
(1.3)

and let $L:{\mathbb{R}}\times{\mathbb{R}}_{0}^{+}\to{\mathbb{R}}_{0}^{+}$ be
given by

$L(z,y)=\begin{cases}F\left(\frac{z}{y}\right)y,&(z,y)\in{\mathbb{R}}\times{\mathbb{R}}^{+},\\\
+\infty,&z\neq 0,\,y=0,\\\ 0,&z=0,\,y=0.\end{cases}$ (1.4)

The constrained variational problem is

###### Problem 2.

Suppose that $m_{0}\in\mathcal{P}({\mathbb{R}})$, $H$ is uniformly convex, and
$Q$, $V$, and $u_{T}$ are continuous. Find
$\varphi:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ that minimizes the functional

$\varphi\mapsto\int_{0}^{T}\int_{{\mathbb{R}}}L(\varphi_{t},\varphi_{x})-V(x)\varphi_{x}-u^{\prime}_{T}(x)\varphi_{t}~{}{\rm
d}x{\rm d}t,$

over the set of functions such that $\varphi_{x}(t,\cdot)$ is a probability
density on ${\mathbb{R}}$ for $t\in[0,T]$,
$\varphi_{x}(0,\cdot)=m_{0}(\cdot)$, and satisfying

$\int_{{\mathbb{R}}}\varphi(t,x)-M_{0}(x){\rm d}x=-\int_{0}^{t}Q(s){\rm
d}s,\quad t\in[0,T].$ (1.5)

We work under assumptions similar to those in [31] used to prove the existence
and uniqueness of solutions to Problem 1. The precise statement of our
assumptions is presented in Section 3. We rigorously study Problem 2 in
Section 4, where we show that its formulation is independent of the solution
$(u,m,\varpi)$ of Problem 1, and relies only on problem data. In Section 5, we
obtain the existence of a price $\varpi$ in (1.1) as a Lagrange multiplier,
and we establish the following connection between solutions of Problems 1 and
2:

###### Theorem 1.1.

Suppose that $\varphi\in C^{2}([0,T]\times{\mathbb{R}})$ solves Problem 2 .
Then, the solution $(u,m,\varpi)$ of Problem 1 admits the representation

$\displaystyle\begin{cases}u(t,x)=u_{T}(x)-\int_{t}^{T}H\left(F^{\prime}\left(\frac{\varphi_{t}(s,x)}{\varphi_{x}(s,x)}\right)\right){\rm
d}s-(T-t)V(x),\quad&(t,x)\in[0,T]\times{\mathbb{R}}\\\
m(t,x)=\varphi_{x}(t,x),\quad&(t,x)\in[0,T]\times{\mathbb{R}}\\\
\varpi(t)=w_{T}-\int_{t}^{T}w(s){\rm d}s,\quad&t\in[0,T],\end{cases}$

where
$w_{T}=\int_{{\mathbb{R}}}(L^{*}_{z}(T,y)-u^{\prime}_{T}(y))\varphi_{x}(T,y){\rm
d}y$,

$w(s)=\int_{{\mathbb{R}}}\bigg{(}\left(L^{*}_{z}(s,y)\right)_{t}+\left(L^{*}_{y}(s,y)-V(y)\right)_{x}\bigg{)}\varphi_{x}(s,y){\rm
d}y,\quad s\in[0,T],$

and $L^{*}_{z}$ and $L^{*}_{y}$ are defined in (5.10) and (5.11),
respectively.

Because Problem 2 is a convex minimization problem, we approximate its
solution $\varphi$ by using standard optimization methods. Furthermore, using
the approximations for $\varphi$ and Theorem 1.1, we obtain efficient
approximation methods for the solution to Problem 1. In Section 6, we
illustrate the implementation of our approach for the linear-quadratic
setting, for which explicit formulas are provided in [31] that can be used as
benchmarks. For all these benchmarks, our numerical method provides accurate
approximations.

##  2\. Derivation of the variational problem

In this section, we present a formal derivation of the variational problem for
the potential function using the solution of the MFGs system. The precise
assumptions we work with are stated in Section 3. The rigorous statement of
the variational problem is given in Section 4, where we no longer rely on the
solution of the MFGs system.

Let $(u,m,\varpi)$ solve Problem 1 with $m>0$. Then, the second equation in
(1.1) can be written as

$\text{div}_{(t,x)}\left(m,-H^{\prime}(\varpi+u_{x})m\right)=0,\quad[0,T]\times{\mathbb{R}}.$

The previous equation combined with Poincaré lemma (see [19], Theorem 1.22)
gives the existence of a function (the potential)
$\varphi:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ such that

$\begin{cases}m=\varphi_{x},\\\
H^{\prime}(\varpi+u_{x})m=\varphi_{t}.\end{cases}$ (2.1)

Because $H$ is uniformly convex, $H^{\prime}$ is strictly monotone. Therefore,
by (1.3), we have

$F^{\prime}(y)=\left(H^{\prime}\right)^{-1}(y).$ (2.2)

Hence, from the second equation in (2.1), we deduce that

$u_{x}=F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)-\varpi.$

If $V\in C^{1}({\mathbb{R}})$, and $u$ is twice differentiable, we
differentiate the Hamilton-Jacobi equation in (1.1) with respect to $x$ to
obtain

$-(u_{x})_{t}+\left(H(\varpi+u_{x})\right)_{x}+V^{\prime}=0.$

Thus, the system (1.1) in terms of $\varphi$ is reduced to the following two
equations

$\begin{cases}-\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)-\varpi\right)_{t}+\left(H\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)\right)_{x}+V^{\prime}=0&[0,T]\times{\mathbb{R}},\\\
-\int_{{\mathbb{R}}}\varphi_{t}+Q\varphi_{x}~{}{\rm
d}x=0&t\in[0,T],\end{cases}$ (2.3)

with initial condition, $\varphi(0,x)=\int_{-\infty}^{x}m_{0}(y){\rm d}y\,\
x\in{\mathbb{R}}$, and terminal condition

$F^{\prime}\left(\frac{\varphi_{t}(T,x)}{\varphi_{x}(T,x)}\right)-\varpi(T)=u^{\prime}_{T}(x)\quad
x\in{\mathbb{R}}.$ (2.4)

###### Remark 2.1.

Notice that the initial condition implies that $\varphi_{x}(0,x)=m_{0}(x)$,
$x\in{\mathbb{R}}$, which is the first equation in (1.2). Moreover, we have
the following explicit formula for $\varphi$ in terms of the solution
$(u,m,\varpi)$ of (1.1) and (1.2)

$\varphi(t,x)=\int_{-\infty}^{x}m_{0}(y){\rm
d}y+\int_{0}^{t}H^{\prime}(\varpi(s)+u_{x}(s,x))m(s,x){\rm
d}s,\quad(t,x)\in[0,T]\times{\mathbb{R}}.$ (2.5)

Therefore, the potential function $\varphi$, which in principle has a closed
formula arising from the solution of (1.1) and (1.2), can be characterized
using the initial condition with $m_{0}$, (2.3) and (2.4), which depend only,
up to $\varpi$, on problem data.

###### Remark 2.2.

Notice that the first equation in (2.3) shows that the expression

$-\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)_{t}+\left(H\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)\right)_{x}+V^{\prime}$

is independent of $x\in{\mathbb{R}}$, so it is a function of time only and
equal to $\dot{\varpi}$. Similarly, (2.4) shows that

$F^{\prime}\left(\frac{\varphi_{t}(T,x)}{\varphi_{x}(T,x)}\right)-u^{\prime}_{T}(x)$

is independent of $x\in{\mathbb{R}}$, and equal to the constant $\varpi(T)$.
Because any numerical method to compute $\varphi$ provides an approximation of
the value $\varphi(t,x)$, we can not expect the numerical approximation to be
independent of $x$ in (2.3) and (2.4). Therefore, we can not rely on these
formulas to recover $\varpi$ using an approximation of $\varphi$. In Section
5, we provide a formula approximating $\varpi$ that averages the dependence on
$x$, and thus, can be implemented with any approximation of the potential.

Next, consider the functional

$\begin{split}\int_{0}^{T}\int_{{\mathbb{R}}}L(\varphi_{t},\varphi_{x})-V\varphi_{x}{\rm
d}x{\rm
d}t-\int_{{\mathbb{R}}}u^{\prime}_{T}(x)(\varphi(T,x)-\varphi(0,x))~{}{\rm
d}x\end{split}$ (2.6)

subject to $\int_{{\mathbb{R}}}-(\varphi_{t}+Q\varphi_{x}){\rm d}x=0$ on
$[0,T]$, and with initial condition
$\varphi(0,x)=\int_{-\infty}^{x}m_{0}(y){\rm d}y$. Using the augmented
functional associated with the constraint
$\int_{{\mathbb{R}}}-(\varphi_{t}+Q\varphi_{x}){\rm d}x=0$, we show that (2.3)
is an Euler-Lagrange equation. Thus, we introduce a Lagrange multiplier
$\varpi:[0,T]\to{\mathbb{R}}$ for the integral constraint, and we define

$\begin{split}\tilde{I}[\varphi,\varpi]:=\int_{0}^{T}\int_{{\mathbb{R}}}L(\varphi_{t},\varphi_{x})-\varpi\left(\varphi_{t}+Q\varphi_{x}~{}\right)-V\varphi_{x}-u^{\prime}_{T}(x)\varphi_{t}~{}{\rm
d}x{\rm d}t\end{split}$ (2.7)

with initial condition $\varphi(0,x)=\int_{-\infty}^{x}m_{0}(y){\rm d}y$. By
considering critical points $(\varphi,\varpi)$ of the previous functional, we
obtain that (2.3) is the corresponding Euler-Lagrange equation, with the
natural boundary condition (2.4).

###### Proposition 2.3.

Let $(\varphi,\varpi)$ be a critical point of the functional (2.7) over
$C^{2}([0,T]\times{\mathbb{R}})\times C^{1}([0,T])$ satisfying
$\varphi(0,x)=\int_{-\infty}^{x}m_{0}(y){\rm d}y$. Assume further that
$\varphi_{x}>0$. Then, the corresponding Euler-Lagrange equation is equivalent
to (2.3)-(2.4).

###### Proof.

Let $(\varphi,\varpi)$ be a critical point of (2.7). Taking
$(\beta^{1},\beta^{2})\in C^{1}_{c}((0,T]\times{\mathbb{R}})\times C([0,T])$,
we have

$\left.\frac{d}{d\varepsilon}\tilde{I}[(\varphi,\varpi)+\varepsilon(\beta^{1},\beta^{2})]\right|_{\varepsilon=0}=0.$
(2.8)

The previous identity implies that

$-\left(L_{z}(\varphi_{t},\varphi_{x})-\varpi\right)_{t}-\left(L_{y}(\varphi_{t},\varphi_{x})\right)_{x}+V^{\prime}=0,$
(2.9)

and

$L_{z}(\varphi_{t}(T,x),\varphi_{x}(T,x))-\varpi(T)-u^{\prime}_{T}(x)=0$
(2.10)

on $[0,T]\times{\mathbb{R}}$. Because $\varphi_{x}>0$, (1.4) gives

$\begin{split}&L_{z}(\varphi_{t},\varphi_{x})=F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right),\\\
&\left(L_{y}(\varphi_{t},\varphi_{x})\right)_{x}=\left(-\frac{\varphi_{t}}{\varphi_{x}}F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)+F\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)_{x}=-\frac{\varphi_{t}}{\varphi_{x}}\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)_{x}.\end{split}$
(2.11)

Notice that, by (2.2), we have

$\left(H\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)\right)_{x}=\frac{\varphi_{t}}{\varphi_{x}}\left(F^{\prime}\left(\frac{\varphi_{t}}{\varphi_{x}}\right)\right)_{x}.$
(2.12)

Combining the identities in (2.11) with (2.12) and using (2.9), we deduce the
first equation in (2.3). Using the first identity of (2.11) in (2.10), we
obtain (2.4).

Finally, taking $\beta^{1}\equiv 0$ in (2.8), we obtain

$\int_{0}^{T}\left(\int_{{\mathbb{R}}}-(\varphi_{t}+Q\varphi_{x}){\rm
d}x\right)\beta^{2}~{}{\rm d}t=0,$

where $\beta^{2}$ is arbitrary. Thus, the continuity of the map
$t\mapsto\int_{{\mathbb{R}}}-(\varphi_{t}+Q\varphi_{x}){\rm d}x$ implies the
second equation in (2.3). ∎

In Section 5, we address the existence of the price $\varpi$ as a Lagrange
multiplier associated with a minimizer of (2.6).

##  3\. Assumptions

In this section, we state the assumptions to prove the existence of minimizers
of the functional (2.6). This set of assumptions is similar to the ones
introduced in [31] to guarantee the existence and uniqueness of $(u,m,\varpi)$
solving (1.1) and (1.2).

The following two assumptions require standard growth and convexity properties
for $H$.

###### Assumption 1.

There exist constants, $c>0$ and $p>1$, such that the Legendre-Fenchel
transform of $H$, the function $F$ in (1.3), satisfies

$F(v)\geqslant c|v|^{p}.$

###### Assumption 2.

For all $x\in{\mathbb{R}}$, the map $p\mapsto H(p)$ is uniformly convex; that
is, there exists a constant $\kappa>0$ such that
$H^{\prime\prime}(p)\geqslant\kappa$ for all $p\in{\mathbb{R}}$. Moreover,
there exists a positive constant, $C$, such that
$|H^{\prime\prime\prime}|\leqslant C$.

For the supply, to simplify, we assume it is a smooth function of time.

###### Assumption 3.

The supply function, $Q$, is $C^{\infty}([0,T])$.

The following assumption is technical and was used in [31] to get bounds for
the price.

###### Assumption 4.

The potential $V$, the terminal cost $u_{T}$, the initial density function
$m_{0}$ are $C^{2}({\mathbb{R}})$ functions and $V$, $u_{T}$ are globally
Lipschitz. Furthermore, there exists a constant $C>0$ such that

$|V^{\prime\prime}|\leqslant C,\quad|u_{T}^{\prime\prime}|\leqslant
C,\quad|m_{0}^{\prime\prime}|\leqslant C.$

The following condition guarantees the uniqueness of solutions of (1.1) and
(1.2).

###### Assumption 5.

The potential $V$ and the terminal cost $u_{T}$ are convex.

Finally, because we are interested in problems where agent’s assets are
bounded, we require the following assumption on $m_{0}$. This assumption
further simplifies some technical points in the presentation.

###### Assumption 6.

The initial density function $m_{0}$ has compact support; that is, there
exists $R_{0}>0$ such that $\operatorname{supp}(m_{0})\subset[-R_{0},R_{0}]$.

##  4\. The variational approach

Here, we examine a variational problem associated with the MFG system
(1.1)-(1.2) continuing the formal derivation in Section 2. This problem is
obtained by minimizing the functional (2.6) in a suitable class of admissible
functions. We study the existence and uniqueness of solutions to this
variational problem. In Section 5, we establish a formula representing the
solution to the MFG system (1.1)-(1.2), in terms of the solution to this
variational problem.

First, we recall that, under Assumptions 1-5, Theorem 1 in [31] gives
existence and uniqueness of solutions $(u,m,\varpi)$ to Problem 1, where
$m\in\mathcal{P}({\mathbb{R}})\cap C([0,T]\times{\mathbb{R}})$. Moreover, $u$
is a viscosity solution to the first equation in (1.1), Lipschitz continuous
and semi-concave in $x$, and $u_{x}$, $u_{xx}$, $m$ are bounded. Furthermore,
by the results in [10], $\varpi$ is Lipschitz continuous.

###  4.1. Preliminary results for the continuity equation

Before we formulate our variational problem, we prove a general result for the
continuity equation (the second equation in (1.1)) that motivates the choice
of the function spaces. We recall the following result from [15] about the
existence and uniqueness of solutions to the continuity equation. Let
$\mu_{0}\in C^{1}({\mathbb{R}})$, $b\in
L^{1}([0,T];W_{loc}^{1,1}({\mathbb{R}}))\cap C([0,T]\times{\mathbb{R}})$ and
$b_{x}\in L^{\infty}([0,T]\times{\mathbb{R}})$. Then, the continuity equation

$\begin{cases}\mu_{t}-(b\mu)_{x}=0&[0,T]\times{\mathbb{R}},\\\
\mu(0,x)=\mu_{0}(x)&x\in{\mathbb{R}}\end{cases}$ (4.1)

has a unique solution $\mu\in L^{\infty}([0,T]\times{\mathbb{R}})$ in
distributional sense. The existence result follows from Theorem 1.1 in [15],
which addresses the existence and uniqueness of distributional solutions to
the continuity equation (4.1) for a vector field $b$ satisfying weaker
conditions.

Now, we prove that if the initial condition $\mu_{0}$ of the continuity
equation is compactly supported, the solution $\mu$ is also compactly
supported.

###### Proposition 4.1.

Let $\mu_{0}\in C^{1}({\mathbb{R}})$, $b\in
L^{1}([0,T];W_{loc}^{1,1}({\mathbb{R}}))\cap C([0,T]\times{\mathbb{R}})$ and
$b_{x}\in L^{\infty}([0,T]\times{\mathbb{R}})$. Assume further that
$\mu_{0}\in C_{c}^{1}({\mathbb{R}})$. Then, the unique solution to the
continuity equation (4.1) has compact support; that is, $\mu\in
L_{c}^{\infty}([0,T]\times{\mathbb{R}})$.

###### Proof.

From the results in [15], it follows that there exists a unique, $\mu\in
L^{\infty}([0,T]\times{\mathbb{R}})$ solving (4.1) in the distributional
sense. Let $b^{\varepsilon}$ be a sequence of functions in
$C^{\infty}([0,T]\times{\mathbb{R}})$ satisfying:

  * •

$b^{\varepsilon}$ is Lipschitz continuous w.r.t. $x$, and its Lipschitz
constant satisfies $\mbox{Lip}(b^{\varepsilon})\leqslant\mbox{Lip}(b)$,

  * •

$b^{\varepsilon}\to b$ uniformly on every compact set of
$[0,T]\times{\mathbb{R}}$.

We can obtain such sequence $b^{\varepsilon}$ by considering the convolution
with standard mollifiers in $x$ and a partition of unity construction in $t$.
Next, we consider the continuity equation with the vector field
$b^{\varepsilon}$

$\begin{cases}\mu^{\varepsilon}_{t}-(b^{\varepsilon}\mu^{\varepsilon})_{x}=0&[0,T]\times{\mathbb{R}},\\\
\mu^{\varepsilon}(0,x)=\mu_{0}(x)&x\in{\mathbb{R}}.\end{cases}$ (4.2)

Because $b^{\varepsilon},\,b_{x}^{\varepsilon}\in
C^{1}([0,T]\times{\mathbb{R}})$, by Theorem 6.3 in [38], (4.2) has a unique
solution $\mu^{\varepsilon}\in C^{1}([0,T]\times{\mathbb{R}})$ given by

$\mu^{\varepsilon}(t,X^{\varepsilon}(t;y))J(t;y)=\mu_{0}(y),\quad(t,y)\in[0,T]\times{\mathbb{R}},$
(4.3)

where

$J(t;y)=\exp\left(\int_{0}^{t}b^{\varepsilon}_{x}(s,X^{\varepsilon}(s;y))~{}{\rm
d}s\right),$ (4.4)

and $X^{\varepsilon}$ solves the following initial value problem

$\begin{cases}\dot{X}^{\varepsilon}(t;y)=b^{\varepsilon}(t,X^{\varepsilon}(t;y))&\quad(t,y)\in(0,T]\times{\mathbb{R}},\\\
X^{\varepsilon}(0;y)=y&\quad y\in{\mathbb{R}}.\end{cases}$ (4.5)

Because $b^{\varepsilon}\in C^{1}([0,T]\times{\mathbb{R}})$, the map $y\mapsto
X^{\varepsilon}(t;y)$ is a diffeomorphism (see [37], Chapter 3). Moreover,
because $b^{\varepsilon}\in C^{1}([0,T]\times{\mathbb{R}})$ is Lipschitz
continuous w.r.t. $x$, we have $|b^{\varepsilon}(t,x)|\leqslant
C_{\varepsilon}\left(1+|x|\right)$, where, by the uniform convergence of
$b^{\varepsilon}$ to $b$ on compact sets,

$C_{\varepsilon}\leqslant\max\\{1+|b(t,0)|,\mbox{Lip}(b)\\}.$ (4.6)

Applying Grönwall’s inequality to (4.5), provides

$|X^{\varepsilon}(t;y)|\leqslant\left(|y|+C_{\varepsilon}T\right)\left(1+C_{\varepsilon}Te^{C_{\varepsilon}T}\right).$
(4.7)

Because $|y|>R_{0}$ implies $\mu_{0}(y)=0$ for some $R_{0}>0$, (4.3) shows
that $\mu^{\varepsilon}$ may have non-zero values only for those $y$
satisfying $|y|\leqslant R_{0}$, for which (4.7) implies

$|X^{\varepsilon}(t;y)|\leqslant\left(R_{0}+C_{\varepsilon}T\right)\left(1+C_{\varepsilon}Te^{C_{\varepsilon}T}\right)=:R_{\varepsilon}.$

Thus,
$\mbox{supp}(\mu^{\varepsilon})\subset[0,T]\times[-R_{\varepsilon},R_{\varepsilon}]$,
which, by (4.6), provides the existence of $R>0$, depending on $T$ and
$\mbox{Lip}(b)$, such that $\operatorname{supp}\mu^{\varepsilon}\subset[-R,R]$
for every $1\gg\varepsilon>0$. Furthermore, (4.3) and (4.4) imply that there
exists $C\geqslant 0$, depending on $T$, $\mbox{Lip}(b)$ and $\mu_{0}$, such
that $||\mu^{\varepsilon}||_{L^{\infty}([0,T]\times{\mathbb{R}})}\leqslant C$
for all $1\gg\varepsilon>0$. Hence, by Banach-Alaoglu theorem, there exists
$\bar{\mu}\in L^{\infty}([0,T]\times{\mathbb{R}})$ such that

$\mu^{\varepsilon}\overset{\ast}{\rightharpoonup}\bar{\mu}\quad\mbox{as}\quad\varepsilon\to
0\quad\text{in}\quad L^{\infty}([0,T]\times{\mathbb{R}}).$

Consequently, $\operatorname{supp}\bar{\mu}\subset[-R,R]$ as well. On the
other hand, $\mu^{\varepsilon}$ also solves (4.2) in the sense of
distributions; that is,

$-\int_{0}^{T}\int_{{\mathbb{R}}}\mu^{\varepsilon}\left(\phi_{t}-b^{\varepsilon}\phi_{x}\right){\rm
d}x{\rm d}t=\int_{{\mathbb{R}}}\mu_{0}\phi~{}{\rm d}x,$ (4.8)

for any $\phi\in C^{1}_{c}([0,T)\times{\mathbb{R}})$.

Thus, given $\phi\in C^{1}_{c}([0,T)\times{\mathbb{R}})$, we write

$\begin{split}&-\int_{0}^{T}\int_{{\mathbb{R}}}\bar{\mu}\left(\phi_{t}-b\phi_{x}\right){\rm
d}x{\rm d}t\\\
&=-\int_{0}^{T}\int_{{\mathbb{R}}}\mu^{\varepsilon}\left(\phi_{t}-b^{\varepsilon}\phi_{x}\right)+\left(\mu^{\varepsilon}-\bar{\mu}\right)\left(\phi_{t}-b\phi_{x}\right)+\mu^{\varepsilon}\phi_{x}\left(b-b^{\varepsilon}\right){\rm
d}x{\rm d}t.\end{split}$ (4.9)

Because $\phi_{t}$, $\phi_{x}$, $b\phi_{x}\in L^{1}([0,T]\times{\mathbb{R}})$,
the second term on the right-hand side of (4.9) vanishes as $\varepsilon\to
0$. Furthermore, using the uniform bound for $\mu^{\varepsilon}$, and because
$b^{\varepsilon}$ converges uniformly to $b$ in the compact support of $\phi$,
we obtain that the third term on the right-hand side of (4.9) also vanishes as
$\varepsilon\to 0$. Thus, using (4.8), we get

$-\int_{0}^{T}\int_{{\mathbb{R}}}\bar{\mu}\left(\phi_{t}-b\phi_{x}\right){\rm
d}x{\rm d}t=\int_{{\mathbb{R}}}\mu_{0}\phi~{}{\rm d}x,$

and since $\phi$ is arbitrary, we conclude that $\bar{\mu}$ is a solution to
(4.1) in the distributional sense.

To conclude the proof, it is enough to recall that the results in [15] provide
uniqueness for the initial value problem in (4.1), in the sense of the
distributions. ∎

Applying the previous result to the MFG system (1.1)-(1.2), we obtain the
following.

###### Corollary 4.2.

Suppose that Assumptions 1-5 hold. Let $(u,m,\varpi)$ be the solution to
(1.1). Assume further that Assumption 6 holds with $R_{0}>0$. Then, $m$ is
compactly supported; that is, there exists a constant $R_{m}\geqslant R_{0}$,
such that $\operatorname{supp}m(t,\cdot)\subseteq[-R_{m},R_{m}]$ for
$t\in[0,T]$. Moreover, $R_{m}$ is bounded by a constant that depends only on
the problem data.

###### Proof.

Let $b(t,x)=H^{\prime}(\varpi(t)+u_{x}(t,x))$ denote the vector field of the
continuity equation in (1.1). By Proposition 8 in [31], $|u_{xx}|\leqslant
C(T,V,u_{T})$, which implies that $u_{x}$ is Lipschitz w.r.t. $x$. By
Assumptions 1 and 2, for $p>2$, $|H^{\prime\prime}|\leqslant C(F)$, for some
$C(F)>0$. Thus, $b(t,\cdot)\in C^{1}({\mathbb{R}})$ is Lipschitz continuous in
${\mathbb{R}}$ uniformly with respect to $t$. Furthermore, the Lipchitz
constant satisfies

$\mbox{Lip}(b(t,\cdot))\leqslant C_{0},$

where $C_{0}=C_{0}(T,V,F,u_{T})$. Therefore, Proposition 4.1 implies the first
part of the result. Moreover, (4.7) shows that

$R_{m}\leqslant\left(R_{0}+C_{0}T\right)\left(1+C_{0}Te^{C_{0}T}\right),$
(4.10)

which concludes the proof. ∎

###### Remark 4.3.

Consider the potential $\varphi$ associated with the solution $(u,m,\varpi)$
of the MFG system (1.1)-(1.2), as given by (2.1). By Corollary 4.2, we deduce
that the gradient of the potential $\varphi$ has compact support; that is,
$\operatorname{supp}(\varphi_{t}(t,\cdot)),\operatorname{supp}(\varphi_{x}(t,\cdot))\subseteq[-R_{m},R_{m}]$
for all $t\in[0,T]$. Thus, (4.10) shows that, by selecting

$R>\left(R_{0}+C_{0}T\right)\left(1+C_{0}Te^{C_{0}T}\right),$

we obtain a compact set $[-R,R]$ that depends only on problem data, and which
contains the support of the gradient of $\varphi$ when (2.5) holds. Thus,
using this compact set, we can formulate our variational problem independently
of the solution $(u,m,\varpi)$ of the MFGs system (1.1) and (1.2). Notice that
(2.5) already suggests a candidate for a minimizer. However, we study the
existence of solutions to the variational problem independently of solutions
to the MFGs system. Moreover, if uniqueness holds and we have existence for
both problems, then (2.5) is the unique minimizer.

###  4.2. Statement of the variational problem

In this subsection, we present our variational approach rigorously using only
problem data. We start with the notations and the definition of admissible
functions. Then, we formulate and study the variational problem.

Let $R_{0}$ be given by Assumption 6 and let

$R>\max\left\\{\left(R_{0}+C_{0}T\right)\left(1+C_{0}Te^{C_{0}T}\right),R_{0}+\|Q\|_{L^{1}([0,T])}\right\\}.$
(4.11)

Notice that, by (4.10), $R$ is an upper bound for $R_{m}$, as required,
according to Remark 4.3. The additional requirement
$R>R_{0}+\|Q\|_{L^{1}([0,T])}$ guarantees that the set of admissible functions
that we define below is not empty. Set

$\Omega_{R}=[0,T]\times[-R,R],\quad\Omega=[0,T]\times{\mathbb{R}}.$

We denote by $\mathcal{M}(\Omega_{R})$ ($\mathcal{M}(\Omega)$) the set of
Radon measures on $\Omega_{R}\subset{\mathbb{R}}^{2}$
($\Omega\subset{\mathbb{R}}^{2}$) and by $BV(\Omega_{R})$ ($BV(\Omega)$) the
set of functions with bounded variation on $\Omega_{R}$ ($\Omega$) (see [23],
[9]).

To define the admissible set for our variational problem, we rewrite the
balance condition, the second equation in (2.3). Recall that
$\operatorname{supp}(m_{0})\subset[-R_{0},R_{0}]$ and $R>R_{0}$. Let

$M_{0}(x)=\int_{-\infty}^{x}m_{0}(y){\rm d}y=\int_{-R}^{x}m_{0}(y){\rm
d}y,\quad x\in{\mathbb{R}},$ (4.12)

be the cumulative density function of $m_{0}$. Note that after integrating the
balance condition over $[0,t]$, and requiring that
$\int_{{\mathbb{R}}}\varphi_{x}(t,x){\rm d}x=1$ for $t\in[0,T]$ (which follows
in case that (2.1) holds), we get

$\int_{0}^{t}\int_{{\mathbb{R}}}\varphi_{t}~{}{\rm d}x{\rm
d}s=-\int_{0}^{t}Q(s){\rm d}s,\quad t\in[0,T].$

Therefore, we write the balance condition as

$\int_{{\mathbb{R}}}\varphi(t,x)-M_{0}(x)~{}{\rm d}x=-\int_{0}^{t}Q(s){\rm
d}s,\quad t\in[0,T].$ (4.13)

Relying on (4.13) and taking into account the discussion in Remark 4.3, for
any set $A\subset{\mathbb{R}}^{2}$ satisfying $[0,T]\times[-R,R]\subseteq A$,
we denote

$\displaystyle\mathcal{B}_{R}(A)=$ $\displaystyle\left\\{(\varphi-M_{0})\in
W^{1,1}(A):~{}\operatorname{supp}(\varphi_{t}(t,\cdot)),\operatorname{supp}(\varphi_{x}(t,\cdot))\subseteq(-R,R),~{}t\in[0,T]\right\\},$
$\displaystyle\mathcal{B}(A)=$
$\displaystyle\left\\{\varphi\in\mathcal{B}_{R}(A):~{}\varphi_{x}\geqslant
0,~{}\varphi(0,x)=\int_{-R}^{x}m_{0}(y){\rm d}y,~{}x\in{\mathbb{R}}\right.$
$\displaystyle\quad\left.\int_{{\mathbb{R}}}\varphi(t,x)-M_{0}(x)~{}{\rm
d}x=-\int_{0}^{t}Q(s)~{}{\rm d}s,~{}\int_{-R}^{R}\varphi_{x}(t,x){\rm
d}x=1,~{}t\in[0,T]\right\\},$

which are convex sets. Before proceeding, we prove a crucial property of the
set $\mathcal{B}(\Omega)$.

###### Proposition 4.4.

For any function $\varphi\in\mathcal{B}(\Omega)$, we have, for $t\in[0,T]$,

$\varphi(t,x)=\begin{cases}0\quad&x\in(-\infty,-R],\\\
\varphi(t,x)\quad&x\in(-R,R),\\\ 1\quad&x\in[R,+\infty).\end{cases}$

###### Proof.

Because $(\varphi-M_{0})\in W^{1,1}(\Omega)$ and
$\lim\limits_{x\to+\infty}M_{0}(x)=1$, for each $t\in[0,T]$, there exists a
sequence ${x_{k}}$ such that $x_{k}\to\infty$ and
$\lim\limits_{k\to+\infty}\varphi(t,x_{k})=1$. On the other hand, recalling
that

$\operatorname{supp}(\varphi_{t}(t,\cdot)),\operatorname{supp}(\varphi_{x}(t,\cdot))\subset(-R,R),$

we have that $\varphi$ is constant on $\Omega\setminus[0,T]\times(-R,R)$.
Consequently, $\varphi(t,x)=1$ for $x\in[R,+\infty)$. Similarly, we can prove
that $\varphi(t,x)=0$, $x\in(-\infty,-R]$. ∎

Finally, the set of admissible functions for our variational problem is given
by

$\begin{split}\mathcal{A}(\Omega_{R})=\left\\{\varphi\in\mathcal{B}(\Omega_{R}):~{}\varphi(t,-R)=0\right\\}.\end{split}$
(4.14)

As a result of Proposition 4.4, we obtain the following relation between the
admissible set $\mathcal{A}(\Omega_{R})$ and the set $\mathcal{B}(\Omega)$.

###### Corollary 4.5.

For any function $\varphi\in\mathcal{B}(\Omega)$ there exist a function
$\tilde{\varphi}\in\mathcal{A}(\Omega_{R})$ such that
$\varphi\equiv\tilde{\varphi}$ in $\Omega_{R}$. The opposite is also true.

Under Assumption 4, we have

$\left|\int_{\Omega_{R}}u^{\prime}_{T}(x)\varphi_{t}~{}{\rm d}x{\rm
d}t\right|\leqslant\mbox{Lip}{(u_{T})}\int_{\Omega_{R}}|\varphi_{t}|~{}{\rm
d}x{\rm d}t,$ (4.15)

where $\mbox{Lip}{(u_{T})}$ is the Lipschitz constant of $u_{T}$. Relying on
the previous inequality, we consider the following variational problem

$\begin{split}\inf_{\varphi\in\mathcal{B}(\Omega)}\int_{\Omega_{R}}L(\varphi_{t},\varphi_{x})-V\varphi_{x}-u^{\prime}_{T}(x)\varphi_{t}~{}{\rm
d}x{\rm d}t,\end{split}$

which, by Corollary 4.5, coincides with the following (see (2.6))

$\inf_{\varphi\in\mathcal{A}(\Omega_{R})}I[\varphi],$ (4.16)

where

$I[\varphi]:=\int_{\Omega_{R}}L(\varphi_{t},\varphi_{x})-V\varphi_{x}-u^{\prime}_{T}(x)\varphi_{t}~{}{\rm
d}x{\rm d}t.$

As anticipated in Remark 4.3, (4.11) guarantees that the previous variational
problem does not rely on the solution $(u,m,\varpi)$ to (1.1)-(1.2) but only
on the data of Problem 1. Moreover, the infimum in (4.16) can be attained by
at most one function, as we show next.

###### Proposition 4.6.

Suppose that Assumptions 1-6 hold. Then, at most, one function attains the
infimum in (4.16).

###### Proof.

Let $\varphi^{1}$ and $\varphi^{2}$ attain the infimum in (4.16). By
Proposition 4.7, we denote

$\displaystyle\ell=\min\limits_{(\varphi)\in\mathcal{A}(\Omega_{R})}I[\varphi]\in{\mathbb{R}}.$

Thus, $I[\varphi^{1}]=I[\varphi^{2}]=\mathcal{\ell}$. Setting
$\bar{\varphi}=\frac{1}{2}(\varphi^{1}+\varphi^{2})$, and using the convexity
of $L$, we obtain

$\ell\leqslant
I[\bar{\varphi}]\leqslant\tfrac{1}{2}I[\varphi^{1}]+\tfrac{1}{2}I[\varphi^{2}]=\ell.$
(4.17)

Hence, $\bar{\varphi}$ is also minimizer of (4.16). Let
$\tilde{\varphi}=\frac{\varphi^{1}+\bar{\varphi}}{2}$ and

$\displaystyle{\mathcal{U}}_{1}=\\{(t,x)\in\Omega_{R}:{\varphi}_{x}^{1}>0\\},\quad{\mathcal{U}}_{2}=\\{(t,x)\in\Omega_{R}:{\varphi}_{x}^{2}>0\\},$
$\displaystyle\tilde{{\mathcal{U}}}=\\{(t,x)\in\Omega_{R}:\tilde{\varphi}_{x}>0\\}=\bar{{\mathcal{U}}}=\\{(t,x)\in\Omega_{R}:\bar{\varphi}_{x}>0\\}={\mathcal{U}}_{1}\cup{\mathcal{U}}_{2}.$

Arguing as in (4.17), we have

$\ell\leqslant
I[\tilde{\varphi}]\leqslant\tfrac{1}{2}I[\varphi^{1}]+\tfrac{1}{2}I[\bar{\varphi}]=\ell.$
(4.18)

This with (4.17), yields that

$\displaystyle
L(\varphi^{1}_{t},\varphi^{1}_{x}),\,L(\varphi^{2}_{t},\varphi^{2}_{x}),\,L(\bar{\varphi}_{t},\bar{\varphi}_{x}),\,L(\tilde{\varphi}_{t},\tilde{\varphi}_{x})<+\infty\enspace\hbox{a.e.~{}in
}\Omega_{R}.$

Therefore,

$\displaystyle\varphi^{1}_{t}=0\enspace\text{ a.e.~{}in
}\Omega_{R}\setminus{\mathcal{U}}_{1},$ (4.19)
$\displaystyle\varphi^{2}_{t}=0\enspace\text{ a.e.~{}in
}\Omega_{R}\setminus{\mathcal{U}}_{2},$
$\displaystyle\bar{\varphi}_{t}=0\enspace\text{ a.e.~{}in
}\Omega_{R}\setminus\bar{{\mathcal{U}}}$
$\displaystyle\tilde{\varphi}_{t}=0\enspace\text{ a.e.~{}in
}\Omega_{R}\setminus\tilde{{\mathcal{U}}}.$

Furthermore, (4.18) implies

$\begin{split}\int_{\Omega_{R}}\Big{(}\tfrac{1}{2}L(\varphi^{1}_{t},\varphi^{1}_{x})+&\tfrac{1}{2}L(\bar{\varphi}_{t},\bar{\varphi}_{x})-L(\tilde{\varphi}_{t},\tilde{\varphi}_{x})\Big{)}\,{\rm
d}x{\rm d}t=0.\end{split}$ (4.20)

The convexity of $L$ and (4.20) implies

$\tfrac{1}{2}L(\varphi^{1}_{t},\varphi^{1}_{x})+\tfrac{1}{2}L(\bar{\varphi}_{t},\bar{\varphi}_{x})-L(\tilde{\varphi}_{t},\tilde{\varphi}_{x})=0,\quad\text{a.e.
in }\Omega_{R}.$

Consequently, the following also holds

$\tfrac{1}{2}L(\varphi^{1}_{t},\varphi^{1}_{x})+\tfrac{1}{2}L(\bar{\varphi}_{t},\bar{\varphi}_{x})-L(\tilde{\varphi}_{t},\tilde{\varphi}_{x})=0,\quad\text{a.e.
in }\quad{\mathcal{U}}_{1}\cap\bar{{\mathcal{U}}}\cap\tilde{{\mathcal{U}}}.$
(4.21)

Because $L$ is strictly convex in ${\mathbb{R}}\times{\mathbb{R}}^{+}$ and
${\mathcal{U}}_{1}\cap\bar{{\mathcal{U}}}\cap\tilde{{\mathcal{U}}}={\mathcal{U}}_{1}\subset{\mathbb{R}}\times{\mathbb{R}}^{+}$,
we obtain from (4.21) that

$\begin{cases}\varphi^{1}_{t}=\bar{\varphi}_{t}\\\
\varphi^{1}_{x}=\bar{\varphi}_{x}.\end{cases}\quad\text{a.e.
in}\quad{\mathcal{U}}_{1}$

Hence,

$\begin{cases}\varphi^{1}_{t}=\varphi^{2}_{t}\\\
\varphi^{1}_{x}=\varphi^{2}_{x}.\end{cases}\quad\text{a.e.
in}\quad{\mathcal{U}}_{1}$ (4.22)

Taking $\varphi^{2}$ instead of $\varphi^{1}$ in (4.18) and arguing as before,
we obtain

$\begin{cases}\varphi^{1}_{t}=\varphi^{2}_{t}\\\
\varphi^{1}_{x}=\varphi^{2}_{x}.\end{cases}\quad\text{a.e.
in}\quad{\mathcal{U}}_{2}$ (4.23)

Combing (4.19), (4.22) and (4.23), we conclude that $\varphi^{1}=\varphi^{2}$.
∎

Next, we prove that the infimum in (4.16) is bounded.

###### Proposition 4.7.

Assume that Assumptions 1-6 hold. Then, there exist positive constants,
$C_{1}$ and $C_{2}$, depending only on the problem data such that

$-C_{2}\leqslant\inf_{\varphi\in\mathcal{A}(\Omega_{R})}I[\varphi]\leqslant
C_{1}.$ (4.24)

Furthermore, there exists a positive constant, $C$, depending only on problem
data, such that for every minimizing sequence
$\\{\varphi^{n}\\}_{n\in{\mathbb{N}}}$ of the variational problem (4.16), we
have

$\int_{\Omega_{R}}\tfrac{\left|\varphi_{t}^{n}\right|^{p}}{(\varphi^{n}_{x})^{p-1}}{\rm
d}x{\rm d}t,\quad\|\varphi^{n}_{t}\|_{L^{1}(\Omega_{R})}\leqslant C.$ (4.25)

###### Proof.

First, we prove the upper bound in (4.24). Let

$\varphi^{0}(t,x)=M_{0}(x-q(t)),$

where $M_{0}$ is defined by (4.12) and $q(t)=\int_{0}^{t}Q(\tau)~{}d\tau$.
Therefore, since $q(0)=0$, we have

$\varphi^{0}(0,x)=\int_{-R}^{x}m_{0}(y){\rm
d}y,\quad\varphi^{0}_{x}=m_{0}(x-q(t)),\quad\varphi^{0}_{t}=-m_{0}(x-q(t))Q(t).$
(4.26)

Thus,

$\varphi^{0}\in\mathcal{A}(\Omega_{R}).$ (4.27)

Taking into account (4.26), we have

$\displaystyle\inf_{\mathcal{A}(\Omega)}I[\varphi]\leqslant I[\varphi^{0}]$
$\displaystyle\leqslant\int_{\Omega_{R}}L(\varphi^{0}_{t},\varphi^{0}_{x})+||V||_{L^{\infty}([-R,R])}\varphi_{x}^{0}+||u_{T}^{\prime}||_{L^{\infty}(\Omega_{R})}|\varphi_{t}^{0}|~{}{\rm
d}x{\rm d}t=:C_{1}.$ (4.28)

Next, relying on this bound, we prove (4.25), implying the lower bound in
(4.24). By Assumption 1 and 2.2, $L\geqslant 0$. Thus, for all
$\varphi\in\mathcal{A}(\Omega_{R})$, we have

$-\int_{\Omega_{R}}V\varphi_{x}+u^{\prime}_{T}\varphi_{t}{\rm d}x{\rm
d}t\leqslant I[\varphi].$ (4.29)

Recalling that $V$ is continuous and taking into account Assumption 4 by
(4.15) and (4.29), we get

$-C(u_{T},R,V)-\mbox{Lip}(u_{T})\int_{\Omega_{R}}\left|\varphi_{t}(t,x)\right|{\rm
d}x{\rm d}t\leqslant I[\varphi].$ (4.30)

From (4.28) follows that for any minimizing sequence
$\\{\varphi^{n}\\}_{n\in{\mathbb{N}}}$, there exists $N$ such that $n\geqslant
N$ implies $I[\varphi^{n}]\leqslant C_{1}+1$. Consequently, recalling the
definition of $L$ in (1.4), by Assumption 1 and (4.15), we deduce that

$\begin{split}c\int_{\Omega_{R}}\frac{\left|\varphi_{t}^{n}\right|^{p}}{(\varphi^{n}_{x})^{p-1}}{\rm
d}x{\rm
d}t\leqslant&C_{1}+1+||V||_{L^{\infty}([-R,R])}\int_{\Omega_{R}}\varphi^{n}_{x}{\rm
d}x{\rm
d}t+\mbox{Lip}(u_{T})\int_{\Omega_{R}}\left|\varphi^{n}_{t}(t,x)\right|{\rm
d}x{\rm d}t\\\
\leqslant&C+\mbox{Lip}(u_{T})\int_{\Omega_{R}}\left|\varphi^{n}_{t}(t,x)\right|{\rm
d}x{\rm d}t,\end{split}$ (4.31)

for all $n\in{\mathbb{N}}$. On the other hand, by Young’s inequality, we have

$\int_{\Omega_{R}}\left|\varphi_{t}^{n}\right|{\rm d}x{\rm
d}t=\int_{\Omega_{R}}\frac{\left|\varphi_{t}^{n}\right|}{(\varphi^{n}_{x})^{\frac{p-1}{p}}}(\varphi^{n}_{x})^{\frac{p-1}{p}}{\rm
d}x{\rm
d}t\leqslant\varepsilon\int_{\Omega_{R}}\frac{\left|\varphi_{t}^{n}\right|^{p}}{(\varphi^{n}_{x})^{p-1}}{\rm
d}x{\rm d}t+C(\varepsilon)\int_{\Omega_{R}}\varphi_{x}^{n}{\rm d}x{\rm d}t,$
(4.32)

where $\varepsilon=\frac{c}{2\mbox{Lip}(u_{T})}$. Recalling that
$\int_{-R}^{R}\varphi_{x}(\cdot,x){\rm d}x=1$, the preceding inequality and
(4.31) imply (4.25). Finally, (4.25) and (4.30) yield the lower bound in
(4.24). ∎

Thus, for all minimizing sequences of the variational problem (4.16), we
obtain uniform bounds in $W^{1,1}(\Omega_{R})$. Therefore, any minimizing
sequence has a weakly convergent sub-sequence in $BV(\Omega_{R})$ ([23],
Chapter 5). However, it is not guaranteed that the infimum in (4.16) is
attained in ${\mathcal{A}}(\Omega_{R})$. Therefore, we enlarge the set of
admissible functions by relaxing the conditions defining
$\mathcal{A}(\Omega_{R})$, as we present in the next section.

###  4.3. Relaxed variational problem

Here, we relax the variational problem (4.16) to ensure the existence of
minimizers in the set of admissible functions.

First, we extend the functional in (4.16) to the convex set

$BV_{0}^{+}(\Omega_{R})=\\{\psi\in BV(\Omega_{R}):\psi_{x}\geqslant 0\\}.$

For that, let $W:BV_{0}^{+}(\Omega_{R})\to{\mathbb{R}}\cup\\{+\infty\\}$ be
given by

$W[\varphi]=\begin{cases}\int_{\Omega_{R}}L(\varphi_{t},\varphi_{x})-V\varphi_{x}-u^{\prime}_{T}(x)\varphi_{t}~{}{\rm
d}x{\rm d}t,~{}\varphi\in W^{1,1}(\Omega_{R})\cap BV_{0}^{+}(\Omega_{R})\\\
+\infty,\quad\mbox{otherwise}.\end{cases}$

In $BV(\Omega_{R})$ we consider the intermediate convergence; that is,
$(\varphi_{k})_{k\in{\mathbb{N}}}\subset BV(\Omega_{R})$ converges to
$\varphi\in BV(\Omega_{R})$ in the intermediate (or strict) sense if

$\varphi_{k}\to\varphi\mbox{ in
}L^{1}(\Omega_{R})\quad\mbox{and}\quad\|D\varphi_{k}\|(\Omega_{R})\to\|D\varphi\|(\Omega_{R}),$

where $\|D\varphi\|(\Omega_{R})$ is the total variation of the measure
$D\varphi$ on $\Omega_{R}$ (see [9]). We recall that $W^{1,1}(\Omega_{R})$ is
dense in $BV(\Omega_{R})$ with respect to the intermediate convergence (see
Theorem 10.1.2 in [11]). We aim to define a functional ${\mathcal{W}}$, the
sequential lower semicontinuous envelope of $W$ w.r.t. intermediate
convergence on $BV$ (Chapter 3, [20]); that is

$\displaystyle{\mathcal{W}}[\varphi]=$
$\displaystyle\sup\left\\{G[\varphi]:~{}G\leqslant W,~{}G\mbox{ is
sequentially lower semicontinuous on }BV(\Omega_{R})\right.$
$\displaystyle\quad\quad\left.\mbox{ w.r.t. intermediate
convergence}\right\\},$

which is the greatest functional below $W$ that is sequentially lower semi-
continuous w.r.t intermediate convergence in $BV(\Omega_{R})$. Let

$\displaystyle\mathcal{J}[\varphi]=$
$\displaystyle\inf\Biggl{\\{}\Biggr{.}\liminf\limits_{n\to\infty}I[\varphi^{n}]:\,\\{\varphi^{n}\\}\subset
W^{1,1}(\Omega_{R})\cap BV_{0}^{+}(\Omega_{R}),$
$\displaystyle\quad\quad\quad\varphi^{n}\to\varphi\text{ in the sense of the
intermediate convergence in }BV(\Omega_{R})\Biggl{.}\Biggr{\\}}.$

Next, we prove that actually ${\mathcal{W}}=\mathcal{J}$ and obtain explicit
expression for $\mathcal{J}$.

Assuming that Assumption 1 holds for some $c>0$, $p>1$ and arguing as in
(4.32) by using Young’s inequality, we obtain

$cp(|v_{1}|+v_{2})\leqslant
L(v_{1},v_{2})+c(2p-1)v_{2},\quad(v_{1},v_{2})\in{\mathbb{R}}\times{\mathbb{R}}^{+}_{0}.$
(4.33)

Let

$f(v_{1},v_{2})=L(v_{1},v_{2})-u^{\prime}_{T}v_{1}-Vv_{2}=f_{N}(v_{1},v_{2})+f_{L}(v_{1},v_{2}),$
(4.34)

where $f_{N}(v_{1},v_{2})=L(v_{1},v_{2})+c(2p-1)v_{2}$ and
$f_{L}(v_{1},v_{2})=-u^{\prime}_{T}v_{1}-(V+c(2p-1))v_{2}$. According to
Lemmas 8.1 and 8.3 in [12], if Assumption 2 holds, the function $L$ defined in
(1.4) is convex and lower semicontinuous in
${\mathbb{R}}\times{\mathbb{R}}_{0}^{+}$, therefore, $f_{N}$ as well.

For the next result, we compute the recession function, $\bar{f}_{N}$, of
$f_{N}$, which is given by

$\bar{f}_{N}(z,y):=\sup\left\\{f(w_{1}+z,w_{2}+y)-f(w_{1},w_{2}):~{}(w_{1},w_{2})\in\mbox{dom}_{e}(f_{N})\right\\},\quad(z,y)\in{\mathbb{R}}\times{\mathbb{R}},$

where

$\mbox{dom}_{e}(f_{N}):=\left\\{(w_{1},w_{2})\in{\mathbb{R}}\times{\mathbb{R}}_{0}^{+}:~{}f_{N}(w_{1},w_{2})<+\infty\right\\}.$

Because $f_{N}$ is convex, from Theorem 4.70 in [26], we have

$\bar{f}_{N}(z,y)=\lim\limits_{t\to\infty}\frac{f_{N}((z,y)t+(w_{1},w_{2}))-f_{N}(w_{1},w_{2})}{t},$

for any $(w_{1},w_{2})\in{\mathbb{R}}\times{\mathbb{R}}_{0}$. Taking
$(w_{1},w_{2})=(0,0)$ in the preceding equation and considering (1.4), we
deduce that $f_{N}$ is equal to its recession function; that is,

$\bar{f}_{N}(z,y)=f_{N}(z,y)=\begin{cases}F\left(\frac{z}{y}\right)y+c(2p-1)y&(z,y)\in{\mathbb{R}}\times{\mathbb{R}}^{+},\\\
+\infty&z\neq 0,y=0,\\\ 0&z=0,y=0,\end{cases}$

where constants $c$ and $p$ are given by Assumption 1. Using the preceding
observation, we prove that the first integrant of the functional in (2.6) is
sequentially lower semi-continuous w.r.t. the weak -$*$ convergence of
measures.

###### Proposition 4.8.

Suppose that Assumption 2 holds. Let $(v_{1}^{n},v_{2}^{n})\in
L^{1}(\Omega_{R})\times L^{1}(\Omega_{R};{\mathbb{R}}_{0}^{+})$ and
$\mu=(\mu_{1},\mu_{2})\in{\mathcal{M}}(\Omega_{R})\times{\mathcal{M}}(\Omega_{R};{\mathbb{R}}_{0}^{+})$
be such that

$(v_{1}^{n},v_{2}^{n}){\mathcal{L}}^{2}\lfloor\Omega_{R}\overset{\ast}{\rightharpoonup}(\mu_{1},\mu_{2}),\quad\text{
in
}{\mathcal{M}}(\Omega_{R})\times{\mathcal{M}}(\Omega_{R};{\mathbb{R}}_{0}^{+}).$

Then,

$\displaystyle\liminf\limits_{n\to\infty}\int_{\Omega_{R}}f_{N}(v_{1}^{n},v_{2}^{n}){\rm
d}x{\rm d}t$
$\displaystyle\geqslant\int_{\Omega_{R}}f_{N}\left(\frac{d\mu}{d{\mathcal{L}}^{2}}(t,x)\right){\rm
d}x{\rm
d}t+\int_{\Omega_{R}}f_{N}\left(\frac{d\mu_{s}}{d||\mu_{s}||}(t,x)\right)d||\mu_{s}||(t,x),$

where $\mu=\frac{d\mu}{d{\mathcal{L}}^{2}}\lfloor\Omega_{R}+\mu_{s}$ is the
Radon-Nikodym decomposition of $\mu$ and $\frac{d\mu_{s}}{d||\mu_{s}||}$ is
the Radon-Nikodym derivative of $\mu_{s}$ with respect to its total variation.

###### Proof.

By Assumption 2, $f_{N}$ is convex and lower semi-continuous in
${\mathbb{R}}\times{\mathbb{R}}_{0}^{+}$ and $\bar{f}_{N}=f_{N}$. Hence, the
proof follows from Theorem 5.19 in [26]. ∎

###### Proposition 4.9.

Suppose that Assumption 4 holds. Let $v_{1}^{n}\to v_{1}$ and $v_{2}^{n}\to
v_{2}$ weakly in ${\mathcal{M}}(\Omega_{R})$. Then,

$\lim\limits_{n\to\infty}\left(-\int_{\Omega_{R}}(V+C)v_{1}^{n}+u_{T}^{\prime}v^{n}_{2}{\rm
d}x{\rm d}t\right)=-\int_{\Omega_{R}}(V+C)v_{1}+u^{\prime}_{T}(x)v_{2}~{}{\rm
d}xdt,$

for any $C\in{\mathbb{R}}$.

###### Proof.

It is enough to notice that the functions $V$ and $u^{\prime}_{T}(x)$ are
continuous. ∎

Now, we are ready to prove that ${\mathcal{W}}=\mathcal{J}$.

###### Theorem 4.10.

Suppose that Assumptions 2 and 4 hold for some $c>0$ and $p>1$. Then, for
every $\varphi\in BV_{0}^{+}(\Omega_{R})$

$\begin{split}{\mathcal{W}}[\varphi]=\mathcal{J}[\varphi]&=\int_{\Omega_{R}}f_{N}\left(\frac{d(D_{t,x}\varphi)}{d{\mathcal{L}}^{2}}(t,x)\right)~{}{\rm
d}x{\rm
d}t-\int_{\Omega_{R}}(V+c(2p-1))\varphi_{x}-\int_{\Omega_{R}}u^{\prime}_{T}(x)\varphi_{t}\\\
&+\int_{\Omega_{R}}f_{N}\left(\frac{d(D_{t,x}\varphi)_{s}}{d||(D_{t,x}\varphi)_{s}||}(t,x)\right)d||(D_{t,x}\varphi)_{s}||(t,x).\end{split}$

###### Proof.

Taking into account (4.33) and the definition of $f_{N}$, (4.34), we have

$cp(|v_{1}|+v_{2})\leqslant
f_{N}(v_{1},v_{2}),\quad(v_{1},v_{2})\in{\mathbb{R}}\times{\mathbb{R}}^{+}_{0}.$

Recalling that $f_{N}$ is convex and using the preceding estimate, the proof
follows from Remark 5.37 in [26] and Proposition 4.9. ∎

Next, relying on Theorem 4.10, we state the relaxed variational problem. We
set

$\displaystyle\mathcal{K}_{0}(\Omega_{R})=$ $\displaystyle\left\\{\varphi\in
BV_{0}^{+}(\Omega_{R}):~{}\operatorname{supp}(\varphi_{t}(t,\cdot)),\operatorname{supp}(\varphi_{x}(t,\cdot))\subseteq(-R,R)\right\\},$
$\displaystyle\mathcal{K}(\Omega_{R})=$
$\displaystyle\left\\{\varphi\in\mathcal{K}_{0}(\Omega_{R}):~{}\varphi(0,x)=\int_{-R}^{x}m_{0}(y){\rm
d}y,~{}\varphi(t,-R)=0,\right.$
$\displaystyle\quad\left.~{}\int_{-R}^{R}\varphi_{x}(t,x)=1,~{}\int_{-R}^{R}\varphi(t,x)-M_{0}(x)~{}{\rm
d}x=-\int_{0}^{t}Q(s)~{}{\rm d}s,~{}t\in[0,T]\right\\}.$

Note that
$\mathcal{A}(\Omega_{R})\subset\mathcal{B}(\Omega_{R})\subset\mathcal{K}(\Omega_{R})$,
so (4.27) guarantees that $\mathcal{K}(\Omega_{R})$ is a nonempty convex set.
Our relaxed variational problem is

$\min_{\varphi\in\mathcal{K}(\Omega_{R})}{\mathcal{I}}[\varphi],$ (4.35)

where

$\displaystyle{\mathcal{I}}[\varphi]=$
$\displaystyle\int_{\Omega_{R}}f_{N}\left(\frac{d(D_{t,x}\varphi)}{d{\mathcal{L}}^{2}}(t,x)\right){\rm
d}x{\rm
d}t-\int_{\Omega_{R}}(V+c(2p-1))\varphi_{x}-\int_{\Omega_{R}}u^{\prime}_{T}(x)\varphi_{t}$
$\displaystyle+\int_{\Omega_{R}}f_{N}\left(\frac{d(D_{t,x}\varphi)_{s}}{d||(D_{t,x}\varphi)_{s}||}(t,x)\right)d||(D_{t,x}\varphi)_{s}||(t,x).$

The next theorem proves the existence of solutions to the preceding
variational problem.

###### Theorem 4.11.

Suppose that Assumptions 1-5 hold. Then, there exists
$\varphi\in\mathcal{K}(\Omega_{R})$ such that

${\mathcal{I}}[\varphi]=\min_{\psi\in\mathcal{K}(\Omega_{R})}{\mathcal{I}}[\psi].$

###### Proof.

We recall that $W^{1,1}(\Omega_{R})$ is dense in $BV(\Omega_{R})$ with respect
to the intermediate convergence (see Theorem 10.1.2 in [11]). Accordingly, we
can take a minimizing sequence, $\\{\varphi^{n}\\}_{n=1}^{\infty}$, such that
$\varphi^{n}\in W^{1,1}(\Omega_{R})$. Therefore,

$\min_{\psi\in\mathcal{K}(\Omega_{R})}{\mathcal{I}}[\psi]=\lim\limits_{n\to\infty}{\mathcal{I}}[\varphi^{n}]=\liminf\limits_{n\to\infty}I[\varphi^{n}],$

where $I$, ${\mathcal{I}}$ are defined by (4.16) and (4.35), respectively.
Note that because

$\varphi^{n}_{x}\in{\mathcal{P}}(-R,R)\cap L^{1}(-R,R),$

there exists $\mu\in{\mathcal{P}}(-R,R)$, such that

$||\varphi^{n}_{x}||_{L^{1}(\Omega_{R})}\leqslant
C,\quad\varphi^{n}_{x}\rightharpoonup\mu\text{ weakly in
}{\mathcal{M}}(\Omega_{R}).$ (4.36)

Combining these estimates with the argument in Proposition 4.7, we deduce that

$||\varphi^{n}_{t}||_{L^{1}(\Omega_{R})}\leqslant C.$ (4.37)

Consequently, because $\Omega_{R}$ is bounded, Prohorov lemma (see Theorem
2.29 in [32]) gives the existence of $\nu\in{\mathcal{M}}(\Omega_{R})$ such
that $\varphi^{n}_{t}\rightharpoonup\nu$ weakly in
${\mathcal{M}}(\Omega_{R})$. On the other hand, because
$\int_{-R}^{R}\varphi(t,x)-M_{0}(x)~{}{\rm d}x=-\int_{0}^{t}Q(s)~{}{\rm d}s$,
we have that $\left|\int_{-R}^{R}\varphi^{n}{\rm d}x\right|\leqslant C$, where
$C$ does not depend on $\varphi$. Hence, by Poincaré inequality (see Theorem 1
in Section 5.8.1 in [22]) from (4.36) and (4.37), we get

$||\varphi^{n}||_{L^{1}(\Omega_{R})}\leqslant||\varphi^{n}_{x}||_{L^{1}(\Omega_{R})}+||\varphi^{n}_{t}||_{L^{1}(\Omega_{R})}+C\leqslant
C.$

Therefore, $||\varphi^{n}||_{W^{1,1}(\Omega_{R})}\leqslant C$. Consequently,
Rellich-Kondrachov Theorem (see Theorem 1, Section 5.7 in [22]) implies that
there exists $\varphi\in L^{\alpha}(\Omega_{R})$ for $\alpha\in[1,2)$, such
that $\varphi^{n}$ converges to $\varphi$ strongly in
$L^{\alpha}(\Omega_{R})$. In particular, $\varphi^{n}$ converges to $\varphi$
strongly in $L^{1}(\Omega_{R})$. This convergence combined with (4.36) and
(4.37) implies that there exists $\varphi\in BV(\Omega_{R})$, such that
$\varphi^{n}\to\varphi$ in the sense of intermediate convergence in
$BV(\Omega_{R})$. Finally, relying on this and recalling that
$\varphi^{n}\in\mathcal{K}(\Omega_{R})\cap W^{1,1}(\Omega_{R})$ from [11,
Theorem 10.2.2], we deduce that $\varphi\in\mathcal{K}(\Omega_{R})$. Moreover,
recalling the definition $f_{N}$ and using Propositions 4.8 and 4.9, we get

$\min_{\psi\in\mathcal{K}(\Omega_{R})}{\mathcal{I}}[\psi]=\lim\limits_{n\to\infty}{\mathcal{I}}[\varphi^{n}]=\liminf\limits_{n\to\infty}I[\varphi^{n}]\geqslant{\mathcal{I}}[\varphi]=\min_{\psi\in\mathcal{K}(\Omega_{R})}{\mathcal{I}}[\psi].\qed$

###### Remark 4.12.

Under Assumptions 2-5, Theorem 1 in [31] implies that $\varphi$ given by (2.5)
belongs to $\mathcal{K}(\Omega_{R})$ and is a minimizer of (4.35). Thus, under
Assumptions 1-5, if we have uniqueness, $\varphi$ given by Theorem 4.11, and
$\varphi$ given by (2.5) coincide.

##  5\. Price as Lagrange multiplier

In this section, we provide a representation formula for the price $\varpi$
using the minimizer $\varphi$ of (4.16). This formula shows that the Lagrange
multiplier associated with the balance constraint (4.13) characterizes the
price.

###### Proposition 5.1.

Suppose that Assumptions 2-5 hold. Let $R$ satisfy (4.11). Let $(u,m,\varpi)$
solve (1.1) and let $\varphi\in\mathcal{A}(\Omega_{R})$ attain the minimum in
(4.16). Furthermore, assume that $\varphi\in C^{2}(\Omega_{R})$. Then,
$\varpi$ is given by a Lagrange multiplier $w:[0,T]\to{\mathbb{R}}$ associated
with $\varphi$.

###### Proof.

The existence and uniqueness of the solution, $(u,m,\varpi)$, to Problem 1
follows from Theorem 1 in [31]. Because $L$ is convex by Remark 2.1, we have
that $\varphi$ minimizes (2.6). Therefore, recalling Proposition 4.6, we
deduce that $\varphi$ is the unique minimizer of (4.16). Furthermore, by
Remark 2.1 and Corollary 4.2 follows that there exists $0<R_{1}<R$ such that

$\operatorname{supp}(\varphi_{x}(t,\cdot))\subseteq[-R_{1},R_{1}]\subseteq(-R,R),\quad
t\in[0,T].$ (5.1)

Because $\varphi$ is the minimizer of (4.16)

$\varphi_{t}(t,x)=0\quad
a.e.\quad\text{in}\\{\varphi_{x}(t,x)=0:(t,x)\in\Omega_{R}\\}.$ (5.2)

Let

$\overline{x}(t):=\int_{{\mathbb{R}}}x\varphi_{x}(t,x){\rm
d}x=R-\int_{-R}^{R}\varphi(t,x){\rm d}x,\quad t\in[0,T].$ (5.3)

Let $\psi\in W^{1,1}(\Omega_{R})$ be such that $\psi(t,\cdot)$ is a cumulative
distribution function on $(-R,R)$ for $t\in[0,T]$, and satisfies

$\psi(0,x)=\varphi(0,x),\quad\operatorname{supp}(\psi_{x}(t,\cdot))\subseteq\operatorname{supp}(\varphi_{x}(t,\cdot))\subseteq[-R_{1},R_{1}]\subset(-R,R),\quad
t\in[0,T],$ (5.4)

and

$\psi_{t}(t,x)=0\quad
a.e.\quad\text{in}\quad\\{(t,x)\in\Omega_{R}:~{}\psi_{x}(t,x)=0\\}.$ (5.5)

Set

$\overline{z}(t):=\int_{{\mathbb{R}}}x\psi_{x}(t,x){\rm
d}x=R-\int_{-R}^{R}\psi(t,x){\rm d}x,\quad t\in[0,T].$ (5.6)

Notice that $|\overline{x}-\overline{z}|\leqslant 4R$. Let
$0<\varepsilon<\min\left\\{\frac{R-R_{1}}{4R},1\right\\}$. Thus,

$[-R_{1},R_{1}]\subset\left(-R-\varepsilon(\overline{x}(t)-\overline{z}(t)),R-\varepsilon(\overline{x}(t)-\overline{z}(t))\right)\cap(-R,R),\quad
t\in[0,T].$ (5.7)

Let

$\varphi^{\varepsilon}(t,x):=(1-\varepsilon)\varphi(t,x-\varepsilon(\overline{x}(t)-\overline{z}(t)))+\varepsilon\psi(t,x-\varepsilon(\overline{x}(t)-\overline{z}(t))).$

We claim that $\varphi^{\varepsilon}\in\mathcal{A}(\Omega_{R})$. Indeed, by
(5.4) and (5.7), we have

$\begin{split}\varphi_{x}^{\varepsilon}\geqslant
0,\quad\int_{-R}^{R}\varphi^{\varepsilon}_{x}{\rm
d}x=1,&\quad\varphi^{\varepsilon}(0,x)=M_{0}(x)\quad x\in[-R,R],\\\
&\varphi^{\varepsilon}(t,-R)=0,\quad t\in[0,T].\end{split}$

It remains to prove that $\varphi^{\varepsilon}$ satisfies the balance
condition; that is,

$\int_{-R}^{R}\varphi^{\varepsilon}(t,x){\rm
d}x=-\int_{0}^{t}Q(s)ds+\int_{-R}^{R}M_{0}(x){\rm d}x,\quad t\in[0,T].$ (5.8)

Because $\varphi\in\mathcal{A}(\Omega_{R})$, (5.3) shows that to prove (5.8)
it is enough to verify that

$\int_{{\mathbb{R}}}x\varphi^{\varepsilon}_{x}(t,x){\rm
d}x=\overline{x}(t),\quad t\in[0,T].$

Computing the left-hand side of the previous identity, we have

$\displaystyle\int_{{\mathbb{R}}}x\varphi^{\varepsilon}_{x}(t,x){\rm d}x$
$\displaystyle=(1-\varepsilon)\int_{{\mathbb{R}}}x\varphi_{x}(t,x-\varepsilon(\overline{x}(t)-\overline{z}(t))){\rm
d}x+\varepsilon\int_{{\mathbb{R}}}x\psi_{x}(t,x-\varepsilon(\overline{x}(t)-\overline{z}(t))){\rm
d}x$ $\displaystyle=(1-\varepsilon)\int_{{\mathbb{R}}}x\varphi_{x}(t,x){\rm
d}x+\varepsilon(\overline{x}(t)-\overline{z}(t))+\varepsilon\int_{{\mathbb{R}}}x\psi_{x}(t,x){\rm
d}x+\varepsilon(\overline{x}(t)-\overline{z}(t))$
$\displaystyle=(1-\varepsilon)\left(\overline{x}(t)+\varepsilon(\overline{x}(t)-\overline{z}(t))\right)+\varepsilon\left(\overline{z}(t)+\varepsilon(\overline{x}(t)-\overline{z}(t))\right)$
$\displaystyle=\overline{x}(t).$

Therefore, $\varphi^{\varepsilon}\in\mathcal{A}(\Omega_{R})$, and the map
$\varepsilon\mapsto I[\varphi^{\varepsilon}]$ has a minimum at
$\varepsilon=0$; that is,

$\lim_{\varepsilon\to
0^{+}}\frac{d}{d\varepsilon}I[\varphi^{\varepsilon}]\geqslant 0.$

To compute the left-hand side of the previous inequality, we notice that

$\displaystyle\varphi^{\varepsilon}_{t}(t,x)=$
$\displaystyle(1-\varepsilon)\bigg{(}\varphi_{t}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right))-\varphi_{x}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right))\varepsilon\left(\dot{\overline{x}}(t)-\dot{\overline{z}}(t)\right)\bigg{)}$
$\displaystyle+\varepsilon\bigg{(}\psi_{t}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right))-\psi_{x}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right))\varepsilon\left(\dot{\overline{x}}(t)-\dot{\overline{z}}(t)\right)\bigg{)},$
$\displaystyle\varphi^{\varepsilon}_{x}(t,x)=$
$\displaystyle(1-\varepsilon)\varphi_{x}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right))+\varepsilon\psi_{x}(t,x-\varepsilon\left(\overline{x}(t)-\overline{z}(t)\right)).$

Note that (5.2) and (5.5) imply

$\varphi^{\varepsilon}_{t}(t,x)=0\quad
a.e.\quad\text{in}\quad\\{\varphi^{\varepsilon}_{x}(t,x)=0:(t,x)\in\Omega_{R}\\}.$
(5.9)

Furthermore,

$\displaystyle\lim_{\varepsilon\to
0^{+}}\dfrac{d}{d\varepsilon}\varphi^{\varepsilon}_{t}$
$\displaystyle=-\varphi_{t}(t,x)-\varphi_{tx}\left(\overline{x}(t)-\overline{z}(t)\right)-\varphi_{x}(t,x)\left(\dot{\overline{x}}(t)-\dot{\overline{z}}(t)\right)+\psi_{t}(t,x)$
$\displaystyle=\dfrac{d}{dt}\bigg{(}-\varphi(t,x)+\psi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\bigg{)},$
$\displaystyle\lim_{\varepsilon\to
0^{+}}\dfrac{d}{d\varepsilon}\varphi^{\varepsilon}_{x}$
$\displaystyle=-\varphi_{x}(t,x)-\varphi_{xx}\left(\overline{x}(t)-\overline{z}(t)\right)+\psi_{x}(t,x)$
$\displaystyle=\dfrac{d}{dx}\bigg{(}-\varphi(t,x)+\psi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\bigg{)}.$

For ease of notation, we denote

$L^{*}_{z}(t,x)=\begin{cases}L_{z}(\varphi_{t}(t,x),\varphi_{x}(t,x)),\quad\varphi_{x}(t,x)>0\\\
0,\quad\text{otherwise},\end{cases}$ (5.10)

and

$L^{*}_{y}(t,x)=\begin{cases}L_{y}(\varphi_{t}(t,x),\varphi_{x}(t,x)),\quad\varphi_{x}(t,x)>0\\\
0,\quad\text{otherwise}.\end{cases}$ (5.11)

Taking into account (5.2) and (5.5), we obtain

$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{d}{d\varepsilon}I[\varphi^{\varepsilon}]$
$\displaystyle=\int_{0}^{T}\int_{-R}^{R}\left(L^{*}_{z}(t,x)-u^{\prime}_{T}(x)\right)\dfrac{d}{dt}\bigg{(}\psi(t,x)-\varphi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\bigg{)}{\rm
d}x{\rm d}t$
$\displaystyle\quad+\int_{0}^{T}\int_{-R}^{R}\left(L^{*}_{y}(t,x)-V(x)\right)\dfrac{d}{dx}\bigg{(}\psi(t,x)-\varphi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\bigg{)}{\rm
d}x{\rm d}t.$

Integrating by parts in the right-hand side of the previous identity, using
that $\overline{x}(0)=\overline{z}(0)$, and recalling (5.1) and (5.4), we
obtain

$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{d}{d\varepsilon}I[\varphi^{\varepsilon}]$
$\displaystyle=\int_{-R}^{R}\left(L^{*}_{z}(T,x)-u^{\prime}_{T}(x)\right)\bigg{(}\psi(T,x)-\varphi(T,x)-\varphi_{x}(T,x)\left(\overline{x}(T)-\overline{z}(T)\right)\bigg{)}{\rm
d}x$
$\displaystyle\quad-\int_{0}^{T}\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}\bigg{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\bigg{(}\psi(t,x)-\varphi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\bigg{)}{\rm
d}x{\rm d}t.$ (5.12)

Recalling (5.3) and (5.6), we have

$\overline{x}(t)-\overline{z}(t)=\int_{-R}^{R}\left(\psi(t,x)-\varphi(t,x)\right){\rm
d}x,\quad t\in[0,T].$

Using the previous identity, we write the first term on the right-hand side of
(5) as follows

$\displaystyle\int_{-R}^{R}\left(L^{*}_{z}(T,x)-u^{\prime}_{T}(x)\right)\bigg{(}\psi(T,x)-\varphi(T,x)-\varphi_{x}(T,x)\left(\overline{x}(T)-\overline{z}(T)\right)\bigg{)}{\rm
d}x$
$\displaystyle=\int_{-R}^{R}\bigg{(}L^{*}_{z}(T,x)-u^{\prime}_{T}(x)-\int_{-R}^{R}(L^{*}_{z}(T,y)-u^{\prime}_{T}(y))\varphi_{x}(T,y){\rm
d}y\bigg{)}\left(\psi(T,x)-\varphi(T,x)\right){\rm d}x.$ (5.13)

Similarly, the second term on the right-hand side of (5) becomes

$\displaystyle\int_{0}^{T}\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}\bigg{)}\left(\psi(t,x)-\varphi(t,x)-\varphi_{x}(t,x)\left(\overline{x}(t)-\overline{z}(t)\right)\right){\rm
d}x{\rm d}t$
$\displaystyle=\int_{0}^{T}\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}$
$\displaystyle\quad\quad\quad\quad\quad-\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,y)\right)_{t}+\left(L^{*}_{y}(t,y)-V(y)\right)_{x}\bigg{)}\varphi_{x}(t,y){\rm
d}y\bigg{)}\left(\psi(t,x)-\varphi(t,x)\right){\rm d}x{\rm d}t.$ (5.14)

Define the Lagrange multiplier by

$\displaystyle
w_{T}=\int_{-R}^{R}(L^{*}_{z}(T,y)-u^{\prime}_{T}(y))\varphi_{x}(T,y){\rm
d}y,$ $\displaystyle
w(t)=\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,y)\right)_{t}+\left(L^{*}_{y}(t,y)-V(y)\right)_{x}\bigg{)}\varphi_{x}(t,y){\rm
d}y,\quad t\in[0,T].$ (5.15)

Then, replacing (5) and (5) in (5), we get

$\displaystyle\int_{-R}^{R}\bigg{(}L^{*}_{z}(T,x)-u^{\prime}_{T}(x)-w_{T}\bigg{)}\left(\psi(T,x)-\varphi(T,x)\right){\rm
d}x$
$\displaystyle+\int_{0}^{T}\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}-w(t)\bigg{)}\left(\psi(t,x)-\varphi(t,x)\right){\rm
d}x{\rm d}t\geqslant 0.$ (5.16)

Notice that, in the previous inequality, the function $\phi=\psi-\varphi$ can
be selected to be strictly positive or negative in any neighborhood of
$(0,T)\times(-R,R)$. Therefore, we can infer the nullity of the functions in
both integrals in (5) as follows. First, select $\psi$ satisfying
$\psi(T,\cdot)=\varphi(T,\cdot)$. Then, (5) shows that

$\displaystyle\int_{0}^{T}\int_{-R}^{R}\bigg{(}\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}-w(t)\bigg{)}\phi(t,x){\rm
d}x{\rm d}t\geqslant 0.$ (5.17)

The regularity of

$(t,x)\mapsto\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}-w(t)$

allows the localization of the integral in (5.17) using $\phi$, and we
conclude that

$\displaystyle\left(L^{*}_{z}(t,x)\right)_{t}+\left(L^{*}_{y}(t,x)-V(x)\right)_{x}-w(t)=0\quad\mbox{a.e.
}(t,x)\in(0,T)\times(-R,R).$ (5.18)

Then, (5) reduces to

$\displaystyle\int_{-R}^{R}\bigg{(}L^{*}_{z}(T,x)-u^{\prime}_{T}(x)-w_{T}\bigg{)}\left(\psi(T,x)-\varphi(T,x)\right){\rm
d}x\geqslant 0,$

and we proceed as before by localizing the integral using $x\mapsto\phi(T,x)$
to conclude that

$\displaystyle L^{*}_{z}(T,x)-u^{\prime}_{T}(x)-w_{T}=0\quad\mbox{a.e.
}x\in(-R,R).$ (5.19)

Recalling (2.9), which characterizes $\varpi$, the identities (5.18) and
(5.19) show that the price, $\varpi$, is given by the Lagrange multiplier (5)
according to

$\dot{\varpi}(t)=w(t),\quad t\in[0,T],\quad\varpi(T)=w_{T}.\qed$

###### Proof of Theorem 1.1.

Because $\varphi$ solves Problem 2, we obtain $\varpi$ according to
Proposition 5.1. Therefore, $(\varphi,\varpi)$ minimizes (2.7), and by
Proposition 2.3 satisfies an Euler-Lagrange equation equivalent to
(2.3)-(2.4). Since the solution $(u,m,\varpi)$ of (1.1) defines a potential
function according to (2.5) which satisfies (2.3)-(2.4), the convexity of
(2.7) implies that this potential is a minimizer of (2.7). Thus, by
Proposition 4.6 , we conclude that the potential function defined by
$(u,m,\varpi)$ coincides with the minimizer $\varphi$. Thus, we can recover
$u$ and $m$ using (2.1); that is,

$u(t,x)=u_{T}(x)-\int_{t}^{T}H\left(F^{\prime}\left(\frac{\varphi_{t}(s,x)}{\varphi_{x}(s,x)}\right)\right){\rm
d}s-(T-t)V(x),\quad(t,x)\in[0,T]\times{\mathbb{R}},$ (5.20)

where the right-hand side of the previous expression is well defined because
$\varphi_{x}$ and $\varphi_{t}$ have the same compact support, and
$m(t,x)=\varphi_{x}(t,x)$, $(t,x)\in[0,T]\times{\mathbb{R}}$. ∎

##  6\. Numerical results

In this section, we provide the results of the potential approach applied to
the price formation MFG system with quadratic cost and oscillating supply. We
use the semi-explicit formulas introduced in [31] to assess the error in our
approximation. We use the standard solver for finite-dimensional convex
problems provided by the software Mathematica to approximate the potential
function in a discrete grid in time and space.

Let $\kappa\in{\mathbb{R}}$, $\eta\geqslant 0$, and $c>0$. For the quadratic
cost configuration, we take

$H(p)=\frac{1}{2c}p^{2},\quad
V(x)=-\frac{\eta}{2}\left(x-\kappa\right)^{2},\quad\mbox{and}\quad
u_{T}\left(x\right)\equiv 0.$

Thus, $F(v)=\frac{c}{2}v^{2}$. As shown in [31] and [30], a feature of the
quadratic setting is the solvability of the Hamilton-Jacobi equation in (1.1)
in the class of quadratic functions of $x$ with time-dependent coefficients

$u(t,x)=a_{0}(t)+a_{1}(t)x+a_{2}(t)x^{2},\quad t\in[0,T],~{}x\in{\mathbb{R}}.$

The coefficients $a_{0}$, $a_{1}$ and $a_{2}$ solve an ODE system that derives
from the Hamilton-Jacobi equation by matching powers of the $x$ variable.
Figure 1(c) shows the value function for $(x,t)\in[0,T]\times[-1,1]$.
Moreover, the price has the following explicit formula

$\varpi(t)=\eta\left(\kappa-\overline{m}_{0}\right)\left(T-t\right)-\eta\int_{t}^{T}\int_{0}^{s}Q(r)drds-
cQ(t),\quad t\in[0,T],$

where $\overline{m}_{0}=\int_{{\mathbb{R}}}xm_{0}(x)dx$. The initial condition
$m_{0}$ is centered at $x=0$ and with compact support $[-0.5,0.5]$ (see Figure
1(a)). The vector-field transporting $m_{0}$ is

$b(t,x)=-\frac{1}{c}\left(\varpi(t)+a_{1}(t)+2a_{2}(t)x\right),\quad
t\in[0,T],~{}x\in{\mathbb{R}},$

which we use to compute $m$ using the method of characteristics (see Figure
2(b)). Thus, recalling (2.1) and (2.5), we have explicit formulas for
$\varphi$, $\varphi_{x}$, and $\varphi_{t}$. We use the previous expressions
as a benchmark for the approximation obtained using (5).

For the discretization of the time variable, we set $T=1$ and $N_{t}=20$ time
steps uniformly spaced. Thus, $h_{t}=0.05$ is the time step size. To
discretize the space variable, the selection of $R$ in (4.11), where
$R_{0}=0.5$, becomes

$R>\max\\{5.57742~{},\quad 0.811579\\}.$

However, to simplify the computational cost, we optimize the selection of $R$
by looking at the support of $m(t,)$ for $t\in[0,T]$, which we illustrate in
Figure 2(b). Thus, we discretize the space variable in the space domain
$[-1,1]$ using $N_{x}=40$ time steps equally spaced. Thus, $h_{x}=0.05$ is the
step size.

Because in several applications the supply function satisfies a mean reversion
assumption, we assume that it follows the ordinary differential equation

$\begin{cases}\dot{Q}(t)=\overline{Q}(t)-\alpha Q(t),&t\in[0,T],\\\
Q(0)=q_{0},\end{cases}$

where $\overline{Q}:[0,T]\to{\mathbb{R}}$ represents the average supply over
time, $\alpha\in{\mathbb{R}}$ measures the tendency to towards the average,
and $q_{0}\in{\mathbb{R}}$ is the initial supply. For numerical purposes, we
select

$\overline{Q}(t)=5\sin(3\pi t),\quad\alpha=4,\quad q_{0}=-0.5.$

While the particular choice of $Q$ does not change the problem substantially,
the preceding choice has oscillatory features, as we want to demonstrate how
price changes and at the same time gives simple analytic expressions. As
Figure 1(b) shows, the price inherits the oscillating behavior from the
supply.

(a) $m_{0}$

(b) Price and supply

(c) value function $u$

Fig. 1. Data $m_{0}$ and $Q$, and solutions $u$ and $\varpi$ for
$\overline{Q}(t)=5\sin(3\pi t)$.

Using the solution $(u,m,\varpi)$, we get $\varphi_{t}$, $\varphi_{x}$ from
(2.1) (see Figure 2(b)), and so (2.5) gives $\varphi$, illustrated in Figure
2. The value of (4.16) is $0.106525$, which we use as an additional benchmark
to assess our numerical approximation.

(a) $\varphi_{t}$

(b) $\varphi_{x}=m$

(c) potential $\varphi$

Fig. 2. Analytic solution $\varphi$ and its partial derivatives for
$\overline{Q}(t)=5\sin(3\pi t)$. The blue lines outline the support of $m$.

We discretize (4.16) over the time-space grid using finite differences to
approximate $\varphi_{t}$ and $\varphi_{x}$; that is

$\varphi_{t}(t_{i},x_{j})=\dfrac{\varphi(t_{i}+h_{t},x_{j})-\varphi(t_{i},x_{j})}{h_{t}},\quad\varphi_{x}(t_{i},x_{j})=\dfrac{\varphi(t_{i},x_{j}+h_{x})-\varphi(t_{i},x_{j})}{h_{x}},$

for $i=1,\ldots,20,$ and $j=1,\ldots,40$. We obtain a finite-dimensional
convex optimization problem with the following constraints

$\displaystyle\varphi_{x}(t_{i},x_{j})\geqslant
0,\quad\sum_{j=1}^{N_{x}}\left(\varphi(t_{i},x_{j})-M_{0}(x_{j})\right)h_{x}+\sum_{k=0}^{i}Q(t_{k})h_{t},$
$\displaystyle\varphi(0,x_{j})-M_{0}(x_{j})=0,\quad\varphi(t_{i},-1)=0,\quad\varphi(t_{i},1)=1,\quad
i=1,\ldots,N_{t},\;j=1,\ldots,N_{x},$

which correspond to the discretization of the admissible set
$\mathcal{A}(\Omega_{R})$ (see (4.14)). The results are depicted in Figure 3.
The approximated value of (4.16) is $0.103765$, in good agreement with the
theoretical value $0.106525$.

(a) $\varphi_{t}$ approximation

(b) approximation error

(c) $\varphi_{x}=m$ approximation

(d) approximation error

(e) $\varphi$ approximation

(f) approximation error

Fig. 3. Approximated solution $\varphi$ for $\overline{Q}(t)=5\sin(3\pi t)$.

Using (5), we obtain the corresponding approximation of $\varpi$, illustrated
in Figure 4. Because of the implementation of finite differences, we can
compute the price on the time horizon $[2h_{t},T]$. The plots show good
agreement between the values of our numerical results with a small discrepancy
that improves as the grid size increases (here, we show the results for the
finest grid we used).

(a) $\varpi$ approximation

(b) approximation error

Fig. 4. Approximated solution $\varpi$ for $\overline{Q}(t)=5\sin(3\pi t)$.

As the last benchmark, we consider the value function $u$. To compute $u$, we
round the approximation to avoid indeterminate expressions and we use (5.20).
The result is depicted in Figure 5. Again, we obtain a good agreement with the
exact solution.

(a) $u$ approximation

(b) approximation error

Fig. 5. Approximated solution $u$ for $\overline{Q}(t)=5\sin(3\pi t)$.

##  7\. Conclusions and further directions

In this paper, we presented a variational approach based on Poincaré Lemma,
reducing one variable in the MFG price formation model. We studied the
variational approach independently of the MFG problem. We obtained existence
for a relaxed formulation using bounded variation functions, and we proved
uniqueness of the potential function. We showed that price existence follows a
Lagrange multiplier rule associated with the balance constrained, an integral
equation for the MFG model depending on a supply function. For the price
problem, the variational formulation allows an efficient computation without
solving the backward-forward coupled problem with integral constraints. The
convexity of the variational approach allows the use of standard optimization
tools to solve its discrete formulation. Our numerical method shows promising
results and good agreement with the explicit solutions. We consider we can
apply a similar approach to the price formation model with common noise, which
corresponds to the case of a stochastic supply function. One challenge is the
dependence of the variational problem formulation on the supply, requiring the
discretization of time, state variables, and the common noise. We plan to
investigate this case in future works.

## References

  * [1] Y. Achdou, F. Camilli, and I. Capuzzo-Dolcetta.  Mean field games: numerical methods for the planning problem.  SIAM J. Control Optim., 50(1):77–109, 2012.
  * [2] Y. Achdou and I. Capuzzo-Dolcetta.  Mean field games: numerical methods.  SIAM J. Numer. Anal., 48(3):1136–1162, 2010.
  * [3] Y. Achdou and M. Laurière.  Mean Field Games and Applications: Numerical Aspects, pages 249–307.  Springer International Publishing, Cham, 2020.
  * [4] R. Aïd, A. Cosso, and H. Pham.  Equilibrium price in intraday electricity markets, 2020.
  * [5] R. Aïd, R. Dumitrescu, and P. Tankov.  The entry and exit game in the electricity markets: A mean-field game approach.  Journal of Dynamics & Games, 8(4):331–358, 2021.
  * [6] C. Alasseur, I. Ben Taher, and A. Matoussi.  An extended mean field game for storage in smart grids.  Journal of Optimization Theory and Applications, 184(2):644–670, 2020.
  * [7] C. Alasseur, L. Campi, R. Dumitrescu, and J. Zeng.  Mfg model with a long-lived penalty at random jump times: application to demand side management for electricity contracts, 2021.
  * [8] A. Alharbi, T. Bakaryan, R. Cabral, S. Campi, N. Christoffersen, P. Colusso, O. Costa, S. Duisembay, R. Ferreira, D. Gomes, S. Guo, J. Gutierrez, P. Havor, M. Mascherpa, S. Portaro, R. Ricardo de Lima, F. Rodriguez, J. Ruiz, F. Saleh, S. Calum, T. Tada, X. Yang, and Z. Wróblewska.  A price model with finitely many agents.  Bulletin of the Portuguese Mathematical Society, 2019.
  * [9] L. Ambrosio, N. Fusco, and D. Pallara.  Functions of bounded variation and free discontinuity problems.  Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
  * [10] Y. Ashrafyan, T. Bakaryan, D. Gomes, and J. Gutierrez.  A duality approach to a price formation mfg model, 2021.
  * [11] H. Attouch, G. Buttazzo, and G. Michaille.  Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization.  SIAM, 2014.
  * [12] T. Bakaryan, R. Ferreira, and D. Gomes.  A Potential Approach for Planning Mean-Field Games in One Dimension .  Submitted to Communications on Pure and Applied Analysis, 2021.
  * [13] T. Basar and R. Srikant.  Revenue-maximizing pricing and capacity expansion in a many-users regime.  In Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, volume 1, pages 294–301 vol.1, 2002.
  * [14] F. Bonnans, P. Lavigne, and L. Pfeiffer.  Discrete potential mean field games, 2021.
  * [15] L. Caravenna and G. Crippa.  Uniqueness and lagrangianity for solutions with lack of integrability of the continuity equation.  Comptes Rendus Mathematique, 354(12):1168–1173, 2016.
  * [16] P. Cardaliaguet.  A short course on mean field games.  2018\.
  * [17] R. Carmona and M. Laurière.  Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: the ergodic case.  SIAM Journal on Numerical Analysis, 59(3):1455–1485, 2021.
  * [18] R. Carmona and M. Laurière.  Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games II: The finite horizon case.  To appear in Annals of Applied Probability, 2021.
  * [19] G. Csató, B. Dacorogna, and O. Kneuss.  The Pullback Equation for Differential Forms.  Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, New York, 2012.
  * [20] G. Dal Maso.  An introduction to G-convergence.  Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Boston, 1 edition, 1993.
  * [21] B. Djehiche, J. Barreiro-Gomez, and H. Tembine.  Price Dynamics for Electricity in Smart Grid Via Mean-Field-Type Games.  Dynamic Games and Applications, 10(4):798–818, December 2020.
  * [22] L. C. Evans.  Partial Differential Equations.  Graduate Studies in Mathematics. American Mathematical Society, 1998.
  * [23] L. C. Evans and R. F. Gariepy.  Measure Theory and Fine Properties of Functions, Revised Edition.  Textbooks in Mathematics. CRC Press, 2015.
  * [24] O. Féron, P. Tankov, and L. Tinsi.  Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player.  Risks, 8(4):1–1, December 2020.
  * [25] O. Féron, P. Tankov, and L. Tinsi.  Price formation and optimal trading in intraday electricity markets, 2021\.
  * [26] I. Fonseca and G. Leoni.  Modern methods in the calculus of variations: $L^{p}$ spaces.  Springer Monographs in Mathematics. Springer, New York, 2007.
  * [27] M. Fujii and A. Takahashi.  A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition.  Papers 2003.03035, arXiv.org, March 2020.
  * [28] M. Fujii and A. Takahashi.  Equilibrium price formation with a major player and its mean field limit, 2021.
  * [29] D. Gomes, J. Gutierrez, and R. Ribeiro.  A mean field game price model with noise.  Math. Eng., 3(4):Paper No. 028, 14, 2021.
  * [30] D. Gomes, J. Gutierrez, and R. Ribeiro.  A random-supply mean field game price model, 2021.
  * [31] D. Gomes and J. Saúde.  A Mean-Field Game Approach to Price Formation.  Dyn. Games Appl., 11(1):29–53, 2021.
  * [32] W. Kirsch.  A survey on the method of moments.  2015\.
  * [33] J.-M. Lasry and P.-L. Lions.  Jeux à champ moyen. II. Horizon fini et contrôle optimal.  C. R. Math. Acad. Sci. Paris, 343(10):679–684, 2006.
  * [34] A. Lin, S. Fung, W. Li, L. Nurbekyan, and S. Osher.  Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games.  Proceedings of the National Academy of Sciences, 118(31), 2021.
  * [35] C. Mou, X. Yang, and C. Zhou.  Numerical methods for mean field games based on gaussian processes and fourier features, 2021.
  * [36] C. Orrieri, A. Porretta, and G. Savaré.  A variational approach to the mean field planning problem.  J. Funct. Anal., 277(6):1868–1957, 2019.
  * [37] L. Perko.  Differential Equations and Dynamical Systems, Third Edition.  3rd edition, 2006.
  * [38] B. Perthame.  Transport equations in biology.  Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.
  * [39] L. Ruthotto, S. Osher, W. Li, L. Nurbekyan, and S. Fung.  A machine learning framework for solving high-dimensional mean field game and mean field control problems, 2020.
  * [40] H. Shen and T. Basar.  Pricing under information asymmetry for a large population of users.  Telecommun. Syst., 47(1-2):123–136, 2011.
  * [41] A. Shrivats, D. Firoozi, and S. Jaimungal.  A Mean-Field Game Approach to Equilibrium Pricing, Optimal Generation, and Trading in Solar Renewable Energy Certificate Markets.  Papers 2003.04938, arXiv.org, March 2020.