# Laplace principle for large population games with control interaction

Peng Luo and Ludovic Tangpi

###### Abstract.

This work investigates continuous time stochastic differential games with a
large number of players, whose costs and dynamics interact through the
empirical distribution of both their states and their controls. The control
processes are assumed to be open-loop. We give regularity conditions
guaranteeing that if the finite-player game admits a Nash equilibrium, then
both the sequence of equilibria and the corresponding state processes satisfy
a Sanov-type large deviation principle. The results require existence of a
Lipschitz continuous solution of the master equation of the corresponding mean
field game and it carries over to cooperative (i.e. central planer) games. We
study the linear-quadratic case of such games in details.

###### Key words and phrases:

Large population games, mean field games, interaction through controls, large
deviation principle, FBSDE, McKean-Vlasov equations, concentration of measure,
PDEs on Wasserstein space.

###### 2010 Mathematics Subject Classification:

60F25, 91A06, 91A13, 60J60, 28C20, 60H20, 35B40.

##  1\. Introduction and main results

This paper is a sequel to [34] in which the convergence of symmetric,
continuous time stochastic differential games to mean field games was
analyzed. Here the goal is to complement the convergence results by deriving
large deviation principles (in Laplace form) for the sequence of Nash
equilibrium and the associated state processes. Let us briefly describe the
stochastic differential game we consider, in its _non-cooperative_ version.
The cooperative version is discussed in the last section of the article. Let
$d\in\mathbb{N}$ be fixed and let $(\Omega,\mathcal{F},\mathbb{P})$ be a
probability space which is rich enough to carry a (fixed) sequence of
$d$-dimensional independent Brownian motions $(W^{i})_{i\geq 1}$. For every
$N\in\mathbb{N}$, consider the filtration
$\mathcal{F}^{N}:=(\mathcal{F}_{t}^{N})_{t\in[0,T]}$, which is the completion
of $\sigma(W^{i}_{s},s\leq t,i=1,\dots,N)$. Without further mention, we will
always use the identification

$W\equiv W^{1}\quad\text{and}\quad\mathcal{F}\equiv\mathcal{F}^{1}.$

Given two functions $f$ and $g$, the cost that agent $i$ seeks to minimize,
when the strategy profile of the $N$ players is
$\underline{\alpha}=(\alpha^{1},\dots,\alpha^{N})$, is

$J(\alpha^{i};\underline{\alpha}^{-i}):=J^{i}(\underline{\alpha}):=\mathbb{E}\bigg{[}\int_{0}^{T}f(t,X^{i,\underline{\alpha}}_{t},\alpha^{i}_{t},L^{N}(\underline{X}^{\underline{\alpha}}_{t},\underline{\alpha}_{t}))dt+g(X^{i,\underline{\alpha}}_{T},L^{N}(\underline{X}^{\underline{\alpha}}_{T}))\bigg{]},$

where we denote

$\underline{x}^{-i}:=(x^{1},\dots,x^{i-1},x^{i+1},\dots,x^{N})\quad\text{and}\quad\underline{x}:=(x^{1},\dots,x^{N}),$

and where the position $X^{i,\underline{\alpha}}$ of player $i$ is given by
the controlled diffusion

(1)
$dX^{i,\underline{\alpha}}_{t}=b\big{(}t,X^{i,\underline{\alpha}}_{t},\alpha^{i}_{t},L^{N}(\underline{X}_{t}^{\underline{\alpha}},\underline{\alpha}_{t})\big{)}dt+\sigma
dW^{i}_{t},\quad X^{i,\underline{\alpha}}_{0}=x\in\mathbb{R}^{\ell}.$

The term
$L^{N}(\underline{X}_{t}^{\underline{\alpha}},\underline{\alpha}_{t})$ models
the _interaction_ between the players. We assume that the players are in weak
interaction through the empirical measure of both the states and the
strategies of the whole system. That is, we let

$L^{N}(\underline{x}):=\frac{1}{N}\sum_{i=1}^{N}\delta_{x^{i}}.$

It is interesting to notice that in this model, the players interact not only
through their respective states (or positions) $X^{i,\underline{\alpha}}$, but
through their _controls_ $\alpha^{i}$ as well. The admissible set
$\mathcal{A}$ of the controls is defined as follows: given $m\in\mathbb{N}$
and $\mathbb{A}\subseteq\mathbb{R}^{m}$ a closed convex set and put

$\mathcal{A}:=\left\\{\alpha:[0,T]\times\Omega\to\mathbb{A},\,\,\mathcal{F}^{N}\text{-progressive
such that
}\mathbb{E}\Big{[}\int_{0}^{T}|\alpha_{t}|^{2}\,dt\Big{]}<\infty\right\\}.$

As usual, one is interested in a Nash equilibrium
$\hat{\underline{\alpha}}:=(\hat{\alpha}^{i},\dots,\hat{\alpha}^{N})$. That
is, admissible strategies $(\hat{\alpha}^{1},\dots,\hat{\alpha}^{N})$ such
that for every $i=1,\dots,N$ and $\alpha\in\mathcal{A}$ it holds that

$J^{i}(\hat{\underline{\alpha}})\leq
J(\alpha^{i};\hat{\underline{\alpha}}^{-i}).$

One formally associates the above $N$-player game to the following mean field
game: Given a flow of distributions $(\xi_{t})_{t\in[0,T]}$ with
$\xi_{t}\in\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})$ with first
marginal $\mu_{t}\in\mathcal{P}_{2}(\mathbb{R}^{\ell})$, consider a solution
$\hat{\alpha}^{\xi}$ of

$\begin{cases}\inf_{\alpha\in\mathfrak{A}}\mathbb{E}\left[\int_{0}^{T}f(t,X^{\alpha}_{t},\alpha_{t},\xi_{t})dt+g(X^{\alpha}_{T},\mu_{T})\right]\\\
dX^{\alpha}_{t}=b(t,X^{\alpha}_{t},\alpha_{t},\xi_{t})dt+\sigma dW_{t},\qquad
X^{\alpha}_{0}=x.\end{cases}$

Hereby, the set $\mathfrak{A}$ of admissible controls for the mean field game
is defined as

$\mathfrak{A}:=\left\\{\alpha:[0,T]\times\Omega\to\mathbb{A},\,\,\mathcal{F}\text{-progressive
such that
}\mathbb{E}\Big{[}\int_{0}^{T}|\alpha_{t}|^{2}\,dt\Big{]}<\infty\right\\}.$

A mean field equilibrium is a strategy $\hat{\alpha}^{\xi}\in\mathfrak{A}$
satisfying the fixed point (or consistency) condition

$\xi_{t}=\mathcal{L}(X^{\hat{\alpha}^{\xi}}_{t},\hat{\alpha}^{\xi}_{t})\quad\text{for
all }t.$

Such mean field game is usually referred to as ”mean field game of controls”
or ”extended mean field game” and has been introduced by Gomes et al. [29,
28]. The interest in this type of mean field games quickly blossomed, mainly
due to their natural applications in economic and finance, see e.g. [7; 15;
31] or [27, Section 3.3.1] and energy production models [30; 1].

When the interaction among the players is through the state only, a rigorous
connection between the $N$-player and the mean field game was first
established by Lacker [32] and Fischer [24], proving compactness results for
sequences of Nash equilibria, using relaxed controls. Cardaliaguet et al. [9]
used existence and regularity properties of solutions of the so-called master
equation, a PDE on the space of probability measures characterizing the mean
field game to prove convergence results for the value functions. Moreover,
under proper smoothness conditions on the solution (when it exists) of the
master equation, Delarue et al. [18, 17] further analyzed the mean field game
limit for the state processes at equilibrium of the game with common noise.
Notably, Delarue et al. [18] derive the large deviation principle for _closed-
loop_ equilibria in the _uniqueness regime_ (i.e. when the limiting mean field
game admits a unique mean field equilibrium).

The mean field game limit in the present context, where interaction is also
through the control, was first proved by Laurière and Tangpi [34]. In this
setting we should also mention results by Possamaï and Tangpi [38] for games
in the weak formulation and Djete [21] using relaxed controls. In the
nutshell, it was proved in [34] that when the coefficients of the game are
sufficiently regular (see the proper statement below) and the $N$-player game
admits at least one Nash equilibrium, then this Nash equilibrium converges in
the $\mathbb{L}^{2}$–sense to a mean field equilibrium. The aim of the present
work is to make a finer analysis of this convergence result by deriving large
deviation principles both for the Nash equilibrium and the state processes of
the players at equilibrium. We will recall the concept of large deviation
principle (LDP) in Section 2; but note already that this allows to quantify
the probability that the Nash equilibrium does not converge to the mean field
equilibrium, a rare event in view of the results in [34]. Our main results
(stated in the next section) state that if in addition to the convergence
assumptions of [34] we assume that a certain (master) equation admits a
Lipschitz continuous solutions, then the Nash equilibrium satisfies the LDP.
In contrast to Delarue et al. [18], our model includes interaction of the
players through their controls, but also deals with _open-loop_ equilibria.
Moreover, the assumptions of the our second LDP result (Theorem 3) do not
require or a priori guarantee uniqueness of the mean field equilibrium. In
addition, we propose a different proof, one that is based on forward backward
stochastic differential equations (FBSDE) and their decoupling fields. We will
elaborate more on the idea of proof in the discussion following the statement
of Theorem 3. Note that this method seems versatile enough to be applied to
the LDP of _cooperative games_.

When the players jointly optimize the cost function

$\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\left[\int_{0}^{T}f(t,X^{i,\underline{\alpha}}_{t},\alpha^{i}_{t},L^{N}(\underline{X}^{\underline{\alpha}}_{t},\underline{\alpha}_{t}))dt+g(X^{i,\underline{\alpha}}_{T},L^{N}(\underline{X}^{\underline{\alpha}}_{T}))\right],$

we obtain a ”central planner” problem (or cooperative game) which has been
showed to converge to a stochastic control problem of McKean-Vlasov type, see
[33; 34; 20]. The analysis of the LDP in this case follows the same steps as
in the non-cooperative game described above. To avoid repeating the proof, in
the case of cooperative games we focus on a linear quadratic game and again
derive an LDP, see Section 4 for details.

In the next section we make precise the assumptions used in [34] as well as
the additional assumptions needed for the LDP and state the main results of
the paper, Theorems 2 and 3. The proof of these results is the subject of
Section 3. In the last section of the article we consider the case of
cooperative stochastic differential games, and restrict ourselves to the
linear-quadratic case.

##  2\. Setting and main results

This section is dedicated to the presentation of the main result of this
article. That is, the large deviation principle for stochastic differential
games. We will focus here on the case of non-cooperative games. The
cooperative case will be discussed in the last section. The setting is exactly
the same as that of [34]. We recall it here for the reader’s convenience, we
also keep the same notation used in that article. Throughout, we denote the
set of probability measures with finite second moment on a Polish space $E$ by
$\mathcal{P}_{2}(E)$, and we equip it with the second order Wasserstein
distance denoted $\mathcal{W}_{2}(\xi,\xi^{\prime})$ for
$\xi,\xi^{\prime}\in\mathcal{P}_{2}(E)$, and defined as

$\mathcal{W}_{2}^{2}(\xi,\xi^{\prime}):=\inf_{\pi}\iint_{E\times
E}d_{E}(x,y)\pi(dx,dy)$

where $d_{E}$ is the distance on $E$ and the infimum is taken over coupling of
$(\xi,\xi^{\prime})$, i.e. probability measures $\pi$ on $E\times E$ with
first and second marginals $\xi$ and $\xi^{\prime}$ respectively. We also
denote by $\mathcal{P}(E)$ the set of probability measures on $E$ and equip
this set with the weak topology.

We will say that a function
$\Gamma:\mathcal{P}_{2}(\mathbb{R}^{e})\to\mathbb{R}$ (with $e\in\mathbb{N}$)
is $L$–differentiable if there is a continuous function
$\partial_{\xi}\Gamma:\mathcal{P}_{2}(\mathbb{R}^{e})\times\mathbb{R}^{e}\to\mathbb{R}$
satisfying the following two properties:

  * •

for every $\xi,\xi^{\prime}\in\mathcal{P}_{2}(\mathbb{R}^{e})$ it holds

$\Gamma(\xi)-\Gamma(\xi^{\prime})=\int_{0}^{1}\int_{\mathbb{R}^{e}}\partial_{\xi}\Gamma\big{(}(1-t)\xi+t\xi^{\prime}\big{)}(x)(\xi^{\prime}-\xi)(dx)\,dt$

  * •

$\partial_{\xi}\Gamma$ is uniformly of quadratic growth on compacts: That is,
for every compact set $K\subseteq\mathcal{P}_{2}(\mathbb{R}^{e})$, there
exists a constant $C>0$ such that $|\partial_{\xi}\Gamma(\xi)(x)|\leq
C(1+|x|^{2})$ for all $\xi\in K$ and $x\in\mathbb{R}^{e}$.

see for instance [2; 35] or [12, Chapter 5] for further details. In
particular, the derivative is uniquely defined, up to an additive constant. In
the rest of the paper, we will use the notation
$\partial_{\xi}h,\partial_{\mu}h$ and $\partial_{\nu}h$ for the $L$-derivative
of a function $h$ in the variable of the probability measure
$\xi\in\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})$,
$\mu\in\mathcal{P}_{2}(\mathbb{R}^{\ell})$ and
$\nu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, respectively.

Throughout the paper, $C$ denotes a generic strictly positive constant. In the
computations, the constant $C$ can change from line to line, but this will not
always be mentioned. _However, $C$ will never depend on $N$._ Consider the
following conditions:

  1. (A1)

The function
$b:[0,T]\times\mathbb{R}^{\ell}\times\mathbb{R}^{m}\times\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})\to\mathbb{R}^{\ell}$
is continuously differentiable in its last three arguments and satisfies the
Lipschitz continuity and linear growth conditions

$\begin{split}&|b(t,x,a,\xi)-b(t,x^{\prime},a^{\prime},\xi^{\prime})|\leq
L_{b}\big{(}|x-x^{\prime}|+|a-a^{\prime}|+\mathcal{W}_{2}(\xi,\xi^{\prime})\big{)}\\\
&|b(t,x,a,\xi)|\leq
C\bigg{(}1+|x|+\Big{(}\int_{\mathbb{R}^{\ell}}|v|^{2}\,\mu(dv)\Big{)}^{1/2}\bigg{)}\end{split}$

for some $L_{b},C>0$ and all $x,x^{\prime}\in\mathbb{R}^{\ell}$,
$a,a^{\prime}\in\mathbb{R}^{m}$, $t\in[0,T]$ and
$\xi,\xi^{\prime}\in\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})$,
where $\mu$ is the first marginal of $\xi$.

The function
$f:[0,T]\times\mathbb{R}^{\ell}\times\mathbb{R}^{m}\times\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})\to\mathbb{R}$
is continuously differentiable in its last three arguments and of quadratic
growth:

$|f(t,x,a,\xi)|\leq
C\Big{(}1+|x|^{2}+|a|^{2}+\int_{\mathbb{R}^{\ell+m}}|v|^{2}\,\xi(dv)\Big{)}.$

The function
$g:\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})\to\mathbb{R}$ is
continuously differentiable and of quadratic growth:

$|g(x,\mu)|\leq
C\Big{(}1+|x|^{2}+\int_{\mathbb{R}^{\ell}}|v|^{2}\,\mu(dv)\Big{)}.$

  1. (A2)

The functions $b$ and $f$ can be decomposed as

(2) $b(t,x,a,\xi):=b_{1}(t,x,a,\mu)+b_{2}(t,x,\xi)\quad\text{and}\quad
f(t,x,a,\xi)=f_{1}(t,x,a,\mu)+f_{2}(t,x,\xi)$

for some functions $b_{1},b_{2}$, $f_{1}$ and $f_{2}$.

  1. (A3)

Considering the function

(3) $H(t,x,y,a,\xi)=f(t,x,a,\xi)+b(t,x,a,\mu)y,$

there is $\gamma>0$ such that

(4)
$H(t,x,y,a,\xi)-H(t,x,y,a^{\prime},\xi)-(a-a^{\prime})\partial_{a}H(t,x,y,a,\xi)\geq\gamma|a-a^{\prime}|^{2}$

for all $a,a^{\prime}\in\mathbb{A}$ and the functions $x\mapsto g(x,\mu)$ and
$(x,a)\mapsto H(t,x,y,a,\xi)$ are convex, where $\mu$ is the first marginal of
$\xi$. In addition, the functions

$\partial_{a}H(t,\cdot,\cdot,\cdot,\cdot)$,
$\partial_{x}H(t,\cdot,\cdot,\cdot,\cdot)$ and $\partial_{x}g(\cdot,\cdot)$
are $L_{f}$-Lipschitz–continuous

and of the linear growth.

$\begin{cases}|\partial_{x}H(t,x,a,y,\xi)|\leq
C\Big{(}1+|x|+|y|+\big{(}\int_{\mathbb{R}^{\ell}}|v|^{2}\mu(dv)\big{)}^{1/2}\Big{)}\\\
|\partial_{a}H(t,x,a,y,\xi)|\leq
C\Big{(}1+|x|+|a|+|y|+\big{(}\int_{\mathbb{R}^{\ell+m}}|v|^{2}\xi(dv)\big{)}^{1/2}\Big{)}\\\
|\partial_{x}g(x,\mu)\leq
C\Big{(}1+|x|+(\int_{\mathbb{R}^{\ell}}|v|^{2}\xi(dv))^{1/2}\Big{)}\end{cases}$

where $\mu$ is the first marginal of $\xi$.

  1. (A4)

For every
$(t,x,a,\xi)\in[0,T]\times\mathbb{R}^{\ell}\times\mathbb{A}\times\mathcal{P}_{2}(\mathbb{R}^{\ell\times
m})$ and $(u,v)\in\mathbb{R}^{\ell}\times\mathbb{R}^{m}$ we have

$\begin{cases}|\partial_{\mu}b(t,x,a,\mu)(u)|\leq C\\\
|\partial_{\xi}f(t,x,a,\xi)(u,v)|\leq
C\Big{(}1+|u|+|x|+\Big{(}\int_{\mathbb{R}^{\ell}}|v|^{2}\mu(dv)\Big{)}^{1/2}\Big{)}\\\
|\partial_{\mu}g(x,\mu)(u)|\leq
C\Big{(}1+|u|+|x|+\Big{(}\int_{\mathbb{R}^{\ell}}|v|^{2}\mu(dv)\Big{)}^{1/2}\Big{)}\end{cases}$

where $\mu$ is the first marginal of $\xi$.

  1. (A5)

The matrix $\sigma$ is uniformly elliptic. That is, there is a constant $c>0$
such that

$\langle\sigma\sigma^{\prime}x,x\rangle\geq c|x|^{2}\quad\text{for every
$x\in\mathbb{R}^{\ell}$.}$

The conditions (A1)-(A5) are essentially regularity and structural conditions
on the coefficients of the game. In [34], these conditions where proved to
guarantee convergence of the Nash equilibrium of the $N$-player game to the
mean field equilibrium of the associated mean field game. The following is
[34, Theorem 1].

######  Theorem 1.

Let conditions (A1)-(A5) be satisfied. Assume that the $N$-player game admits
a Nash equilibrium $\hat{\underline{\alpha}}^{N}\in\mathcal{A}^{N}$. Then
there is $\delta>0$ such that if $T\leq\delta$, for each $i=1,\dots,N$ the
sequence $(\hat{\alpha}^{i,N})$ converges to an admissible control
$\hat{\alpha}$ which is a mean field equilibrium and it holds that

$E[|\hat{\alpha}^{i,N}_{t}-\hat{\alpha}_{t}|^{2}]\leq Cr_{N,m,\ell}$

for all $N\in\mathbb{N}$ large enough and some constant $C>0$ where,
$r_{N,m,\ell}$ is a rate depending on $N,m,\ell$ such that
$r_{N,m,\ell}\downarrow 0$ as $N\to\infty$.

The case of arbitrary large time is treated in that paper under additional
monotonicity conditions as in Theorem 3 below. In order to derive a large
deviation principle, we will strengthen the growth conditions on the
derivatives into boundedness conditions, and more importantly, require the
master equation to admit a Lipschitz–continuous classical solution.

  1. (A6)

The functions $\partial_{x}g$, $\partial_{\mu}g$, $\partial_{x}f$ and
$\partial_{\mu}f$ are bounded.

  1. (A7)

There is a measurable function such that

$\Lambda(t,x,y,\mu)=\operatorname*{arg\,max}_{a\in A}H(t,x,y,\xi,a),$

where $\mu$ is the first marginal of $\xi$. Consider the function
$\varphi_{t}:\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{\ell})\to\mathcal{P}_{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{m})$
given by

(5) $\varphi_{t}(\xi):=\xi\circ(id_{\ell},\Lambda_{t}(\cdot,\cdot,\mu))^{-1}$

where $id_{\ell}$ is the projection on $\mathbb{R}^{\ell}$ and $\mu$ the first
marginal of $\xi$ and the functions

$B(t,x,y,\xi):=b\big{(}t,x,\Lambda_{t}(x,y,\mu),\varphi_{t}(\xi)\big{)},\quad
G(x,\mu)=\partial_{x}g(x,\mu).$

and

$\displaystyle F(t,x,y,\xi)$
$\displaystyle:=\partial_{x}f\big{(}t,x,\Lambda_{t}(x,y,\mu),\varphi_{t}(\xi)\big{)}+\partial_{x}b\big{(}t,x,\Lambda_{t}(x,y,\mu),\varphi_{t}(\xi)\big{)}y$

where $\mu$ is the first marginal of $\xi$. The following PDE admits a
(classical) solution
$V:[0,T]\times\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})\to\mathbb{R}^{\ell}$
which is Lipschitz–continuous in its second and third arguments uniformly in
$t$:

(6)
$\displaystyle\begin{cases}\partial_{t}V(t,x,\mu)+B(t,x,V(t,x,\mu),\xi)\partial_{x}V(t,x,\mu)+\frac{1}{2}\text{tr}(\partial_{xx}V(t,x,\mu)\sigma\sigma^{\prime})\\\
\quad+F(t,x,V(t,x,\mu),\xi)\displaystyle+\int_{\mathbb{R}^{d}}\partial_{\mu}V(t,x,\mu)(y)\cdot
B(t,y,V(t,x,\mu),\xi)\mu(dy)\\\
\quad+\int_{\mathbb{R}^{d}}\frac{1}{2}\text{tr}\left(\partial_{y}\partial_{\mu}V(t,x,\mu)(y)\sigma\sigma^{\prime}\right)\,\mu(dy)=0\\\
V(T,x,\mu)=G(x,\mu)\end{cases}$

with
$(t,x,\mu)\in[0,T)\times\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})$,
$\xi$ is the joint law of $(\chi,V(t,\chi,\mu))$ when $\mathcal{L}(\chi)=\mu$.
Note that the term $\text{tr}(\partial_{xx}V(t,x,\mu)\sigma\sigma^{\prime})$
is to be understood coordinate-wise, that is, putting
$V=(V^{1},\dots,V^{\ell})^{\top}$ we write
$\text{tr}(\partial_{xx}V(t,x,\mu)\sigma\sigma^{\prime})=\left(\text{tr}(\partial_{xx}V^{i}(t,x,\mu)\sigma\sigma^{\prime})\right)_{i}$.

The main result of the present work is a refinement of Theorem 1 into a LDP
both for a Nash equilibrium and the state processes at equilibrium. It is
well-known by the celebrated Varadhan-Bryc equivalence, given in [5; 19] (see
also [22, Section 1.2]) that the LDP is equivalent to the so-called Laplace
principle which can be stated as follows: Given a function
$\mathcal{I}:\mathcal{P}(E)\to[0,\infty]$ with (weakly) compact sublevel sets
$\\{\mu\in\mathcal{P}(E):\mathcal{I}(\mu)\leq a\\}$ called a (good) rate
function, a sequence of measures $(\mu^{N})_{N\in\mathbb{N}}$ on the Polish
space $E$ satisfies the Laplace principle (in the weak topology) if for every
bounded continuous function $F:\mathcal{P}(E)\to\mathbb{R}$ it holds

$\lim_{N\to\infty}-\frac{1}{N}\log(E\left[\exp(-NF(\mu^{N}))\right])=\inf_{\mu\in\mathcal{P}(E)}\left(F(\mu)-\mathcal{I}(\mu)\right).$

In the statement of the result we use the following notation:

  * •

The set $\mathcal{U}$ is defined as the set of
$((\Omega,\mathcal{F},P),(\mathcal{F}_{t}),u,W)$ such that the pair
$((\Omega,\mathcal{F},P),(\mathcal{F}_{t}))$ forms a stochastic basis
satisfying the usual conditions and carrying the $d$-dimensional Brownian
motion $W$ and $u$ is an $\mathbb{R}^{d}$-valued
$(\mathcal{F}_{t})$-progressive square integrable process.

  * •

By $\mathcal{C}^{e}$ we denote the space of continuous maps from $[0,T]$ to
$\mathbb{R}^{e}$.

  * •

The map
$\overline{B}:[0,T]\times\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})\to\mathbb{R}^{\ell}$
is defined as

(7)
$\overline{B}(t,x,\mu):=B\Big{(}t,x,V(t,x,\mu),\mathcal{L}\big{(}\chi,V(t,\chi,\mu)\big{)}\Big{)}$

where $\chi\in\mathbb{L}^{2}(\Omega,\mathcal{F},\mathbb{P})$ is a random
variable with law $\mathcal{L}(\chi)=\mu$, with $B$ defined in (A7).

  * •

The map
$\Psi:[0,T]\times\mathcal{P}_{2}(\mathbb{R}^{\ell})\to\mathcal{P}_{2}(\mathbb{R}^{m})$
is defined as

(8)
$\Psi(t,\mu)=\mu\circ\Lambda\big{(}t,\cdot,V(t,\cdot,\mu),\mu\big{)}^{-1}.$

######  Theorem 2.

If the conditions (A1)-(A7) are satisfied and the $N$-player game admits a
Nash equilibrium $\underline{\hat{\alpha}}$, then, there is a constant
$c(L_{b},L_{f},L_{g})$ depending on (the Lipschitz constants of) $f,$ $b$ and
$g$ such that if $T\leq c(L_{b},L_{f},L_{g})$, then the following hold:

  1. (i)

The sequence $(L^{N}(\underline{X}^{\underline{\hat{\alpha}}}))_{N}$ satisfies
the LDP with rate function111We use the convention $\inf\emptyset=\infty.$

(9)
$\mathcal{I}(\theta)=\inf_{u\in\mathcal{U}:\mathrm{law}(X^{u})=\theta}\mathbb{E}\bigg{[}\frac{1}{2}\int_{0}^{T}|u_{t}|^{2}\,dt\bigg{]},\quad\theta\in\mathcal{P}(\mathcal{C}^{\ell})$

where $dX^{u}_{t}=\overline{B}(t,X^{u}_{t},\mathcal{L}(X^{u}_{t}))+\sigma
u_{t}\,dt+\sigma\,dW_{t}$.

  2. (ii)

If in addition, the functions
$\Lambda:[0,T]\times\mathbb{R}^{\ell}\times\mathbb{R}^{\ell}\times\mathcal{P}(\mathbb{R}^{\ell})\to\mathbb{R}^{m}$
and
$V:[0,T]\times\mathbb{R}^{\ell}\times\mathcal{P}(\mathbb{R}^{\ell})\to\mathbb{R}^{\ell}$
are continuous for the weak topology on $\mathcal{P}(\mathbb{R}^{\ell})$, then
the sequence $(L^{N}(\underline{\hat{\alpha}}_{t}))_{N}$ satisfies the LDP
with rate function

$\mathcal{\widetilde{I}}_{t}(\nu):=\inf_{\theta\in\mathcal{P}_{2}(\mathcal{C}^{l}):\Psi(t,\theta_{t})=\nu}\mathcal{I}(\theta),\quad\nu\in\mathcal{P}(\mathbb{R}^{m}),$

where $\theta_{t}$ is the time $t$ marginal of $\theta$.

In summary, Theorem 2 tells us that when the coefficients $b,f$ and $g$ of the
game are sufficiently regular, the Hamiltonian satisfies a certain convexity
condition and the master equation corresponding to the mean field game admits
a Lipschitz–continuous solution, then, not only that any sequence of Nash
equilibria converges to a mean field equilibrium, but in addition the sequence
of Nash equilibria satisfies the LDP. The condition pertaining to existence
and Lipschitz–continuity of the solution of the master equation (6) (i.e.
(A7)) is the only hard-to-check condition we impose here. That being said,
conditions guarantying existence and uniqueness of the master equation are
given by Chassagneux et al. [16] and Cardaliaguet et al. [9]. These authors
study for instance the equation arising from a mean field game (albeit without
interaction through the controls) in [16, Section 5]. In section 3.2 below we
discuss an example for which the equation is known (from the work [16]) to
have a Lipschitz–continuous solution. Let us also refer to [25; 8; 36; 3] for
more recent results on the existence of the master equation.

The limitation in the above theorem is to assume $T$ small enough, a condition
which is needed to guarantee some FBSDE estimations. We can get around the
smallness condition by imposing additional monotonicity–type conditions on the
parameters. In fact, consider the following condition:

  1. (A8)

The drift $b$ satisfies the monotonicity condition

(10) $(x-x^{\prime})\cdot\Big{(}b(t,x,a,\xi)-b(t,x^{\prime},a,\xi)\Big{)}\leq-
K_{b}|x-x^{\prime}|^{2}$

and the function $\Lambda$ defined in (A7) as well as the functions $b,H$ and
$g$ satisfy

(11)
$\begin{cases}(y-y^{\prime})\cdot\Big{(}b(t,x,\Lambda(t,x,y,\mu),\xi)-b(t,x,\Lambda(t,x,y^{\prime},\mu),\xi)\Big{)}\leq-K|y-y^{\prime}|^{2}\\\
(x-x^{\prime})\cdot\Big{(}\partial_{x}H(t,x^{\prime},\Lambda(t,x^{\prime},y,\mu),\xi)-\partial_{x}H(t,x,\Lambda(t,x,y,\mu),\xi)\Big{)}\leq-K|x-x^{\prime}|^{2}\\\
(x-x^{\prime})\cdot\Big{(}\partial_{x}g(x,\mu)-\partial_{x}g(x^{\prime},\mu)\Big{)}\geq
K|x-x^{\prime}|\end{cases}$

for all $t\in[0,T]$, $x,x^{\prime},y,y^{\prime}\in\mathbb{R}^{\ell}$, $a\in A$
and $\xi\in\mathcal{P}_{2}(\mathbb{R}^{\ell\times m})$, and for some strictly
positive constants $K,K_{b}$.

Assuming monotonicity of $\Lambda$ is not an abstract condition. In many cases
(e.g. in the linear quadratic case) $\Lambda$ is a linear function of $y$.
Note moreover that by (4) the function $\Lambda$ is Lipschitz–continuous (see
e.g. [34]). Let us denote by $L_{\Lambda}$ its Lipschitz constant. Under the
above additional assumption, we have the following LDP:

######  Theorem 3.

If the conditions (A1)-11 are satisfied and the $N$-player game admits a Nash
equilibrium $\underline{\hat{\alpha}}$, then, there is a constant
$c(L_{b},L_{\Lambda})>0$ depending only on $L_{b}$ and $L_{\Lambda}$ such that
if $K_{b}>c(L_{b},L_{\Lambda})$ then the conclusions $(i)$ and $(ii)$ of
Theorem 2 hold.

The idea of the proofs is inspired from [18], but the details and techniques
are wholly different. The proof starts by identifying a weakly interacting
particle system for which the LDP is known, and that is exponentially close
(see Definition 4) to the state processes, then use the closeness property to
”transfer” the LDP to the state processes. Since the state processes of the
$N$ agents at equilibrium are characterized by a system of forward-backward
SDEs, the identification of the suitable auxiliary particle system whose LDP
is known uses the well-known technique of _decoupling fields_. However, the
proper decoupling field turns out to be the solution (when it exists) of the
_master equation_. The main difficulty lies in the proof of the exponential
closeness property. This is based on a priori bounds for systems of FBSDEs at
least provided that the solution of the master equation is
Lipschitz–continuous. In addition to the fact that we consider games allowing
control interaction and open-loop controls in the non-uniqueness MFG case, the
essential difference with [18] is the assumptions made on the PDEs. On the one
hand, we do not make use of the $N$-player PDE, and on the other hand we
require the solution of the master equation to be _Lipschitz–continuous_ ; we
do not make assumptions pertaining to its second derivative. This is an
interesting by-product of the convergence method of [34] which is based on
Pontryagin’s stochastic maximum principle. The (very) rough intuition for this
gain of smoothness is that in the present case, the FBSDE (resp. the master
equation) allows to represent the control (or the ”derivative of the value
function”). In contrast, the Hamilton-Jacobi-Bellman equation used in [18]
represents the value function itself.

##  3\.  Laplace principle for non-cooperating games

This section is mostly dedicated to the proof of Theorem 2. We will start by
shortly recalling the main idea of the proof of [34, Theorem 1]. This will
prepare the terrain for us here for the proof of the main results of the
present paper. The proof starts with key representation results for the Nash
equilibrium and the mean field equilibrium. In fact, let $\hat{\alpha}^{i,N}$
be a Nash equilibrium. It follows from [34] that there is a Lipschitz
continuous function
$\widehat{\Lambda}:[0,T]\times\mathbb{R}^{\ell}\times\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})\times\mathbb{R}\to\mathbb{A}$
such that

(12)
$\hat{\alpha}_{t}^{i,N}=\widehat{\Lambda}(t,X^{i,\underline{\hat{\alpha}}}_{t},Y^{i,i}_{t},L^{N}(\underline{X}_{t}^{\underline{\hat{\alpha}}}),\zeta^{N}_{t}),$

and $(Y^{i,j},Z^{i,j,k})$ satisfies the system of adjoint equations

(13)
$\begin{cases}dX^{i,\underline{\hat{\alpha}}}_{t}=b(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}_{t}^{i,N},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))\,dt+\sigma\,dW_{t}^{i}\\\
dY^{i,j}_{t}=-\partial_{x^{j}}H^{N,i}(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{i,N}_{t},\underline{Y}^{i,\cdot}_{t})\,dt+\sum_{k=1}^{N}Z^{i,j,k}_{t}dW^{k}_{t}\\\
X^{i,\underline{\hat{\alpha}}}_{0}=x,\quad\hat{\alpha}^{i,N}_{t}=\Lambda\Big{(}t,X^{i,\underline{\hat{\alpha}}}_{t},Y^{i,i}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t}),\zeta^{N}_{t}\Big{)},\quad
Y^{i,j}_{T}=\partial_{x^{j}}g^{N,i}(\underline{X}^{i,\underline{\hat{\alpha}}}_{T})\end{cases}$

with

$H^{N,i}(t,\underline{x},\underline{\alpha},\underline{y}):=f(t,x^{i},\alpha^{i},L^{N}(\underline{x},\underline{\alpha}))+\sum_{j=1}^{N}b(t,x^{j},\alpha^{j},L^{N}(\underline{x},\underline{\alpha}))y^{i,j}\text{
and }g^{N,i}(\underline{x})=g(x^{i},L^{N}(\underline{x}))$

and

$\begin{cases}\varepsilon^{i,N}_{t}:=\frac{1}{N}\partial_{\mu}f(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{i,N}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))(X^{i,\underline{\hat{\alpha}}}_{t})+\frac{1}{N}\sum_{j=1}^{N}\partial_{\mu}b(t,X^{j,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{j,N}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))(X^{i,\underline{\hat{\alpha}}}_{t})Y^{i,j}_{t}\\\
\zeta^{i,N}_{t}:=-\frac{1}{N}\partial_{\nu}f_{2}(t,X_{t}^{i,\underline{\hat{\alpha}}},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))(\hat{\alpha}^{i,N}_{t})-\frac{1}{N}\sum_{k=1}^{N}\partial_{\nu}b_{2}(t,X_{t}^{i,\underline{\hat{\alpha}}},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))(\hat{\alpha}^{k,N}_{t})Y^{i,k}_{t}\\\
\gamma^{i,N}:=\frac{1}{N}\partial_{\mu}g(X^{i,\underline{\hat{\alpha}}}_{T},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{T}))(X^{i,\underline{\hat{\alpha}}}_{T}).\end{cases}$

We will particularly be interested in the diagonal term
$(X^{i,\underline{\hat{\alpha}}},Y^{i,i},Z^{i,i,k})$ which, by a quick
verification, can be shown to satisfy

(14)
$\begin{cases}dX^{i,\underline{\hat{\alpha}}}_{t}=b(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}_{t}^{i},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))\,dt+\sigma\,dW_{t}^{i}\\\
dY^{i,i}_{t}=-\Big{\\{}\partial_{x}f(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{i}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))+\partial_{x}b(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{i}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t},\underline{\hat{\alpha}}_{t}))Y^{i,i}_{t}+\varepsilon^{i,N}_{t}\Big{\\}}\,dt+\sum_{k=1}^{N}Z^{i,i,k}_{t}dW^{k}_{t}\\\
X^{i,\underline{\hat{\alpha}}}_{0}=x,\quad\hat{\alpha}^{i}_{t}=\widehat{\Lambda}\Big{(}t,X^{i,\underline{\hat{\alpha}}}_{t},Y^{i,i}_{t},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{t}),\zeta^{i,N}_{t}\Big{)},\quad
Y^{i,i}_{T}=\partial_{x}g(X^{i,\underline{\hat{\alpha}}}_{T},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{T}))+\gamma^{i,N}.\end{cases}$

On the other hand, the limiting mean field equilibrium $\hat{\alpha}$
satisfies

(15)
$\hat{\alpha}^{i}_{t}=\widehat{\Lambda}(t,X^{i}_{t},Y^{i}_{t},\mathcal{L}(X^{i}_{t}),0)\equiv\Lambda(t,X^{i}_{t},Y^{i}_{t},\mathcal{L}(X^{i}_{t}))$

where222For ease of notation we omit the superscript $i$ and write $(X,Y,Z)$
instead of $(X^{i},Y^{i},Z^{i})$. $(X,Y,Z)$ solves the Mckean–Vlasov FBSDE

(16)
$\begin{cases}dX_{t}=b(X_{t},\hat{\alpha}_{t},\mathcal{L}(X_{t},\alpha_{t}))\,dt+\sigma\,dW_{t}\\\
dY_{t}=-\Big{\\{}\partial_{x}f(X_{t},\hat{\alpha}_{t},\mathcal{L}(X_{t},\hat{\alpha}_{t}))+\partial_{x}b(X_{t},\hat{\alpha}_{t},\mathcal{L}(X_{t},\hat{\alpha}_{t}))Y_{t}\Big{\\}}\,dt+Z_{t}dW_{t}\\\
X_{0}=x,\quad\hat{\alpha}^{i}_{t}=\Lambda\Big{(}t,X_{t},Y_{t},\mathcal{L}(X_{t})\Big{)},\quad
Y_{T}=\partial_{x}g(X_{T},\mathcal{L}(X_{T})).\end{cases}$

More precisely, it follows by propagation of chaos arguments that
$(X^{i,\hat{\alpha}},Y^{i,i})$ converges to $(X,Y)$ in
$\mathcal{S}^{2}(\mathbb{R}^{\ell}\times\mathbb{R}^{\ell})$, where, given a
normed space $E$ we denote by $\mathcal{S}^{p}(E)$ the space of adapted
processes $X$ such that

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}\|X_{t}\|^{p}_{E}\Big{]}<\infty.$

With this preparation out of the way, we are now ready for the proofs of the
main results of this article. The next section focuses on the non-cooperative
$N$-player game described in the introduction, and Section 4 will deal with
the linear-quadratic case for cooperative games.

###  3.1. LDP for small time horizons: Proof of Theorem 2

As announced in the previous section, the proof of Theorem 2 builds upon LDP
for uncontrolled, interacting (forward) particle systems. In order to exploit
such results, we will introduce a forward-backward particle system that is
”similar” to (but considerably more tractable than) (14). Using this auxiliary
particle system and well-known decoupling techniques from the theory of
forward-backward SDEs allow to construct an uncontrolled forward particle
system for which LDP results are well-known. The last step of the proof is to
show that the auxiliary particle system for which the LDP is known in the
literature is ”close enough” to our original particle system. This is achieved
using a priori estimations for FBSDEs. Here, close enough should be understood
in the following sense put forth in [19, Definition 4.2.10]:

######  Definition 4.

Let $(\mathcal{Y},d)$ be a metric space. The probability measures
$\mu_{\varepsilon}$ and $\tilde{\mu}_{\varepsilon}$ on $\mathcal{Y}$ are
called exponentially equivalent if there exist probability spaces
$(\Omega,\mathcal{B}_{\varepsilon},P_{\varepsilon})$ and two families of
$\mathcal{Y}$-valued random variables $Z_{\varepsilon}$ and
$\tilde{Z}_{\varepsilon}$ with joint laws $(P_{\varepsilon})$ and marginals
$\mu_{\varepsilon}$ and $\tilde{\mu}_{\varepsilon}$, respectively, such that
the following condition is satisfied:

For each $\delta>0$, the set
$\\{\omega:(\tilde{Z}_{\varepsilon},Z_{\varepsilon})\in\Gamma_{\delta}\\}$ is
$\mathcal{B}_{\varepsilon}$ measurable, and

$\limsup_{\varepsilon\to 0}\varepsilon\log
P_{\varepsilon}(\Gamma_{\delta})=-\infty,$

where
$\Gamma_{\delta}=\\{(\tilde{y},y):d(\tilde{y},y)>\delta\\}\subseteq\mathcal{Y}\times\mathcal{Y}$.

###### Proof of Theorem 2.

At equilibrium, the state process of player $i$ is given by the SDE

$dX^{i,\underline{\hat{\alpha}}}_{t}=b(t,X^{i,\underline{\hat{\alpha}}}_{t},\hat{\alpha}^{i,N}_{t},L^{N}(\underline{X}_{t}^{\underline{\hat{\alpha}}},\hat{\underline{\alpha}}_{t}))\,dt+\sigma\,dW^{i}_{t}$

with $\hat{\alpha}^{i,N}$ given by (12). Since $\Lambda$ is
Lipschitz–continuous (this follows from (4)), it can be shown that
$\varphi_{t}$ defined in (5) is Lipschitz continuous with respect to the
second order Wasserstein distance, see for instance (the proof of) [34,
Theorem 1]. Thus, the functions $B,F$ and $G$ introduced in (A7) are
Lipschitz–continuous and of linear growth. Further observe that using these
functions, the McKean–Vlasov equation (16) characterizing the mean field
equilibrium reads

(17)
$\begin{cases}dX_{t}=B(t,X_{t},Y_{t},\mathcal{L}(X_{t},Y_{t}))\,dt+\sigma\,dW_{t}\\\
dY_{t}=-F(t,X_{t},{Y}_{t},\mathcal{L}(X_{t},Y_{t}))\,dt+Z_{t}\,dW_{t}\\\
X_{0}=x,\quad Y_{T}=G(X_{T},\mathcal{L}(X_{T})).\end{cases}$

We will now introduce two auxiliary interacting particle systems that will
allow us to derive the LDP for the sequence of interest. First consider the
equation

(18)
$\begin{cases}d\widetilde{X}^{i,N}_{t}=B(t,\widetilde{X}^{i,N}_{t},\widetilde{Y}^{i,N}_{t},L^{N}(\widetilde{\underline{X}}_{t},\widetilde{\underline{Y}}_{t}))\,dt+\sigma\,dW_{t}^{i}\\\
d\widetilde{Y}^{i,N}_{t}=-F(t,\widetilde{X}^{i,N}_{t},\widetilde{Y}^{i,N}_{t},L^{N}(\widetilde{\underline{X}}_{t},\widetilde{\underline{Y}}_{t}))\,dt+\sum_{k=1}^{N}\widetilde{Z}^{i,k}_{t}\,dW_{t}^{k}\\\
X_{0}=x,\quad\widetilde{Y}_{T}^{i,N}=G(\widetilde{X}^{i,N}_{T},L^{N}(\underline{\widetilde{X}}_{T}))\end{cases}$

which simply corresponds to (14) after taking
”$\varepsilon^{i,N}=\gamma^{i,N}=\zeta^{i,N}=0$” for all $i$. Note that by
Lipschitz-continuity of $B,F$ and $G$, it follows e.g. from [12, Theorem 4.2]
that if $T$ is small enough, then Equation (18) admits a unique solution.

The second auxilliary equation is introduced through a decoupling argument.
Since $V$ is a classical solution of the PDE (6), applying Itô’s formula to
$\bar{Y}_{t}:=V(t,X_{t},\mathcal{L}(X_{t}))$ shows that there is $\bar{Z}$
such that $(\bar{Y},\bar{Z})$ solves the backward equation in (17). Thus, by
uniqueness, we have $Y_{t}=V(t,X_{t},\mathcal{L}(X_{t}))$. The function $V$ is
often called a decoupling field for the system (17) because it allows to write
the system as two decoupled equations, where $X$ satisfies

$\displaystyle dX_{t}$
$\displaystyle=B\Big{(}t,X_{t},V(t,X_{t},\mathcal{L}(X_{t})),\mathcal{L}\big{(}X_{t},V(t,X_{t},\mathcal{L}(X_{t})))\big{)}\Big{)}\,dt+\sigma\,dW_{t}$
(19)
$\displaystyle=\overline{B}(t,X_{t},\mathcal{L}(X_{t}))\,dt+\sigma\,dW_{t}$

where $\overline{B}$ is the function defined in (7). Since $V(t,\cdot,\cdot)$
is Lipschitz–continuous on
$\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})$ uniformly in
$t\in[0,T]$, it follows that $\overline{B}$ is Lipschitz continuous on
$\mathbb{R}^{\ell}\times\mathcal{P}_{2}(\mathbb{R}^{\ell})$ as well. The
second auxiliary particle system is then

(20)
$d\mathcal{X}^{i,N}_{t}=\overline{B}(t,\mathcal{X}^{i,N}_{t},L^{N}(\underline{\mathcal{X}}_{t}))\,dt+\sigma\,dW^{i}_{t},\quad\mathcal{X}^{i,N}_{0}=x.$

Note that $(\mathcal{X}^{i,N})_{i=1,\dots,N}$ is well-defined by classical SDE
theory. Moreover, the standard theory of propagation of chaos (see e.g. [40])
shows that the sequence $\mathcal{X}^{i,N}$ converges in
$\mathcal{S}^{2}(\mathbb{R}^{\ell})$ to $X$ and the sequence of empirical
measures $L^{N}(\underline{\mathcal{X}}_{t})$ converges to
$\mathcal{L}(X_{t})$ in $\mathcal{P}_{2}(\mathbb{R}^{\ell})$, see e.g. [13,
Theorem 2.12]. Furthermore, again by Lipschitz–continuity of $\overline{B}$,
it follows by [23, Theorem 5.2] (see also [6, Theorem 3.1]) that the family of
empirical measures $(L^{N}(\underline{\mathcal{X}}))_{N}$ satisfies the
Laplace principle (in $\mathcal{P}(\mathcal{C}^{\ell})$) with rate function
given by (32). [23, Theorem 5.2] shows that this rate function is lower
semicontinuous for the weak topology, and therefore the sublevel sets are
weakly closed. Let us show that the rate function is good, i.e. that it has
weakly compact sublevel sets. By Prokhorov’s theorem, if we show that the
sublevel sets are tight it will follow that they are weakly relatively
compact. Let $\mathbb{Q}^{n}$ be a probability measures on (the Polish space)
$\mathcal{C}^{d}$ such that $\mathcal{I}(\mathbb{Q}^{n})\leq K$ for some
constant $K>0$. Then there is a sequence $(u^{n})$ in
$\mathcal{H}^{2}(\mathbb{R}^{d})$, the space of $\mathbb{R}^{d}$–valued,
square integrable and progressive processes such that

$\mathbb{E}\bigg{[}\frac{1}{2}\int_{0}^{T}|u^{n}_{t}|^{2}\,dt\bigg{]}<K+1/n\quad\text{and}\quad\mathcal{L}(X^{u^{n}})=\mathbb{Q}^{n}.$

Thus, it suffices to show that the sequence $\mathcal{L}(X^{u^{n}})$ is tight.
Observe that by boundedness of the sequence $(u^{n})$ in
$\mathcal{H}^{2}(\mathbb{R}^{d})$ and linear growth of $\overline{B}$ it
follows by standard SDE estimations that
$\sup_{n}\sup_{t}\mathbb{E}[|X^{u^{n}}_{t}|^{2}]<\infty$. Furthermore, for
every $0\leq s\leq t\leq T$, we have

$\displaystyle|X^{u^{n}}_{t}-X^{u^{n}}_{s}|\leq
C\int_{s}^{t}1+|X^{u^{n}}_{r}|+E[|X^{u^{n}}_{r}|^{2}]^{1/2}+|u^{n}_{r}|\,dr+|\sigma||W_{t}-W_{s}|.$

Hence, taking expectation on both sides and applying Cauchy-Schwarz inequality
yields

$\displaystyle\mathbb{E}|X^{u^{n}}_{t}-X^{u^{n}}_{s}|$ $\displaystyle\leq
C|t-s|^{1/2}\bigg{(}1+\mathbb{E}\bigg{[}\int_{0}^{T}|u^{n}_{r}|^{2}\,dr\bigg{]}^{1/2}+\sup_{r}\mathbb{E}\big{[}|X^{u^{n}}_{r}|^{2}\big{]}^{1/2}+|\sigma|\bigg{)}$
$\displaystyle\leq C|t-s|^{1/2}.$

Thus, since $(X^{u^{n}})$ is a sequence of continuous processes, it follows by
Kolmogorov’s tightness criterion that $\mathbb{Q}^{n}$ is tight. Therefore,
$\mathcal{I}$ is a good rate function.

In order to ”transfer” the LDP from the sequence
$(L^{N}(\underline{\mathcal{X}}))$ to the relevant sequence
$(L^{N}(\underline{X}^{\underline{\hat{\alpha}}}))$, we need to show that the
two sequences are exponentially close in the sense of Definition 4. This
follows from Chebyshev’s inequality and Lemma 5 below since we have

$\displaystyle\mathbb{P}\Big{(}\sup_{t\in[0,T]}\mathcal{W}_{2}(L^{N}(\underline{X}_{t}^{\underline{\hat{\alpha}}}),L^{N}(\underline{\mathcal{X}}_{t}))>\varepsilon\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\frac{1}{N}\sup_{t\in[0,T]}\sum_{i=1}^{N}|X^{\underline{\alpha},i}_{t}-\mathcal{X}^{i}_{t}|^{2}\Big{\\}}^{1/2}\geq\varepsilon\Big{)}$
$\displaystyle\leq\mathbb{E}\Big{[}\exp\Big{\\{}\Big{(}\sum_{i=1}^{N}\|X^{\underline{\alpha},i}-\mathcal{X}^{i}\|^{2}_{\infty}\Big{)}\Big{\\}}\Big{]}e^{-\varepsilon^{2}N^{2}}$
$\displaystyle\leq Ce^{-\varepsilon^{2}N^{2}}.$

Therefore,

$\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\Big{(}\sup_{t\in[0,T]}\mathcal{W}_{2}(L^{N}(\underline{X}_{t}^{\underline{\hat{\alpha}}}),L^{N}(\underline{\mathcal{X}}_{t}))>\varepsilon\Big{)}=-\infty.$

It then follows from [19, Theorem 4.2.13] that the sequence
$L^{N}(\underline{X}^{\underline{\hat{\alpha}}})$ satisfies the LDP with rate
function $\mathcal{I}$.

Let us now turn to the large deviation principle for the $N$-Nash equilibrium
$(\hat{\alpha}^{i,N})_{i=1,\dots,N}$. The difficulty here is the fact that
$\hat{\alpha}^{i,N}$ is not a function of $X^{i,\underline{\hat{\alpha}}}$ and
$L^{N}(\underline{X}^{\underline{\hat{\alpha}}})$ only, it also depends on the
process $\zeta_{t}^{i,N}$, see (14). Nevertheless, using the contraction
principle, we will again prove the LDP for an auxiliary sequence that is
exponentially equivalent to $L^{N}(\underline{\hat{\alpha}})$. Define the
auxiliary process

$\alpha^{i,N}_{t}:=\Lambda\Big{(}t,\mathcal{X}^{i,N}_{t},V\big{(}t,\mathcal{X}^{i,N}_{t},L^{N}(\underline{\mathcal{X}}_{t})\big{)},L^{N}(\underline{\mathcal{X}}_{t})\Big{)}.$

It follows by continuity of $\Lambda$ and $V$, the convergence of
$(\mathcal{X}^{i,N})_{N\geq 1}$ to $X$ and the representation (15) of the mean
field equilibrium $\hat{\alpha}$ that

$\alpha^{i,N}_{t}\to\hat{\alpha}_{t}\quad\text{in}\quad\mathbb{L}^{2}.$

We will now use the contraction principle to show that
$(L^{N}(\underline{\alpha}_{t}))_{N\geq 1}$ satisfies the LDP. Consider the
function $\Psi$ mapping $\mathcal{P}(\mathbb{R}^{\ell})$ to
$\mathcal{P}(\mathbb{R}^{\ell})$ and defined as

(21)
$\Psi(t,\mu):=\mu\circ\Lambda\big{(}t,\cdot,V(t,\cdot,\mu),\mu\big{)}^{-1}.$

The function $\Psi$ is continuous. In fact, given a sequence $(\mu^{n})_{n\geq
1}$ converging to $\mu$ in the weak topology, and a bounded, Lipschitz-
continuous test function $f$, we have

$\displaystyle\int_{\mathbb{R}^{\ell}}f(x)\Psi(t,\mu^{n})(dx)$
$\displaystyle=\int_{\mathbb{R}^{\ell}}f\Big{(}\Lambda(t,x,V(t,x,\mu^{n}),\mu^{n})\Big{)}\mu^{n}(dx).$

Since $\Lambda$ and $V$ are continuous on $\mathcal{P}(\mathbb{R}^{\ell})$,
the functions $f^{n}(x):=f\Big{(}\Lambda(t,x,V(t,x,\mu^{n}),\mu^{n})\Big{)}$
define a sequence of bounded continuous functions converging to
$f\Big{(}\Lambda(t,x,V(t,x,\mu),\mu)\Big{)}$ pointwise. Since $f^{n}$ is
uniformly bounded and uniformly Lipschitz, it follows that

$\displaystyle\int_{\mathbb{R}^{\ell}}f(x)\Psi(t,\mu^{n})(dx)$
$\displaystyle=\int_{\mathbb{R}^{\ell}}f^{n}(x)\mu^{n}(dx)\to\int_{\mathbb{R}^{\ell}}f\Big{(}\Lambda(t,x,V(t,x,\mu),\mu)\Big{)}\mu(dx),$

showing that $\Psi$ is continuous. Now, by the definition of $\Psi$, we have

$L^{N}(\underline{\alpha}_{t})=\Psi(t,L^{N}(\underline{\mathcal{X}}_{t}))$

and similarly, since the mean field equilibrium $\hat{\alpha}$ satisfies the
representation
$\hat{\alpha}_{t}=\Lambda(t,X_{t},Y_{t},\mathcal{L}(X_{t}))=\Lambda\big{(}t,X_{t},V(t,X_{t},\mathcal{L}(X_{t})),\mathcal{L}(X_{t})\big{)}$,
it follows that

$\mathcal{L}(\alpha_{t})=\Psi(t,\mathcal{L}(X_{t})).$

Using the fact that the sequence $L^{N}(\underline{\mathcal{X}}_{t})$
satisfies the LDP with rate function

$\mathcal{I}_{t}(\mu):=\inf_{\theta\in\mathcal{P}(\mathcal{C}^{\ell}):\theta_{t}=\mu}\mathcal{I}(\theta)$

and that $\Psi$ is continuous, it follows by the contraction principle that
$L^{N}(\underline{\alpha}_{t})$ satisfies the LDP as well. In fact, for every
bounded continuous function $F:\mathcal{P}(\mathbb{R}^{m})\to\mathbb{R}$, the
function $F\circ\Psi(t,\cdot)$ is again bounded continuous and therefore we
have

$\displaystyle-\frac{1}{N}\log\mathbb{E}\Big{[}\exp(-NF(L^{N}(\underline{\alpha}_{t})))\Big{]}$
$\displaystyle=-\frac{1}{N}\log\mathbb{E}\Big{[}\exp\big{(}-NF\circ\Psi(t,L^{N}(\underline{\mathcal{X}}_{t}))\big{)}\Big{]}$
$\displaystyle\to\inf_{\mu\in\mathcal{P}(\mathbb{R}^{\ell})}(F\circ\Psi(t,\mu)-\mathcal{I}_{t}(\mu))$
$\displaystyle=\inf_{\nu\in\mathcal{P}(\mathbb{R}^{m})}(F(\nu)-\widetilde{\mathcal{I}}(\nu))$

with

$\widetilde{\mathcal{I}}(\nu):=\inf_{\theta\in\mathcal{P}(\mathcal{C}^{\ell}):\Psi(t,\theta_{t})=\nu}\mathcal{I}(\mu).$

Similar to the proof of the LDP for
$L^{N}(\underline{X}^{\underline{\hat{\alpha}}})$, it remains to show that the
sequences $(L^{N}(\underline{\alpha}_{t}))_{N}$ and
$(L^{N}(\underline{\hat{\alpha}}_{t}))_{N}$ are exponentially close and that
the rate function $\mathcal{\widetilde{I}}$ has compact sublevel sets. The
latter property follows from the fact that for every $x\geq 0$ it holds
$\\{\nu\in\mathcal{P}(\mathbb{R}^{m}):\mathcal{\widetilde{I}}(\nu)\leq
x\\}=\Psi\big{(}t,\\{\theta_{t}:\theta\in\mathcal{P}_{2}(\mathbb{R}^{\ell}),\,\,\mathcal{I}(\theta)\leq
x\\}\big{)}$ and $\Psi(t,\cdot)$ is a continuous function for the weak
topology. Let us now show exponential closeness. To this end, we first show
that the function $\Psi(t,\cdot)$ is Lipschitz continuous with respect to the
$2$-Wasserstein topology. By Kantorovich’s duality theorem, see [41, Theorem
5.10], for every $\xi,\xi^{\prime}\in\mathcal{P}_{2}(\mathbb{R}^{\ell})$ we
have

$\displaystyle\mathcal{W}_{2}(\Psi(t,\xi),\Psi(t,\xi^{\prime}))$
$\displaystyle=\sup\bigg{(}\int_{\mathbb{R}^{\ell}}h_{1}(x)\Psi(t,\xi)(dx)-\int_{\mathbb{R}^{\ell}}h_{2}(x^{\prime})\Psi(t,\xi^{\prime})(dx^{\prime})\bigg{)}$
$\displaystyle=\sup\bigg{(}\int_{\mathbb{R}^{\ell}}h_{1}(\Lambda(t,x,V(t,x,\xi),\xi))\xi(dx)$
$\displaystyle\qquad-\int_{\mathbb{R}^{\ell}}h_{2}(\Lambda(t,x^{\prime},V(t,x,\xi^{\prime}),\xi^{\prime}))\xi^{\prime}(dx^{\prime})\bigg{)}$

with the supremum being taken over the set of bounded continuous functions
$h_{1},h_{2}:\mathbb{R}^{\ell}\to\mathbb{R}$ such that
$h_{1}(x)-h_{2}(x^{\prime})\leq|x-x^{\prime}|^{2}$ for every
$x,x^{\prime}\in\mathbb{R}^{\ell}$ which, by Lipschitz continuity of $\Lambda$
and $V$ imply that
$h_{1}\Big{(}\Lambda(t,x,V(t,x,\xi),\xi)\Big{)}-h_{2}\Big{(}\Lambda(t,x^{\prime},V(t,x^{\prime},\xi^{\prime}),\xi^{\prime})\Big{)}\leq
C|x-x^{\prime}|^{2}+C\mathcal{W}_{2}(\xi,\xi^{\prime})$ for some constant
$C>0$. This shows that

$\displaystyle\mathcal{W}_{2}(\Psi(t,\xi),\Psi(t,\xi^{\prime}))\leq
C\sup\bigg{(}\int_{\mathbb{R}^{\ell}}\tilde{h}_{1}(x)\xi(dx)-\int_{\mathbb{R}^{\ell}}\tilde{h}_{2}(x^{\prime})\xi^{\prime}(dx^{\prime})\bigg{)}+C\mathcal{W}_{2}(\xi,\xi^{\prime})$

with the supremum over functions $\tilde{h}_{1},\tilde{h}_{2}$ such that
$\tilde{h}_{1}(x)-\tilde{h}_{2}(x^{\prime})\leq C|x-x^{\prime}|^{2}$. Hence,
applying Kantorovich duality once again yields

(22) $\displaystyle\mathcal{W}_{2}(\Psi(t,\xi),\Psi(t,\xi^{\prime}))\leq
C\mathcal{W}_{2}(\xi,\xi^{\prime}).$

Thus, we have

$\displaystyle\mathbb{P}\Big{(}\mathcal{W}_{2}^{2}\big{(}L^{N}(\underline{\hat{\alpha}}_{t}),L^{N}(\underline{\alpha}_{t})\big{)}>\varepsilon\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\mathcal{W}_{2}^{2}\big{(}L^{N}(\underline{X}^{\underline{\alpha}}_{t},L^{N}(\underline{\mathcal{X}}_{t})\big{)}>\varepsilon\Big{)}$
$\displaystyle\leq Ce^{-\varepsilon^{2}N^{2}}$

for some constant $C>0$.

Therefore, it follows that

$\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\Big{(}\sup_{t\in[0,T]}\mathcal{W}_{2}^{2}(L^{N}(\underline{\hat{\alpha}}),L^{N}(\underline{\alpha_{t}}))>\varepsilon\Big{)}=-\infty.$

This concludes the proof. ∎

###### Proof of Theorem 3.

The proof is almost the same as that of Theorem 2, except for two points.
First, to get well-posedness of the FBSDEs (18) and (17), we rather use the
results in [37] and [4] respectively. Secondly, one should apply the second
part of Lemma 5 below, (i.e. the case of $T$ arbitrary) to get the bound of
$\sum_{i=1}^{N}\|X^{\underline{\alpha},i}-\mathcal{X}^{i}\|^{2}_{\infty}$. ∎

We finish the proof with the following key lemma used to derive the large
deviation principles.

######  Lemma 5.

If the assumptions (A1)-(A7) are satisfied, then it holds that

(23)
$\sup_{N}\Big{\|}\sum_{j=1}^{N}|Y^{i,j}|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}<\infty\quad\text{for
all}\quad t\in[0,T]\quad\text{and for each }i$

and there is a constant $C>0$ such that

(24) $|\varepsilon^{N}|+|\gamma^{N}|+|\zeta^{N}|\leq C/N.$

Moreover, there is $c>0$ such that if $T\leq c$, then

(25)
$\sup_{N}\Big{\|}\sum_{i=1}^{N}|X^{i,\underline{\hat{\alpha}}}-\mathcal{X}^{i,N}|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}+\sup_{N}\Big{\|}\sum_{i=1}^{N}|Y^{i,i}-V(\cdot,\mathcal{X}^{i,N}_{\cdot},L^{N}(\underline{\mathcal{X}}_{\cdot}))|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}<\infty.$

If the conditions (A1)-11 hold, then there is a constant
$c(L_{b},L_{\Lambda})$ such that if $K_{b}>c(L_{b},L_{\Lambda})$, then the
bound (25) holds for all $T>0$.

###### Proof.

Taking the conditional expectation in (13) with respect to
$\mathcal{F}^{N}_{t}$ and recalling the definition of $H^{N,i}$, we get

$\displaystyle|Y^{i,j}_{t}|$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{|}\delta_{i,j}\partial_{x}g\Big{(}X^{i,\underline{\hat{\alpha}}}_{T},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{T})\Big{)}\Big{|}+\frac{1}{N}\Big{|}\partial_{\mu}g\Big{(}X^{i,\underline{\hat{\alpha}}}_{T},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{T})\Big{)}(X^{j,\underline{\hat{\alpha}}}_{T})\Big{|}\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{t}^{T}\Big{(}\Big{|}\delta_{i,j}\partial_{x}f\Big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}\Big{|}+\Big{|}\frac{1}{N}\partial_{\mu}f\Big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}(X^{j,\underline{\hat{\alpha}}}_{s})\Big{|}\Big{)}ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle+\mathbb{E}\bigg{[}\int_{t}^{T}\Big{(}\Big{|}\partial_{x}b\Big{(}s,X^{j,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{j}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}Y^{i,j}_{s}\Big{|}+\frac{1}{N}\sum_{k=1}^{N}\Big{|}\partial_{\mu}b\Big{(}s,X^{k,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{k}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}(X^{j,\underline{\hat{\alpha}}}_{s})Y^{i,k}_{s}\Big{|}\Big{)}ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}.$

Next, using boundedness of the functions $\partial_{x}g$, $\partial_{\mu}g$,
$\partial_{x}f$, $\partial_{\mu}f$, $\partial_{x}b$ and $\partial_{\mu}b$
yields

$\displaystyle|Y^{i,j}_{t}|$ $\displaystyle\leq
C_{1}\delta_{i,j}+C_{2}/N+C_{3}\mathbb{E}\bigg{[}\int_{t}^{T}\Big{(}|Y^{i,j}_{s}|+\frac{1}{N}\sum_{k=1}^{N}|Y^{i,k}_{s}|\Big{)}ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}.$

Summing up over $j$ and using Gronwall’s inequality yields (23).

Let us now show (24). This follows directly from the above bound and the fact
that the measure derivatives of the coefficients $b,f$ and $g$ are bounded. In
fact, it follows by definition of $\varepsilon^{N},\gamma^{N}$ and $\zeta^{N}$
that

$|\varepsilon^{N}|+|\gamma^{N}|+|\zeta^{N}|\leq
C_{1}\frac{1}{N}+C_{2}\frac{1}{N}\sum_{j=1}^{N}|Y^{i,j}_{t}|\leq C/N,$

where the second equality follows by (23).

We will prove the bound (25) only when the monotonicity condition 11 is
satisfied. The case $T$ small follows by similar arguments, we will explain
the different at the end of the proof. Thus, we start by denoting

$\mathcal{Y}^{i,N}_{t}:=V(t,\mathcal{X}^{i,N}_{t},L^{N}(\underline{\mathcal{X}}_{t}))$

recall the function $\bar{B}$ introduced in (7) and the auxiliary control

$\alpha^{i,N}_{t}:=\Lambda\Big{(}t,\mathcal{X}^{i,N}_{t},V\big{(}t,\mathcal{X}^{i,N}_{t},L^{N}(\underline{\mathcal{X}}_{t})\big{)},L^{N}(\underline{\mathcal{X}}_{t})\Big{)}.$

By Lipschitz continuity of $b$ and ${\Lambda}$ and recalling the
representation (12) of the Nash equilibrium $\hat{\alpha}^{i,N}$, we have

$\displaystyle|X^{i,\underline{\hat{\alpha}}}_{t}-\mathcal{X}^{i,N}_{t}|^{2}=2\int_{0}^{t}(X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s})\Big{(}b\big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i,N}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\big{)}-\overline{B}\big{(}s,\mathcal{X}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s})\big{)}\Big{)}ds$
$\displaystyle=2\int_{0}^{t}(X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s})\bigg{(}b\big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i,N}_{s},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\big{)}$
$\displaystyle\qquad-b\big{(}s,\mathcal{X}^{i,N}_{s},\Lambda_{s}(\mathcal{X}^{i,N}_{s},\mathcal{Y}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s})),L^{N}(\underline{\mathcal{X}}_{s},\underline{\alpha}_{s}))\big{)}\bigg{)}ds$
$\displaystyle\leq
2\int_{0}^{t}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|\bigg{(}(L_{b}L_{\Lambda}-K_{b})|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+L_{b}L_{\Lambda}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|+L_{b}L_{\Lambda}\mathcal{W}_{2}(L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s}),L^{N}(\underline{\mathcal{X}}_{s}))$
$\displaystyle\qquad+L_{b}L_{\Lambda}|\zeta^{i,N}_{s}|+L_{b}\mathcal{W}_{2}(L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s}),L^{N}(\underline{\mathcal{X}}_{s},\underline{\alpha}_{s}))\bigg{)}ds$
$\displaystyle\leq
2\int_{0}^{t}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|\bigg{(}(L_{b}L_{\Lambda}-K_{b})|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+L_{b}L_{\Lambda}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|+L_{b}(L_{\Lambda}+1)\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}$
$\displaystyle\qquad+4L_{\Lambda}L_{b}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}\Big{(}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}+|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}+\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}+|\zeta^{j,N}_{s}|^{2}\Big{)}\Big{\\}}^{1/2}$
$\displaystyle\qquad+L_{b}L_{\Lambda}|\zeta^{i,N}_{s}|\bigg{)}ds$
$\displaystyle\leq
2\int_{0}^{t}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|\bigg{(}(L_{b}L_{\Lambda}-K_{b})|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+L_{b}L_{\Lambda}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|+L_{b}(L_{\Lambda}+1)\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}$
$\displaystyle\qquad+8L_{b}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}+4L_{b}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}$
$\displaystyle\qquad+4L_{b}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|\zeta^{j,N}_{s}|^{2}\Big{\\}}^{1/2}+L_{b}L_{\Lambda}|\zeta^{i,N}_{s}|\bigg{)}ds.$

Applying Young’s inequality for some $\varepsilon>0$ and $\eta>0$ to be
determined, we continue the estimation as

$\displaystyle|X^{i,\underline{\hat{\alpha}}}_{t}-\mathcal{X}^{i,N}_{t}|^{2}$
$\displaystyle\leq
2\int_{0}^{t}\bigg{(}\Big{(}L_{b}L_{\Lambda}+\frac{L_{b}+9L_{b}L_{\Lambda}}{2}+\frac{17L_{b}^{2}L_{\Lambda}^{2}}{2\varepsilon}+\eta-
K_{b}\Big{)}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|^{2}$
$\displaystyle\qquad+\frac{\varepsilon}{2}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|^{2}+\frac{9L_{b}L_{\Lambda}+L_{b}}{2}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}$
$\displaystyle\qquad+\frac{L_{b}^{2}L_{\Lambda}^{2}}{2\eta}|\zeta^{i,N}_{s}|+\frac{\varepsilon}{2}\frac{1}{N}\sum_{j=1}^{N}|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}+\frac{8L_{b}^{2}L_{\Lambda}^{2}}{\eta}\frac{1}{N}\sum_{j=1}^{N}|\zeta^{j,N}_{s}|^{2}\bigg{)}ds.$

Let us introduce the quantity

$\delta(\varepsilon,\eta)=10L_{b}L_{\Lambda}+L_{b}+\frac{17L_{b}^{2}L_{\Lambda}^{2}}{2\varepsilon}+\eta-
K_{b}.$

Summing up over $i$ on both sides and using Gronwall’s inequality implies that

(26)
$\sum_{i=1}^{N}|X^{i,\underline{\hat{\alpha}}}_{t}-\mathcal{X}^{i,N}_{t}|^{2}\leq
2e^{2\delta(\varepsilon,\eta)t}\int_{0}^{T}\Big{(}\varepsilon\sum_{i=1}^{N}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|^{2}+\frac{17L_{b}^{2}L_{\Lambda}^{2}}{2\eta}\sum_{i=1}^{N}|\zeta^{i,N}_{s}|^{2}\Big{)}ds.$

On the other hand, since $V$ is a classical solution of the PDE (6), applying
Itô’s formula for functions of laws of diffusions (see [12, Theorem 5.104])
implies that

$\mathcal{Y}^{i,N}_{t}=\partial_{x}g(\mathcal{X}^{i,N}_{T},L^{N}(\underline{\mathcal{X}}_{T}))+\int_{t}^{T}F(s,\mathcal{X}^{i,N}_{s},\mathcal{Y}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s},\underline{\mathcal{Y}}_{s}))ds-\int_{t}^{T}\sum_{k=1}^{N}\mathcal{Z}^{i,N,k}_{s}dW^{k}_{s}$

with the process $\mathcal{Z}^{i,k,N}$ given by

$\mathcal{Z}^{i,k,N}_{t}:=\partial_{x^{k}}V(t,\mathcal{X}^{i}_{t},L^{N}(\underline{\mathcal{X}}_{t})).$

Using the Lipschitz–continuity condition on $\partial_{x}H(t,x,y,a,\xi)$ and
$\partial_{x}g(x,\mu)$ (see (A3)), it follows by (14) and (18) that

$\displaystyle|Y^{i,i}_{t}-\mathcal{Y}^{i,N}_{t}|^{2}$
$\displaystyle=\mathbb{E}\bigg{[}\Big{|}\partial_{x}g(X^{i,\underline{\hat{\alpha}}}_{T},L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{T}))+\gamma^{i,N}-\partial_{x}g(\mathcal{X}^{i,N}_{T},L^{N}(\underline{\mathcal{X}_{T}}))\Big{|}^{2}\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\quad+\mathbb{E}\bigg{[}2\int_{t}^{T}\big{(}Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}\big{)}\bigg{(}\partial_{x}f\Big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i,N}_{s},L^{N}(\underline{X^{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}+\partial_{x}b\Big{(}s,X^{i,\underline{\hat{\alpha}}}_{s},\hat{\alpha}^{i,N}_{s},L^{N}(\underline{X^{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s})\Big{)}Y^{i,i}_{s}+\varepsilon^{i,N}_{s}$
$\displaystyle\quad-\partial_{x}f\Big{(}s,\mathcal{X}^{i,N}_{s},\Lambda_{s}\big{(}\mathcal{X}^{i,N}_{s},\mathcal{Y}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s})\big{)},L^{N}(\underline{\mathcal{X}}_{s},\underline{\alpha}_{s})\Big{)}$
$\displaystyle\quad-\partial_{x}b\Big{(}s,\mathcal{X}^{i,N}_{s},\Lambda_{s}\big{(}\mathcal{X}^{i,N}_{s},\mathcal{Y}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s})\big{)},L^{N}(\underline{\mathcal{X}}_{s},\underline{\alpha}_{s})\Big{)}\mathcal{Y}^{i,N}_{s}\bigg{)}ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}3L_{g}^{2}|X^{i,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{i,N}_{T}|^{2}+\frac{3L_{g}^{2}}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{j,N}_{T}|^{2}+3|\gamma^{i,N}|^{2}\bigg{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\quad+\mathbb{E}\bigg{[}2\int_{t}^{T}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|\bigg{(}L_{f}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+L_{f}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|+|\varepsilon^{i,N}_{s}|$
$\displaystyle\quad+L_{f}\Big{|}\hat{\alpha}^{i,N}_{s}-\Lambda_{s}\big{(}\mathcal{X}^{i,N}_{s},\mathcal{Y}^{i,N}_{s},L^{N}(\underline{\mathcal{X}}_{s})\big{)}\Big{|}+L_{f}\mathcal{W}_{2}\Big{(}L^{N}(\underline{X}^{\underline{\hat{\alpha}}}_{s},\underline{\hat{\alpha}}_{s}),L^{N}(\underline{\mathcal{X}}_{s},\underline{\alpha}_{s})\Big{)}ds\bigg{)}\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}L_{g}^{2}|X^{i,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{i,N}_{T}|^{2}+\frac{3L_{g}^{2}}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{j,N}_{T}|^{2}+3|\gamma^{i,N}|^{2}\bigg{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\quad+\mathbb{E}\bigg{[}2\int_{t}^{T}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|\bigg{(}L_{f}(1+L_{\Lambda})\big{(}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|\big{)}+|\varepsilon^{i,N}_{s}|+L_{f}L_{\Lambda}|\zeta^{i,N}_{s}|$
$\displaystyle\quad+L_{f}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}\Big{(}(1+4L^{2}_{\Lambda})|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}+4L_{\Lambda}^{2}|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}+4L_{\Lambda}^{2}|\zeta^{j,N}_{s}|^{2}+4L_{\Lambda}^{2}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{)}\Big{\\}}^{1/2}$
$\displaystyle\quad+L_{f}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}\bigg{)}\,ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}3L_{g}^{2}|X^{i,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{i,N}_{T}|^{2}+3L_{g}^{2}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{j,N}_{T}|^{2}\Big{\\}}+3|\gamma^{i,N}|^{2}\bigg{|}\mathcal{F}^{N}_{t}\bigg{]}$
$\displaystyle\quad+\mathbb{E}\bigg{[}2\int_{t}^{T}|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|\bigg{(}L_{f}(1+L_{\Lambda})\big{(}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|+|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|\big{)}+|\varepsilon^{i,N}_{s}|+L_{f}L_{\Lambda}|\zeta^{i,N}_{s}|$
$\displaystyle\quad+\Big{(}L_{f}(1+L_{\Lambda})\sqrt{1+8L_{\Lambda}^{2}}\Big{)}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}$
$\displaystyle\quad+2L_{f}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}\Big{\\}}^{1/2}+2L_{f}L_{\Lambda}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|\zeta^{j,N}_{s}|^{2}\Big{\\}}^{1/2}\bigg{)}\,ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]}.$

Applying Young’s inequality for some constants $\varepsilon_{1}>0$ and
$\eta_{1}>0$ to be determined, we continue the estimations as

$\displaystyle|Y^{i,i}_{t}-\mathcal{Y}^{i,N}_{t}|^{2}$
$\displaystyle\leq\mathbb{E}\bigg{[}3L_{g}^{2}|X^{i,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{i,N}_{T}|^{2}+3L_{g}^{2}\Big{\\{}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{T}-\mathcal{X}^{j,N}_{T}|^{2}\Big{\\}}+3|\gamma^{i,N}|^{2}\bigg{|}\mathcal{F}_{t}^{N}\bigg{]}$
$\displaystyle\qquad+\mathbb{E}\bigg{[}\int_{t}^{T}\bigg{(}(2L_{f}+2L_{\Lambda}L_{f}+3\varepsilon_{1}+3\eta_{1})|Y^{i,i}_{s}-\mathcal{Y}^{i,N}_{s}|^{2}+\frac{(L_{f}+L_{f}L_{\Lambda})^{2}}{\varepsilon_{1}}|X^{i,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{i,N}_{s}|^{2}$
$\displaystyle\qquad+\frac{(L_{\Lambda}L_{f}+L_{f}\sqrt{1+8L_{\Lambda}^{2}})^{2}}{\varepsilon_{1}}\frac{1}{N}\sum_{j=1}^{N}|X^{j,\underline{\hat{\alpha}}}_{s}-\mathcal{X}^{j,N}_{s}|^{2}+\frac{4L_{f}^{2}L_{\Lambda}^{2}}{\varepsilon_{1}}\frac{1}{N}\sum_{j=1}^{N}|Y^{j,j}_{s}-\mathcal{Y}^{j,N}_{s}|^{2}$
$\displaystyle\qquad+\frac{3}{\eta_{1}}|\varepsilon^{i,N}_{s}|^{2}+\frac{3L_{f}^{2}L_{\Lambda}^{2}}{\eta_{1}}|\zeta^{i,N}_{s}|^{2}+\frac{6L_{\Lambda}^{2}L_{f}^{2}}{\eta_{1}}\frac{1}{N}\sum_{j=1}^{N}|\zeta^{j,N}_{s}|^{2}\bigg{)}\,ds\Big{|}\mathcal{F}^{N}_{t}\bigg{]},$

where we also used the representation (12) of $\hat{\alpha}^{i,N}$ and the
fact that $\Lambda$ is Lipschitz–continuous. Denoting

$\delta(\varepsilon_{1},\eta_{1}):=2L_{f}+2L_{\Lambda}L_{f}+3\varepsilon_{1}+3\eta_{1}+\frac{4L_{\Lambda}^{2}L_{f}^{2}}{\varepsilon_{1}}$

summing up on both sides and applying Gronwall inequality, we obtain

$\displaystyle\sum_{i=1}^{N}|Y^{i,i}_{t}-\mathcal{Y}^{i,N}_{t}|^{2}$
$\displaystyle\leq
e^{\delta(\varepsilon_{1},\eta_{1})(T-t)}\bigg{\\{}6L_{f}^{2}+\Big{(}\frac{L_{f}(1+L_{\Lambda})^{2}}{\varepsilon_{1}}+\frac{L_{\Lambda}(L_{f}+\sqrt{1+8L_{\Lambda}^{2}})^{2}}{\varepsilon_{1}}\Big{)}T\bigg{\\}}\Big{\|}\sum_{i=1}^{N}|X^{i,\underline{\hat{\alpha}}}-\mathcal{X}^{i,N}|^{2}\Big{\|}_{\mathcal{S}^{\infty}}$
$\displaystyle\quad+3e^{\delta(\varepsilon_{1},\eta_{1})(T-t)}\bigg{\\{}\sum_{i=1}^{N}\|\gamma^{i,N}\|_{L^{\infty}}^{2}+\frac{1}{\eta_{1}}\sum_{i=1}^{N}\|\varepsilon^{i,N}\|_{\mathcal{S}^{\infty}}^{2}+3\frac{L_{\Lambda}^{2}L_{f}^{2}}{\eta_{1}}\sum_{i=1}^{N}\|\zeta^{i,N}\|_{\mathcal{S}^{\infty}}^{2}\bigg{\\}}.$

If $K_{b}$ is large enough, then $\delta(\varepsilon,\eta)\leq 0$. In this
case, combining this with (26), it follows that we can choose $\varepsilon$
small enough that

(27) $2\varepsilon
Te^{2\delta(\varepsilon,\eta)T+\delta(\varepsilon_{1},\eta_{1})T}\bigg{\\{}6L_{f}^{2}+\Big{(}\frac{L_{f}(1+L_{\Lambda})^{2}}{\varepsilon_{1}}+\frac{L_{\Lambda}(L_{f}+\sqrt{1+8L_{\Lambda}^{2}})^{2}}{\varepsilon_{1}}\Big{)}T\bigg{\\}}<1.$

Therefore, we have

$\Big{\|}\sum_{i=1}^{N}|Y^{i,i}-\mathcal{Y}^{i,N}|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}\leq
C\sum_{i=1}^{N}\Big{(}\|\gamma^{i,N}\|_{L^{\infty}}+\|\varepsilon^{i,N}\|_{\mathcal{S}^{\infty}}+\|\zeta^{i,N}\|_{\mathcal{S}^{\infty}}\Big{)}$

for a constant $C>0$. Since the bound of
$\gamma^{i,N},\varepsilon^{i,N},\zeta^{i,N}$ is $O(N^{-1})$, we obtain (25) in
view of (26).

When the monotonicity condition 11 is not assumed, we can use the same
argument (with $K_{b}=0$) and in this case we need $T$ small enough to get
(27). ∎

######  Remark 6.

Observe that in 11, the condition (11) is needed only to guarantee existence
for the FBSDE (18) for arbitrary time horizons. When these FBSDEs are known to
have solutions only the monotonicity condition (10) on $b$ is needed for the
LDP.

###  3.2. Examples: A model of systemic risk

For illustration, consider (as slight modification of) the problem of inter-
bank borrowing and lending studied by Carmona et al. [14]. In the systemic
risk model proposed by these authors, bank i’s reserve is given by the
dynamics

$dX^{i}_{t}=a\Big{(}\frac{1}{N}\sum_{j=1}^{N}X^{j}_{t}-X^{i}_{t}\Big{)}+\alpha^{i}_{t}\,dt+\sigma\,dW^{i}_{t}$

for a given mean-reverting parameter $a\geq 0$. Each bank controls its
borrowing and lending rate $\alpha^{i}$ at time $t$ by choosing it so as to
minimize the cost function

$J^{i}(\underline{\alpha})=\mathbb{E}\Big{[}\int_{0}^{T}f(X_{t}^{i},L^{N}(\underline{X}_{t}),\alpha^{i}_{t})\,dt+g(X^{i}_{T})\Big{]}$

with

$f(x,\mu,\alpha):=\frac{1}{2}\alpha^{2}-q\alpha\Big{(}\int_{\mathbb{R}}z\mu(dz)-x\Big{)}+\frac{\varepsilon}{2}\Big{|}\int_{\mathbb{R}}z\mu(dz)-x\Big{|}\quad
g(x)=\frac{c}{2}\Big{|}\int_{\mathbb{R}}z\mu(dz)-x\Big{|}$

where $q,c$ and $\varepsilon$ are strictly positive parameters such that
$q^{2}\leq\varepsilon$. With these specifications of the coefficients of the
game, the conditions (A1)-(A5) are clearly satisfied. To check (A7), notice
that the function $\Lambda$ therein now takes the form

$\Lambda(x,\mu,y)=q\Big{(}\int_{\mathbb{R}}z\mu(dz)-x\Big{)}-y.$

In particular, the functions $B,F$ and $G$ in (A7) are linear and time-
independent. Thus, it follows from [16, Proposition 5.2] and [11, Lemma 5.6]
that the PDE (6) admits a solution $V$ which is Lipschitz continuous in its
second variable. Therefore, if for each $N$ the finite player game admits a
Nash equilibrium, (see e.g. [14, Section 3.1] for details on the existence)
$\underline{\hat{\alpha}}=(\hat{\alpha}^{1,N},\dots\hat{\alpha}^{N,N})$ then
it follows by Theorem 2 that if $T$ is small enough, then the sequence
$(L^{N}(\underline{X}^{\underline{\hat{\alpha}}}))$ satisfies the LDP with
rate function given by (32), wherein

$\overline{B}(t,x,\mu)=(a+q)\Big{(}\int_{\mathbb{R}}z\mu(dz)-x\Big{)}-V(t,x,\mu).$

For LDP with $T$ arbitrary, one needed additional monotonicity properties
which can be guaranteed under conditions on the constants $\varepsilon,q,c$
and $a$. Note that the large deviation principle for this example was obtained
in [18] for the case of closed-loop controls.

##  4\. Large deviation principle for cooperative games: a case study

As explained in the introduction, the method developed in this article also
applies to cooperative large population games. Such games are important in
several applications, especially when a ”societal goal” should be achieved. We
refer the reader for instance to [10] and [26]. In the general setting, and
keeping the notation of the previous section, a continuous time differential
cooperative game takes the form

(28)
$V^{N}:=\inf_{\underline{\alpha}\in\mathcal{A}^{N}}\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\Big{[}\int_{0}^{T}f(t,X^{i,\underline{\alpha}}_{t},{\alpha^{i}_{t}},L^{N}(\underline{X}^{\underline{\alpha}}_{t},\underline{\alpha}_{t}))\,dt+g(X^{i,\underline{\alpha}}_{T},L^{N}(X^{\underline{\alpha}}_{T}))\Big{]},$

where $\mathcal{A}^{N}$ is the $N$-fold cartesian product of $\mathcal{A}$. In
[34, Theorem 25], conditions on the parameters $b,f$ and $g$ are given to
guarantee that, if the problem (28) admits a solution
$\hat{\alpha}^{N}=(\hat{\alpha}^{1,N},\dots,\hat{\alpha}^{N,N})$, then for
each $i$, and each $t$, the sequence $(\hat{\alpha}^{i,N}_{t})$ converges to
$\hat{\alpha}_{t}$ in $\mathbb{L}^{2}$, where $\hat{\alpha}$ solves the
McKean-Vlasov control problem

(29)
$\begin{cases}\inf_{\alpha\in\mathcal{A}}E\Big{[}\int_{0}^{T}f(t,X^{\alpha}_{t},\alpha_{t},\mathcal{L}(X^{\alpha}_{t},\alpha_{t}))\,dt+g(X^{\alpha}_{T},\mathcal{L}(X^{\alpha}_{T}))\Big{]}\\\
dX^{\alpha}_{t}=b(t,X^{\alpha}_{t},\alpha_{t},\mathcal{L}(X^{\alpha}_{t},\alpha_{t}))\,dt+\sigma\,dW_{t},\quad
X^{\alpha}_{0}=x.\end{cases}$

Since the derivation of the Laplace principle for this game is similar to the
non-cooperative case, in order to avoid repetitions we will focus here on the
linear quadratic case. Thus, let us specify the coefficients as follows:

(30)
$\begin{cases}f(t,x,\alpha,\xi)&=Q|x|^{2}+\bar{Q}|\overline{x}|^{2}+R|\alpha|^{2}+\bar{R}|\overline{\alpha}|^{2}+\bar{S}x\overline{\alpha},\\\
b(t,x,\alpha,\xi)&=Ax+\bar{A}\overline{x}+B\alpha+\bar{B}\overline{\alpha},\\\
g(x,\mu)&=Q_{T}x^{2}+\bar{Q}_{T}\overline{x}^{2}\end{cases}$

for some real numbers
$A,\bar{A},B,\bar{B},R,\bar{R},Q,\bar{Q},Q_{T},\bar{Q}_{T},R$ and $\bar{R}$,
and where $\bar{x}$ and $\bar{\alpha}$ are the mean of the first and second
marginals of $\xi$, respectively. We assume that $x$ and $\alpha$ have the
same dimension denoted $m$. To state the LDP in this case, let us introduce
the following notation: We consider the functions

$\ell(y,\bar{x},\bar{y}):=-\frac{1}{2R}\left[By+\bar{S}\left(1-\frac{\bar{R}}{R+\bar{R}}\right)\bar{x}+\left(\bar{B}-\frac{\bar{R}}{R+\bar{R}}(B+\bar{B})\right)\bar{y}\right]$

and

$\displaystyle\ell_{1}(x,y,\xi)$
$\displaystyle:=Ax+\bar{A}\mathbb{E}^{\xi_{1}}[X]+B\ell\Big{(}y,\mathbb{E}^{\xi_{1}}[X],\mathbb{E}^{\xi_{2}}[Y]\Big{)}+\bar{B}\mathbb{E}^{\xi_{2}}\Big{[}\ell\Big{(}Y,\mathbb{E}^{\xi_{1}}[X],\mathbb{E}^{\xi_{2}}[Y]\Big{)}\Big{]}$
$\displaystyle=Ax+\Big{[}\bar{A}-\frac{1}{2R}(SB+\bar{S}\bar{B})\Big{(}1-\frac{\bar{R}}{R+\bar{R}}\Big{)}\Big{]}\mathbb{E}^{\xi_{1}}[X]$
$\displaystyle\quad-\frac{B^{2}}{2R}y-\frac{1}{2R}\Big{[}\bar{B}B+(B+\bar{B})\Big{(}\bar{B}-\frac{\bar{R}}{R+\bar{R}}(B+\bar{B})\Big{)}\Big{]}\mathbb{E}^{\xi_{2}}[Y].$

where $\xi_{i}$ is the $i^{th}$ marginal of $\xi$, and
$\mathbb{E}^{\xi_{1}}[X]$ and $\mathbb{E}^{\xi_{2}}[Y]$ are the means of
$\xi_{1}$ and $\xi_{2}$, respectively. We also consider

$\ell_{2}(x,y,\xi)=2Qx+\Big{[}2\bar{Q}-\frac{\bar{S}^{2}}{2(R+\bar{R})}\Big{]}\mathbb{E}^{\xi_{1}}[X]+Ay+\Big{[}\bar{A}-\frac{\bar{S}(B+\bar{B})}{2(R+\bar{R})}\Big{]}\mathbb{E}^{\xi_{2}}[Y].$

With these notation, consider the PDE

(31)
$\begin{cases}\partial_{t}V(t,x,\mu)+\partial_{x}V(t,x,\mu)\ell_{1}(x,V(t,x,\mu),\xi)+\ell_{2}(x,V(t,x,\mu),\xi)\\\
\quad+\int_{\mathbb{R}^{m}}\partial_{\mu}V(t,x,\mu)\cdot\ell_{1}(y,V(t,x,\mu),\xi)d\mu(y)\\\
+\frac{1}{2}tr\Big{[}\partial_{xx}V(t,x,\mu)\sigma(x)\sigma^{\prime}(x)+\int_{\mathbb{R}^{m}}\partial_{y}\partial_{\mu}V(t,x,\mu)(y)\sigma(x)\sigma^{\prime}(x)\,d\mu(y)\Big{]}=0\\\
\quad V(T,x,\mu)=2Q_{T}x+2\bar{Q}_{T}\int_{\mathbb{R}^{m}}x\mu(dx)\end{cases}$

with $\xi=\text{law}(\chi,V(t,\chi,\mu))$ when $\chi\sim\mu$. The main result
of this section is the following:

######  Theorem 7.

Assume that $R\neq 0$, $R+\bar{R}\neq 0$. If the $N$-player problem (28)
admits an optimal control $(\hat{\alpha}^{i,N})_{i=1,\dots,N}$, then, there is
$\delta>0$ such that if $T\leq\delta$, the sequence
$(L^{N}(\underline{X}^{\hat{}\underline{\alpha}}))$ satisfies the LDP with
rate function

(32)
$\mathcal{J}(\theta):=\inf_{u\in\mathcal{U}:\,\mathrm{law}(X^{u})=\theta}\mathbb{E}\bigg{[}\frac{1}{2}\int_{0}^{T}|u_{t}|^{2}\,dt\bigg{]},$

where
$dX^{u}_{t}=\overline{\ell}_{1}(t,X^{u}_{t},\mathcal{L}(X^{u}_{t}))+\sigma
u_{t}\,dt+\sigma\,dW_{t}$ and $\overline{\ell}_{1}$ is the function defined by

(33)
$\overline{\ell}_{1}(t,x,\xi):=\ell_{1}\Big{(}x,V(t,x,\xi_{1}),\mathcal{L}(\chi,V(t,\chi,\xi_{1}))\Big{)}\quad\text{with}\quad\chi\sim\xi_{1}\
$

and $\xi_{1}$ the first marginal of $\xi$.

###### Proof.

Let us put

$\overline{x}^{N}:=\frac{1}{N}\sum_{j=1}^{N}x^{j},\quad\text{and}\quad\overline{\alpha}^{N}:=\frac{1}{N}\sum_{j=1}^{N}\alpha^{j}.$

It was shown in [34, Section 5.3.2] that, if there is an optimal control
$(\hat{\alpha}^{1,N},\dots,\hat{\alpha}^{N,N})$, then it satisfies

(34) $\displaystyle\hat{\alpha}^{i,N}_{t}$
$\displaystyle=-\frac{1}{2R}\left[BY^{i}_{t}+\bar{S}\left(1-\frac{\bar{R}}{R+\bar{R}}\right)\overline{X_{t}}^{N}+\left(\bar{B}-\frac{\bar{R}}{R+\bar{R}}(B+\bar{B})\right)\overline{Y_{t}}^{N}\right]$
$\displaystyle=\ell(Y_{t}^{i},\overline{X}_{t}^{N},\overline{Y}_{t}^{N})$

for every $i=1,\dots,N$, where the processes $(X^{i,N},Y^{i,N},Z^{i,j,N})$
satisfy the FBSDE system

$\displaystyle dX^{i}_{t}$
$\displaystyle=\left[AX^{i}_{t}+\bar{A}\overline{X_{t}}^{N}+B\hat{\alpha}^{i}_{t}+\bar{B}\overline{\hat{\alpha}_{t}}^{N}\right]\,dt+\sigma\,dW^{i}_{t}$
$\displaystyle dY^{i}_{t}$
$\displaystyle=-\left\\{2QX^{i}_{t}+\left[2\bar{Q}-\frac{\bar{S}^{2}}{2(R+\bar{R})}\right]\overline{X_{t}}^{N}+AY^{i}_{t}+\left[\bar{A}-\frac{\bar{S}(B+\bar{B})}{2(R+\bar{R})}\right]\overline{Y_{t}}^{N}\right\\}dt+\sum_{k=1}^{N}Z^{i,k}_{t}dW^{k}_{t}$
$\displaystyle X^{i}_{0}=x,\quad
Y^{i}_{T}=2Q_{T}X_{T}^{i}+2\bar{Q}_{T}\overline{X}^{N}_{T}.$

On the other hand, the optimal control $\hat{\alpha}$ of the McKean–Vlasov
control problem satisfies

$\hat{\alpha}_{t}=-\frac{1}{2R}\left[BY_{t}+\bar{S}\left(1-\frac{\bar{R}}{R+\bar{R}}\right)\mathbb{E}[X_{t}]+\left(\bar{B}-\frac{\bar{R}}{R+\bar{R}}(B+\bar{B})\right)\mathbb{E}[Y_{t}]\right]$

where $(X,Y,Z)$ solves the McKean-Vlasov FBSDE

$\displaystyle dX_{t}$
$\displaystyle=\Big{\\{}AX_{t}+\bar{A}\mathbb{E}[X_{t}]+B\hat{\alpha}_{t}+\bar{B}\mathbb{E}[\hat{\alpha}_{t}]\Big{\\}}\,dt+\sigma\,dW_{t}$
(35) $\displaystyle dY_{t}$
$\displaystyle=-\left\\{2QX_{t}+\Big{[}2\bar{Q}-\frac{\bar{S}^{2}}{2(R+\bar{R})}\Big{]}\mathbb{E}[X_{t}]+AY_{t}+\Big{[}\bar{A}-\frac{\bar{S}(B+\bar{B})}{2(R+\bar{R})}\Big{]}\mathbb{E}[Y_{t}]\right\\}dt+Z_{t}dW_{t}$
$\displaystyle X_{0}=x,\quad Y_{T}=2Q_{T}X_{T}+2\bar{Q}_{T}\mathbb{E}[X_{T}],$

see [34]. Essentially, this is a fully coupled McKean-Vlasov FBSDE system with
linear coefficients. The existence of a unique solution of this equation is
guaranteed e.g. by [12, Theorem 4.2] if $T$ is small enough. Furthermore, it
follows by [16, Theorem 2.7, Proposition 5.2] that if $T$ is sufficiently
small, then the solution $V$ of Equation 31 exists, is Lipschitz–continuous in
$(x,\mu)$ and it satisfies $V(t,\chi,\mu)=Y_{t}^{t,\chi,\mu}$, which is the
solution of (35) such that $X_{t}=\chi$ and $\mathcal{L}(\chi)=\mu$.
Therefore, the process $X$ satisfies the (decoupled) equation

$\displaystyle dX_{t}$
$\displaystyle=\ell_{1}\big{(}X_{t},V(t,X_{t},\mathcal{L}(X_{t})),\mathcal{L}(X_{t},V(t,X_{t},\mathcal{L}(X_{t})))\big{)}\,dt+\sigma\,dW_{t}$
$\displaystyle=\overline{\ell}_{1}(t,X_{t},\mathcal{L}(X_{t}))+\sigma\,dW_{t}.$

Now, consider the interacting particle system

$d\mathcal{X}^{i,N}_{t}=\overline{\ell}_{1}\big{(}\mathcal{X}_{t}^{i,N},L^{N}(\underline{\mathcal{X}}_{t}))\big{)}+\sigma\,dW_{t}^{i}$

which is well defined, since $\overline{\ell}_{1}$ is Lipschitz–continuous, as
the composition of two Lipschitz–continuous functions. It follows by [6,
Theorem 3.1] that the sequence $(L^{N}(\underline{\mathcal{X}}))_{N}$
satisfies the LDP with rate function $\mathcal{J}$. It is checked exactly as
in the proof of Theorem 2 that $\mathcal{J}$ has compact sublevel sets. Thus,
it remains to show that $X^{i,N}$ and $\mathcal{X}^{i,N}$ are exponentially
close. It follows from a similar argument as in Lemma 5 that

(36)
$\sup_{N}\Big{\|}\sum_{i=1}^{N}|X^{i,N}-\mathcal{X}^{i,N}|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}+\sup_{N}\Big{\|}\sum_{i=1}^{N}|Y^{i,i}-V(\cdot,\mathcal{X}^{i,N},L^{N}(\underline{\mathcal{X}}))|\Big{\|}_{\mathcal{S}^{\infty}(\mathbb{R}^{\ell})}<\infty.$

Subsequently using Chebyshev’s inequality and (36), we have

$\displaystyle\mathbb{P}\Big{(}\sup_{t\in[0,T]}\mathcal{W}_{2}(L^{N}(\underline{X}_{t}),L^{N}(\underline{\mathcal{X}}_{t}))>\varepsilon\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\frac{1}{N}\sup_{t\in[0,T]}\sum_{i=1}^{N}|X^{i,N}_{t}-\mathcal{X}^{i}_{t}|^{2}\Big{\\}}^{1/2}\geq\varepsilon\Big{)}$
$\displaystyle\leq\mathbb{E}\Big{[}\exp\Big{\\{}\Big{(}\sum_{i=1}^{N}\|X^{\underline{\alpha},i}-\mathcal{X}^{i}\|^{2}_{\infty}\Big{)}\Big{\\}}\Big{]}e^{-\varepsilon^{2}N^{2}}$
$\displaystyle\leq Ce^{-\varepsilon^{2}N^{2}}$

from which we deduce that

$\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\Big{(}\sup_{t\in[0,T]}\mathcal{W}_{1}(L^{N}(\underline{X}_{t}),L^{N}(\underline{\mathcal{X}}_{t}))\geq\varepsilon\Big{)}=-\infty.$

Therefore, it follows by [19, Theorem 4.2.13] that the sequence
$(L^{N}(\underline{X}))$ satisfies the LDP with rate function $\mathcal{J}$. ∎

We conclude the paper with a remark about the small time assumption.

######  Remark 8.

Observe that the smallness assumption made on $T$ in Theorem 7 is needed only
to guarantee existence of a classical solution to the PDE (31). By [16,
Proposition 5.2], this PDE admits a (Lipschitz-continuous) solution on $[0,T]$
for all $T>0$ provided that the FBSDE solution satisfies

(37)
$\mathbb{E}\big{[}|Y^{t,\chi,\mathcal{L}(\chi)}_{t}-Y_{t}^{t,\chi^{\prime},\mathcal{L}(\chi^{\prime})}|^{2}\big{]}^{1/2}\leq
C\mathbb{E}[|\chi-\chi^{\prime}|^{2}]^{1/2}$

for every $\chi,\chi^{\prime}\in L^{2}(\Omega,\mathcal{F}_{t},\mathbb{P})$,
and for some constant $C>0$ that does not depend on $t$, $\chi$ and
$\chi^{\prime}$. The latter property has been established for instance in [11,
Lemma 5.6], [4] or [39, Corollary 2.4], the latter references assuming
monotonicity properties such as those of condition 11.

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Peng Luo: School of Mathematical Sciences, Shanghai Jiao Tong University,
Shanghai 200240, China<EMAIL_ADDRESS>
Ludovic Tangpi: Department of Operations Research and Financial Engineering,
Princeton University, Princeton, 08540, NJ; USA<EMAIL_ADDRESS>
Financial suupport by NSF grant DMS-2005832 is gratefully acknowledged.