∎

11institutetext: A. Bensoussan 22institutetext: International Center for
Decision and Risk Analysis,Jindal School of Management, The University of
Texas at Dallas
Department of Systems Engineering and Engineering Management, College of
Science and Engineering, City University of Hong Kong
22email<EMAIL_ADDRESS>33institutetext: M. H. M. Chau, S. C. P.
Yam44institutetext: Department of Statistics, The Chinese University of Hong
Kong
44email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Mean Field Games with a Dominating Player

A. Bensoussan  M. H. M. Chau  S. C. P. Yam

(Received: date / Accepted: date)

###### Abstract

In this article, we consider mean field games between a dominating player and
a group of representative agents, each of which acts similarly and also
interacts with each other through a mean field term being substantially
influenced by the dominating player. We first provide the general theory and
discuss the necessary condition for the optimal controls and equilibrium
condition by adopting adjoint equation approach. We then present a special
case in the context of linear-quadratic framework, in which a necessary and
sufficient condition can be asserted by stochastic maximum principle; we
finally establish the sufficient condition that guarantees the unique
existence of the equilibrium control. The proof of the convergence result of
finite player game to mean field counterpart is provided in Appendix.

Keywords: Mean field games; Dominating player; Wasserstein Metric; Adjoint
equation approach/Stochastic maximum principle; Stochastic Hamilton-Jacobi-
Bellman equations; Linear quadratic; Separation principle; Banach fixed point
theorem.

##  1 Introduction

For long, modeling the joint interactive behaviour of individual
objects(agents) in a large population in various dynamic systems has been one
of the major problems. For instance, physicists often apply the traditional
variational methods in Lagrangian and/or Hamiltonian mechanics to study
interacting particle systems, which left a shortcoming of extremely high
computational cost that made this microscopic approach almost mathematically
intractable. To resolve this matter, a completely different macroscopic
approach from statistical physics had been gradually developed, which
eventually leads to the primitive notion of mean field theory. The novelty of
this approach is that particles interact through a medium, namely the mean
field term, which aggregates by action and reaction on other particles.
Moreover, by passing the number of particles to the infinity in these
macroscopic models, the mean field term become a functional of the density
function which represents the whole population of particles. This leads to
mathematical problems of much less computational complexity.

From the economic perspective, due to the dramatic population growth and rapid
urbanization, urgent needs of in-depth understanding of collective strategic
interactive behavior of a huge group of decision makers is crucial in order to
maintain a sustainable economic growth. Since the vector of fair prices is
determined by both demand and supply, it is natural to utilize the aggregation
effect from the players’ states as a canonical candidate of mean-field term,
and then we employ the mean-field models in place of the corresponding
classical equilibrium models; moreover, as the decision makers control the
evolution of a dynamic system, it is necessary to also incorporate the theory
of stochastic differential games (SDGs) in these mean-field models. Over the
past few decades, the theory of SDGs has been a major research topic in
control theory and financial economics, especially in studying the continuous-
time decision making problem between non-cooperative investors; in regard to
the one-dimensional setting the theory of two person zero-sum games is quite
well-developed via the notion of viscosity solutions, see for example Fleming
and Souganidis (1989). Unfortunately, most interesting SDGs are $N$-player
non-zero sum SDGs; see Bensoussan and Frehse BF1 ; BF2  and Bensoussan et al.
BFV , yet there are still relatively few results in the literature.

As a macroscopic equilibrium model, et al. HCM1 ; HCM2  investigated
stochastic differential game problems involving infinitely many players under
the name “Large Population Stochastic Dynamic Games”; and independently, Lasry
and Lions LL1 ; LL2 ; LL3  studied similar problems from the viewpoint of the
mean-field theory in physics and termed “Mean-Field Games (MFGs)”. As an
organic combination of mean field theory and theory of stochastic differential
games, MFGs provide a more realistic interpretation of individual dynamics at
the microscopic level, so that each player will be able to optimize his
prescribed objectives, yet with the mathematical tractability in a macroscopic
framework. To be more precise, the general theory of MFGs has been built by
combining various consistent assumptions on the following modeling aspects:
(1) a continuum of players; (2) homogeneity in strategic performance of
players; and (3) social interactions through the impact of mean field term.
The first aspect is describing the approximation of a game model with a huge
number of players by a continuum one yet with a sufficient mathematical
tractability. The second aspect is assuming that all players obey the same set
of rules of the interactive game, which provide guidance on their own behavior
that potentially leads them to optimal decisions. Finally, due to the
intrinsic complexity of the society in which the players participate in, the
third aspect is explaining the fact that each player is so negligible and can
only affect others marginally through his own infinitesimal contribution to
the society. In a MFG, each player will base his decision making purely on his
own criteria and certain summary statistics (that is, the mean field term)
about the community; in other words, in explanation of their interactions, the
pair of personal and mean-field characteristics of the whole population is
already sufficient and exhaustive. Mathematically, each MFG will possess the
following forward-backward structure: (1) a forward dynamic describes the
individual strategic behavior; (2) a backward equation describes the evolution
of individual optimal strategy, such as those in terms of the individual value
function via the usual backward recursive techniques. For the detail of the
derivation of this system of equations with forward-backward feature, one can
consult from the works of Huang et al. HCM2 , Lasry and Lions LL1 ; LL2 ; LL3
and Bensoussan et al. BFY .

In this article, we consider a class of MFG problems, in which there is a
‘significantly big’ player playing together with a huge group of ‘small’
players. The first work along this direction under a Linear Quadratic setting
has been investigated by Huang H0 . In their following work HN1 , the authors
regard the mean field term, represented by the conditional expectation of the
small agent, as exogenous to the whole control problem for both the big (the
authors called it,‘major’) and small (minor) players. Nourian and Caines NC
consider a similar problem under a generalized framework. However, the authors
also consider the mean field term, which is represented by a conditional
probability measure, as exogenous to the control problem for the major player.
In contrast, we here consider the mean field term as endogenous for the big
(we rephrase as ‘dominating’ in order to emphasize our distinction from the
previous works) player. That is to say, changes in the control of the big
(dominating) player would directly affect and even essentially determine the
mean field term. Our present setting appears to be natural in the economic
literature related to ‘actual’ governance, as the governor can often take up
the initiative or key role on setting up rubrics and regulations to be
followed by citizens. To avoid ambiguity, we here regard the ‘dominating’
major player as a “Dominating Player”, and all other minor players as
“Representative Agents” throughout the whole paper. In our work, we assume
that this dominating player can influence both the mean field term and
representative agents directly. We first discuss the necessary condition for
the optimality under the most general setting in which both the state
coefficients and the objective functions are sufficiently regular (e.g.
differentiable); we then consider the Linear-Quadratic case by applying the
results obtained in the general theory, which results in three adjoint
equations. It is noted that Huang et al. HN2  also considered the non-
stationary case and obtained the intermediary result with only two adjoint
equations, which represents a particular case of our present theory. Besides,
concerning the related fixed point issue in any standard MFG problem in order
to achieve the equilibrium strategy, we here only need to involve one single
affine map, that simplifies much than that in HN2 , in which the authors need
a couple of two similar mappings; apart from the simplicity of the sufficient
condition provided here, it is also directly expressed in terms of the data
(coefficients) of the underlying model.

The paper is organized as follows: In Section 2, we present the general theory
of the Mean Field Games in the presence of a dominating Player, in which both
the state coefficients and the objective functions are sufficiently regular.
The necessary condition for optimality and equilibrium is also provided there.
Firstly, solving for the control problem of the representative agent, and then
the equilibrium condition leads to a coupled Hamilton-Jacobi-Bellman and
Fokker Planck equations. As the mean field term is endogenous to the
dominating Player, in order to achieve an optimal control, he/she should take
into account of the coupled equations when deciding his own controlling
strategy. The related fixed point problem is described by six equations. In
Section 3, we study a special case with linear states together with linear
quadratic objective functions. Due to natural coerciveness of the problem
formulation, a necessary and sufficient condition for the optimality can be
guaranteed. We write down both the stochastic maximum principle and the
corresponding adjoint equations. In Section 4, the corresponding fixed point
problem is then tackled by considering the related Riccati equation, with
which the equilibrium could be achieved. We then provide a ‘practical’
sufficient condition, which only involves the data (coefficients) of the model
without referring to any specific solution of any Riccati equations, for the
existence of the equilibrium strategy. In Appendix, proof of the approximate
Nash equilibrium for the general setting is also provided.

##  2 General Theory

Consider a probability space $(\Omega,\mathcal{F},P)$, a fixed terminal time
$T$ and two independent standard Brownian motion $W_{0}(t)$ and $W_{1}(t)$
taking values in $\mathbb{R}^{d_{0}}$ and $\mathbb{R}^{d_{1}}$ respectively.
Also consider two independent initial square integrable random variables
$\xi_{0}\in\mathbb{R}^{n_{0}}$ and $\xi_{1}\in\mathbb{R}^{n_{1}}$, which are
also assumed to be independent of both $W_{0}(t)$ and $W_{1}(t)$. Define the
filtrations as follows, in which $\mathcal{F}_{t}^{0}$ and
$\mathcal{F}_{t}^{1}$ are clearly independent to each other,

$\begin{split}\mathcal{F}_{t}^{0}&:=\sigma(\xi_{0},W_{0}(s),s\leq t),\\\
\mathcal{F}_{t}^{1}&:=\sigma(\xi_{1},W_{1}(s),s\leq t),\\\
\mathcal{G}_{t}&:=\mathcal{F}_{t}^{0}\vee\mathcal{F}_{t}^{1}.\end{split}$

Let $\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$ be the space of probability measures
equipped with the $2^{nd}$ Wasserstein metric (for example, see villani ),
$W_{2}(\cdot,\cdot)$ such that for any $\mu$ and $\nu$ in
$\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$,

$W_{2}(\nu_{1},\nu_{2}):=\inf_{\gamma\in\Gamma(\nu_{1},\nu_{2})}\bigg{(}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}|x-y|^{2}d\gamma(x,y)\bigg{)}^{\frac{1}{2}},$

where the infimum is taken over the family $\Gamma(\nu_{1},\nu_{2})$, the
collection of all joint measures with respective marginals $\nu_{1}$ and
$\nu_{2}$. Denote $d\lambda$ to be the Lebesgue measure on
$\mathbb{R}^{n_{1}}$.

Denote $x_{0}(t)\in\mathbb{R}^{n_{0}}$ and $x_{1}(t)\in\mathbb{R}^{n_{1}}$ the
state evolutions for the dominating player and a representative agent
respectively whose dynamics are given by the following stochastic differential
equations (SDEs),

$\begin{split}\left\\{\begin{array}[]{rcl}dx_{0}&=&g_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(x_{0}(t),t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\
dx_{1}&=&g_{1}\Big{(}x_{1}(t),x_{0}(t),\mu(t),u_{1}(x_{1}t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}(t)\Big{)}dW_{1}(t),\\\
x_{1}(0)&=&\xi_{1}.\end{array}\right.\end{split}$ (1)

The functional coefficients are defined as follows:

$\left\\{\begin{array}[]{rl}&g_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{0}}\rightarrow\mathbb{R}^{n_{0}},\\\
&g_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{1}}\rightarrow\mathbb{R}^{n_{1}},\\\
&\sigma_{0}:\mathbb{R}^{n_{0}}\rightarrow\mathbb{R}^{n_{0}\times d_{0}},\\\
&\sigma_{1}:\mathbb{R}^{n_{1}}\rightarrow\mathbb{R}^{n_{1}\times d_{1}}.\\\
\end{array}\right.$ (2)

The dominating player and the representative agents also possess the following
objective functionals respectively:

$\begin{array}[]{rcl}J_{0}(u_{0})&=&\mathbb{E}\left[\int_{0}^{T}f_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(t)\Big{)}dt+h_{0}\Big{(}x_{0}(T),\mu(T)\Big{)}\right],\\\
J_{1}(u_{1},x_{0},\nu)&=&\mathbb{E}\left[\int_{0}^{T}f_{1}\Big{(}x_{1}(t),x_{0}(t),\mu(t),u_{1}(t)\Big{)}dt+h_{1}\Big{(}x_{1}(T),x_{0}(T),\mu(T)\Big{)}\right].\end{array}$

The functions are defined as follows:

$\left\\{\begin{array}[]{rl}&f_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{0}}\rightarrow\mathbb{R},\\\
&f_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{1}}\rightarrow\mathbb{R},\\\
&h_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\rightarrow\mathbb{R},\\\
&h_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\rightarrow\mathbb{R}.\\\
\end{array}\right.$ (3)

Here $u_{0}\in\mathbb{R}^{m_{0}}$ and $u_{1}\in\mathbb{R}^{m_{1}}$ represent
the respective controls of the dominating player and the representative agent.
The controls $u_{0}$ and $u_{1}$ are respectively adapted to the filtrations
$\mathcal{F}_{t}^{0}$ and $\mathcal{G}_{t}$. We further assume that the
functional form (being a function of $(x_{1}(t),t)$) of $u_{1}$ is adapted to
$\mathcal{F}_{t}^{0}$ and uniformly Lipschitz in $x_{1}(t)$, even though its
value evaluated at $x_{1}(t)$ would be adapted to $\mathcal{G}_{t}$ instead.
Loosely speaking the dominating player takes his own privilege of setting up
the framework to be followed by the representative agent. We shall then define
the classes of admissible controls for the dominating player and the
representative agent by $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ respectively,
where $\mathcal{A}_{0}$ (resp. $\mathcal{A}_{1}$) is a subset of
$\mathcal{F}^{0}-$ (resp. $\mathcal{G}-$)progressively measurable process
which are in $\mathcal{L}^{2}(\Omega\times[0,T];\mathbb{R}^{m_{0}})$ (resp.
$\mathcal{L}^{2}(\Omega\times[0,T];\mathbb{R}^{m_{1}})$).

The mean field term, $\mu(t)\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, is the
probability measure of the state of the representative agent at time $t$.
Indeed, the dominating player sets rules for representative agent to take into
account. One natural consideration is that the dominating player is incapable
of tracing the state of each individual’s evolution, but only takes account of
the overall performance of the community subject to the rules he set, that is
his own flow of information, $\mathcal{F}^{0}_{t}$. By the same token, each
agent cannot fully keep track of any other agents’ states and they can only
rely on the summarized information of the community provided by the dominating
player. Thus it is justifiable to assume that the mean field term, $\mu(t)$,
is adapted to $\mathcal{F}^{0}_{t}$. The dominating player can directly
influence both the representative agent and the mean field term, thus we
consider $\mu(t)$ as endogenous in the consideration of optimal behavior of
$x_{0}(t)$ rather than as an exogenous variable commonly found in the
literature such as that of H0 ; NC .

For any probability measure $\mu\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, we
write $M_{2}(\mu)=(\int_{\mathbb{R}^{n_{1}}}|x|^{2}d\mu(x))^{\frac{1}{2}}$. We
first give the following assumptions on the functional coefficients:

(A.1) Lipschitz Continuity
$g_{0}$, $\sigma_{0}$, $g_{1}$ and $\sigma_{1}$ are globally Lipschitz
continuous in all arguments. In particular, there exists $K>0$, such that

$\begin{array}[]{rcl}|g_{0}(x_{0},\mu,u_{0})-g_{0}(x_{0}^{\prime},\mu^{\prime},u_{0}^{\prime})|&\leq&K\bigg{(}|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{0}-u_{0}^{\prime}|\bigg{)};\\\
|\sigma_{0}(x_{0})-\sigma_{0}(x_{0}^{\prime})|&\leq&K|x_{0}-x_{0}^{\prime}|.\\\
|g_{1}(x_{1},x_{0},\mu,u_{1})-g_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime},u_{1}^{\prime})|&\leq&K\bigg{(}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{1}-u_{1}^{\prime}|\bigg{)};\\\
|\sigma_{1}(x_{1})-\sigma_{1}(x_{1}^{\prime})|&\leq&K|x_{1}-x_{1}^{\prime}|.\\\
\end{array}$

(A.2) Linear Growth
$g_{0}$, $\sigma_{0}$, $g_{1}$ and $\sigma_{1}$ are of linear growth in all
arguments. In particular, there exists $K>0$, such that

$\begin{split}|g_{0}(x_{0},\mu,u_{0})|&\leq
K(1+|x_{0}|+M_{2}(\mu)+|u_{0}|);\\\ |\sigma_{0}(x_{0})|&\leq K(1+|x_{0}|).\\\
|g_{1}(x_{1},x_{0},\mu,u_{1})|&\leq
K(1+|x_{0}|+|x_{1}|+M_{2}(\mu)+|u_{1}|);\\\ |\sigma_{1}(x_{1})|&\leq
K(1+|x_{1}|).\\\ \end{split}$

(A.3) Quadratic Condition on the Cost Functional (See (A.5) in Carmona and
Delarue CAR_PROB .)
There exists $K>0$, such that

$\begin{split}|f_{1}(x_{1},x_{0},\mu,u_{1})-f_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime},u_{1}^{\prime})|\leq&K\Big{[}1+|x_{1}|+|x_{1}^{\prime}|+|x_{0}|+|x_{0}^{\prime}|+|M_{2}(\mu)|+|M_{2}(\mu^{\prime})|+|u_{1}|+|u_{1}^{\prime}|\Big{]}\\\
&\qquad\cdot\Big{[}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{1}-u_{1}^{\prime}|\Big{]};\\\
|h_{1}(x_{1},x_{0},\mu)-h_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime})|\leq&K\Big{[}1+|x_{1}|+|x_{1}^{\prime}|+|x_{0}|+|x_{0}^{\prime}|+|M_{2}(\mu)|+|M_{2}(\mu^{\prime})|\Big{]}\\\
&\qquad\cdot\Big{[}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})\Big{]}.\end{split}$
(4)

Under the assumptions A.1-A.3, we show in the Appendix that if we have the
mean field term coincides with the probability measure of $x_{1}(t)$
conditioning on $\mathcal{F}^{0}_{t}$, then the optimization problem for the
representative agent constitutes to a Mean Field Game. In general, it is more
convenient to compare two probability measures if they possess density
functions on $\mathbb{R}^{n_{1}}$. We define the second order operator $A_{1}$
and its adjoint $A_{1}^{*}$ by

$\begin{split}A_{1}\varphi(x,t)&=-\text{tr}\Big{(}a_{1}(x)D^{2}\varphi(x,t)\Big{)},\\\
A_{1}^{*}\varphi(x,t)&=-\sum_{i,j=1}^{n_{1}}\frac{\partial^{2}}{\partial{x_{i}}\partial{x_{j}}}\Big{(}a_{1}^{ij}(x)\varphi(x,t)\Big{)},\end{split}$

where $a_{1}(x)=\frac{1}{2}\sigma_{1}(x)\sigma_{1}(x)^{*}$ is a positive
definite matrix. Let $x_{1}=x_{1}^{u_{1}}$ be the solution of the SDE for the
representative agent with respect to control $u_{1}$. For any test function
$f$, by Itô’s lemma,

$\mathbb{E}^{\mathcal{F}^{0}_{t}}[f(x_{1}^{u_{1}}(t))]=\mathbb{E}^{\mathcal{F}^{0}_{t}}\bigg{[}f(\xi_{1})+\displaystyle\int_{0}^{t}\big{(}\partial_{t}+g\cdot
D+A_{1}\big{)}fds\bigg{]},$ (5)

The conditional density function $p^{u_{1}}(\cdot,t)$ of $x_{1}^{u_{1}}(t)$
(if exists), i.e.
$\mathbb{E}^{\mathcal{F}^{0}_{t}}[f(x_{1}^{u_{1}}(t))]=\int_{\mathbb{R}^{n_{1}}}f(x)p^{u_{1}}(x,t)dx$,
would be given by the Fokker-Planck (FP) equation

$\begin{split}\left\\{\begin{array}[]{rl}\dfrac{\partial p^{u_{1}}}{\partial
t}&=-A_{1}^{*}p^{u_{1}}(x,t)-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),u_{1}(x,t)\Big{)}p^{u_{1}}(x,t)\Big{)},\\\
p^{u_{1}}(x,0)&=\omega(x),\end{array}\right.\end{split}$ (6)

where $\omega(x)$ is the initial density function of $\xi_{1}$. We will
justify the existence and regularities of the conditional density function
$p^{u_{1}}$ later. We first assume
$p^{u_{1}}(\cdot,t)\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$ and
$p^{u_{1}}(\cdot,t)d\lambda\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$. For any
density $m$, we may write $md\lambda=m$ if no ambiguity arises. We impose
further assumptions on the functional coefficients:

(A.4)
$g_{0},f_{0},h_{0},g_{1},f_{1}$ and $h_{1}$ are continuously differentiable in
(if the argument exist) $x_{0}\in\mathbb{R}^{n_{0}}$,
$x_{1}\in\mathbb{R}^{n_{1}}$ $u_{0}\in\mathbb{R}^{m_{0}}$,
$u_{1}\in\mathbb{R}^{m_{1}}$ with bounded derivatives. We will denote, for
example, the derivative of $g_{0}$ with respect to $x_{0}$ by $g_{0,x_{0}}$.
They are also Gâteaux differentiable in
$\mu=md\lambda\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, for example, for
$m\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$,

$\and{\dfrac{d}{d\theta}}{\theta=0}g_{0}(x_{i},(m+\theta\tilde{m})d\lambda,u_{i})=\int_{\mathbb{R}^{n}}\frac{\partial
g_{0}}{\partial m}(x_{i},md\lambda,u_{i})(\xi)\tilde{m}(\xi)d\xi,$

for some $\frac{\partial g_{0}}{\partial
m}(x_{i},md\lambda,u_{i})\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$.

(A.5)
$\sigma_{0}$ (resp. $\sigma_{1}$) is twice continuously differentiable in
$x_{0}\in\mathbb{R}^{n_{0}}$ (resp. $x_{1}\in\mathbb{R}^{n_{1}}$) with bounded
first order and second order derivative.

###### Remark

With the regularities on the coefficients, if we have the initial density
$\omega(x)\in\mathcal{L}^{2}\cap\mathcal{L}^{\infty}(\mathbb{R}^{n_{1}})$,
then the FP equation (6) admits a unique solution
$p^{u_{1}}\in\mathcal{L}^{\infty}([0,T],\mathcal{L}^{2}\cap\mathcal{L}^{\infty}(\mathbb{R}^{n_{1}}))$.
See Proposition $4$ and $5$ in Le Bris and Lions BL  for details.

Define then a pair of mutually dependent control problems for the dominating
player and the representative agent as below:

###### Problem 2.1

Control of Representative Agent
Given the process $x_{0}$ and an exogenous probability measure-valued process
$\nu$ (adapted to $\mathcal{F}^{0}_{t}$), find a control
$u_{1}\in\mathcal{A}_{1}$ which minimizes the cost functional

$J_{1}(u_{1},x_{0},\nu):=\mathbb{E}\left[\int_{0}^{T}f_{1}\Big{(}x_{1}(t),x_{0}(t),\nu(t),u_{1}(t)\Big{)}dt+h_{1}\Big{(}x_{1}(T),x_{0}(T),\nu(T)\Big{)}\right].$
(7)

###### Problem 2.2

Equilibrium Condition
Given an exogenous probability measure-valued process $\nu$, let
$\mathcal{M}(\nu)(t)$ be the measure induced by the corresponding optimal
$x_{1}(t)$ found in Problem 2.1 conditioning on $\mathcal{F}_{t}^{0}$. Find
the probability measure-valued process $\mu$ such that the fixed point
property is satisfied: $\mathcal{M}(\mu)(\cdot)=\mu(\cdot)$.

###### Problem 2.3

Control of the Dominating Player
Find a control $u_{0}\in\mathcal{A}_{0}$ which minimizes the cost functional

$J_{0}(u_{0}):=\mathbb{E}\left[\int_{0}^{T}f_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(t)\Big{)}dt+h_{0}\Big{(}x_{0}(T),\mu(T)\Big{)}\right],$
(8)

where $\mu$ is the solution given in Problem 2.2.

###### Remark

The setting of our problem is different from those mean field related problems
with a major player (not a dominating player) commonly found in the
literature, such as that in H0 ; NC . For example, in H0 , the corresponding
objective functions for the major player and i-th minor player are
respectively

$\begin{array}[]{rcl}J_{0}(u_{0},z)&=&\mathbb{E}\int_{0}^{T}\left\\{|x_{0}-H_{0}z-\eta|^{2}_{Q_{0}}+u_{0}^{*}R_{0}u_{0}\right\\}dt,\\\
J_{i}(u_{i},z)&=&\mathbb{E}\int_{0}^{T}\left\\{|x_{i}-Hx_{0}-\hat{H}_{0}z-\eta|^{2}_{Q}+u^{*}Ru\right\\}dt,\\\
\end{array}$

where the mean field term $z$ is exogenous to both control optimization
problems for $J_{0}$ and $J_{i}$. Instead, we here consider the mean field
term $\nu$, as established in Problem 2.2, as endogenous for the dominating
player in Problem 2.3. In particular, changes in control $u_{0}$ would affect
and even completely determine the mean field term $\nu$ accordingly. Our
setting appears to be natural in the economic context related to governance,
as the governor can sometimes take the initiative to set-up rubrics to be
obeyed and followed by citizens; this latter notion is covered in ORIGINCRISIS
.

We first establish the necessary condition of optimality for the
representative agent (Problem 2.1) by the adjoint equation approach. After
resolving Problem 2.1, we solve for the fixed point in Problem 2.2. Recall
that $x_{0}(t)$, the functional form of $u_{1}$ (conditioning on
$\mathcal{F}_{0}^{t}$, $u_{1}$ is a function of $(x_{1}(t),t)$) and now
together with the input measure-valued process $\nu$ are all adapted to
$\mathcal{F}_{t}^{0}$. We can then rewrite the cost functional (7) for the
representative agent as

$\begin{split}J_{1}(u_{1},x_{0},\nu)=&\mathbb{E}\bigg{[}\int_{0}^{T}\mathbb{E}^{\mathcal{F}_{t}^{0}}f_{1}\Big{(}x_{1}(t),x_{0}(t),\nu(t),u_{1}(x_{1}(t),t)\Big{)}dt+\mathbb{E}^{\mathcal{F}_{T}^{0}}h_{1}\Big{(}x_{1}(T),x_{0}(T),\nu(T)\Big{)}\bigg{]}\\\
=&\mathbb{E}\bigg{[}\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}p_{u_{1}}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt+\int_{\mathbb{R}^{n}}p_{u_{1}}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}.\end{split}$

###### Lemma 2.4

(Necessary condition for Problem 2.1)
Given $x_{0}$ and $\nu$ as in Problem 2.1, the control
$\hat{u}_{1}\in\mathcal{A}_{1}$ is optimal only if it satisfies the following
SHJB:

$\left\\{\begin{array}[]{rl}-\partial_{t}\Psi=&\Big{(}H_{1}(x,x_{0}(t),\nu(t),D\Psi(x,t))-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)=&h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)},\\\
\end{array}\right.$ (9)

where

$\begin{split}H_{1}(x,x_{0},\nu,q)&=\inf_{u_{1}}L(x,x_{0},\nu,u_{1},q),\\\
L(x,x_{0},\nu,u_{1},q)&=f_{1}(x,x_{0},\nu,u_{1})+qg_{1}(x,x_{0},\nu,u_{1}).\\\
\end{split}$

and the infimum is uniquely attained at $\hat{u}_{1}$, i.e.
$H_{1}(x,x_{0},\nu,q)=L(x,x_{0},\nu,\hat{u}_{1},q)$.

###### Remark

As in the work in Carmona et al. CAR_PROB , one convenient set of assumptions
on $g_{1}$,$f_{1}$ and $h_{1}$ which ensures the unique existence of the
minimizer, $\hat{u}_{1}(x,x_{0},\nu,q)={\arg\min}_{u}L(x,x_{0},\nu,u,q)$, is
the affine and convexity assumption. See Lemma $2.1$ in CAR_PROB  for more
details. In particular, for
$x_{1}\in\mathbb{R}^{n_{1}};x_{0}\in\mathbb{R}^{n_{0}};\mu\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$
and $u_{1},u_{1}^{\prime}\in\mathbb{R}^{m_{1}}$, there exists $K>0$ such that

  1. 1.

$g_{1}(x_{1},x_{0},\mu,u_{1})=g_{1,1}(x_{1},x_{0},\mu)+g_{1,2}\cdot u_{1}$,

  2. 2.

$f_{1}(x_{1}^{\prime},x_{0},\mu,u_{1}^{\prime})\geq
f_{1}(x_{1},x_{0},\mu,u_{1})+f^{*}_{1,x_{1}}(x_{1},x_{0},\mu,u_{1})(x_{1}^{\prime}-x_{1})+f^{*}_{1,u_{1}}(x_{1},x_{0},\mu,u_{1})(u_{1}^{\prime}-u_{1})+K|u_{1}^{\prime}-u_{1}|^{2}$.

Moreover, the minimizer $(x,q)\mapsto\hat{u}_{1}(x,x_{0},\mu,q)$ is Lipschitz
continuous, uniformly in $(x_{0},\mu)$. Similar conditions for $g_{0}$,
$f_{0}$ and $h_{0}$ can be assumed to guarantee a unique minimizer for the
Lagrangian of the control problem for the dominating player in Lemma 2.6.

###### Proof

To express a necessary condition for optimality, we adopt the stochastic
maximum principle. In particular, for any $\tilde{u}_{1}\in\mathcal{A}_{1}$,

$\begin{array}[]{rcl}0&=&\and{\dfrac{d}{d\theta}}{\theta=0}J_{1}(\hat{u}_{1}+\theta\tilde{u}_{1},x_{0},\nu)\\\
&=&\and{\dfrac{d}{d\theta}}{\theta=0}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)+\theta\tilde{u}_{1}(x,t)\Big{)}dxdt\\\
&&\qquad\qquad\qquad+\displaystyle\int_{\mathbb{R}^{n}}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}\\\
&=&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}+p_{\hat{u}_{1}}(x,t)f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{u}_{1}(x,t)dxdt\\\
&&\qquad\qquad\qquad+\displaystyle\int_{\mathbb{R}^{n}}\tilde{p}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]},\end{array}$
(10)

where
$\tilde{p}=\and{\frac{d}{d\theta}}{\theta=0}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}$.
By taking derivative with respect to $\theta$ in the FP equation (6), we have

$\begin{split}\left\\{\begin{array}[]{rl}\dfrac{\partial\tilde{p}}{\partial
t}&=-A_{1}^{*}\tilde{p}(x,t)-\text{div}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}\\\
&\qquad\qquad-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{p}(x,t)\Big{)},\\\
\tilde{p}(x,0)&=0.\end{array}\right.\end{split}$

As an adjoint process, we consider the backward stochastic differential
equation

$\left\\{\begin{array}[]{rcl}-\partial_{t}\Psi&=&\Big{\\{}f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}+D\Psi(x,t)g_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{\\}}dt\\\
&&\qquad-K_{\Psi}(x,t)dW_{0}(t),\\\
\Psi(x,T)&=&h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}.\end{array}\right.$

We consider the inner product

$\begin{array}[]{rcl}&&d\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)\Psi(x,t)dx\\\
&=&\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{\\{}-A_{1}^{*}\tilde{p}(x,t)-\text{div}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}\\\
&&\qquad\qquad-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{p}(x,t)\Big{)}\Big{\\}}\Psi(x,t)dxdt\\\
&&-\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)\Big{\\{}f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}+D\Psi(x,t)g_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{\\}}dxdt\\\
&&+\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)K_{\Psi}(x,t)dxdW_{0}(t)\\\
&=&\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}D\Psi(x,t)dxdt\\\
&&-\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt+\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)K_{\Psi}(x,t)dxdW_{0}(t).\\\
\end{array}$

Taking integration on $[0,T]$ and expectation on both sides yields

$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}\displaystyle\int_{\mathbb{R}^{n}}\tilde{p}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}\\\
&=&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}D\Psi(x,t)dxdt\\\
&&\qquad-\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt\bigg{]}.\end{array}$

Together with the first order condition (10), we have

$0=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{u}_{1}(x,t)\Big{[}g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}D\Psi(x,t)+f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}\Big{]}p_{\hat{u}_{1}}(x,t)dxdt\bigg{]}.$

Recall that the $p_{\hat{u}_{1}}(\cdot,t)$ is a conditional probability
density function and hence non-negative, and $\tilde{u}_{1}$ is an arbitrary
Markovian control, we have $\hat{u}_{1}$ is optimal only if

$g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}D\Psi(x,t)+f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}=0,a.e.(x,t).$

With the definition of $L$ in the theorem, the condition becomes

$L_{u_{1}}(x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t),D\Psi(x,t))=0,a.e.(x,t),$

which provides a necessary condition for the minimization problem. As the
minimizer is assumed to be attained at $\hat{u}_{1}$, which depends on $x$,
$x_{0}$, $\nu$, and $D\Psi$, we arrive for the SHJB Equation. ∎

Replace the arbitrary measure $\nu$ by the mean field measure $\mu$. Equating
$\mu:=m_{x_{0}}d\lambda$ with $p_{\hat{u}_{1}}d\lambda$, the measure of the
optimal state of the representative agent conditioning on
$\mathcal{F}^{0}_{t}$; the couple (6) and (9) give the following corollary.

###### Corollary 2.5

(Necessary condition for Problems 2.1 and 2.2)
The control for the representative agent is optimal and the equilibrium
condition holds only if the SHJB-FP coupled equations are satisfied

$\left\\{\begin{array}[]{rcl}-\partial_{t}\Psi&=&\Big{(}H_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)&=&h_{1}(x,x_{0}(T),m_{x_{0}}(x,T)).\\\
\dfrac{\partial m_{x_{0}}}{\partial
t}&=&-A_{1}^{*}m_{x_{0}}(x,t)-{\text{div}}\Big{(}G_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}m_{x_{0}}(x,t)\Big{)},\\\
m_{x_{0}}(x,0)&=&\omega(x),\\\ \end{array}\right.$ (11)

where $G_{1}(x,x_{0},m,q)=g_{1}(x,x_{0},m,\hat{u}_{1}(x,x_{0},m,q))$.

The SHJB-FP coupled equations (11) allow us to obtain the control of the
representative agent in terms of a given trajectory of the dominating player
$x_{0}$ while the equilibrium condition also holds.

We then turn to the optimal problem for the dominating player. As $m_{x_{0}}$
is not external to the dominating player, the dominating player has to
consider both its own dynamic evolution and (11).

###### Lemma 2.6

Necessary condition for Problem 2.3
The control for the dominating player $\hat{u}_{0}$ is optimal only if

$\begin{split}f_{0}(x_{0},m_{x_{0}},\hat{u}_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},\hat{u}_{0})&=\inf_{u_{0}}\Big{\\{}f_{0}(x_{0},m_{x_{0}},u_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},u_{0})\Big{\\}}\\\ :&=H_{0}(x_{0},m_{x_{0}},p)\\\
\end{split}$

where $p(t)$ satisfies the following adjoint processes

$\begin{array}[]{rcl}-dp&=&\bigg{[}g_{0,x_{0}}^{*}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))p(t)+f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)dx\\\
&&+\displaystyle\int
r(x,t)H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})dx\bigg{]}dt\\\
&&-\sum_{l=1}^{d_{0}}K_{p}^{l}(t)dW_{0}^{l}(t)+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l*}(x_{0}(t))K_{p}^{l}(t)dt,\\\
p(T)&=&h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx;\\\
\end{array}$
$\begin{array}[]{rcl}-\partial_{t}{q}&=&\bigg{[}-A_{1}{q}(x,t)+p(t)\dfrac{\partial
g_{0}}{\partial m}(x_{0}(t),m_{x_{0}}(\xi,t),\hat{u}_{0}(t))(x)\\\
&&+D{q}(x,t)G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+\displaystyle\int D{q}(\xi,t)\dfrac{\partial G_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)m_{x_{0}}(\xi,t)d\xi\\\
&&+\displaystyle\int{r}(\xi,t)\dfrac{\partial H_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\\\ &&+\dfrac{\partial
f_{0}}{\partial m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\bigg{]}dt-
K_{q}(x,t)dW_{0}(t),\\\ {q}(x,T)&=&\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(\xi,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial m}(\xi,x_{0}(T),m_{x_{0}}(\xi,T))(x)d\xi;\end{array}$
$\begin{array}[]{rcl}\dfrac{\partial{r}}{\partial
t}&=&-A_{1}^{*}{r}(x,t)-\text{\rm
div}\bigg{[}{r}(x,t)H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\bigg{]},\\\
{r}(x,0)&=&0.\end{array}$

###### Proof

Again we consider the Gâteaux derivative

$\begin{split}0=&\and{\dfrac{d}{d\theta}}{\theta=0}J_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})\\\
=&\mathbb{E}\left\\{\int_{0}^{T}[f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{x}_{0}(t)+\int\dfrac{\partial
f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\right.\\\
&+f_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)]dt\\\
&+\left.h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\int\dfrac{\partial
h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi\right\\},\end{split}$
(12)

where
$\tilde{x}_{0}=\and{\frac{d}{d\theta}}{\theta=0}x_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})$;
$\tilde{m}_{x_{0}}=\and{\frac{d}{d\theta}}{\theta=0}m_{x_{0}}(\hat{u}_{0}+\theta\tilde{u}_{0})$;
$\tilde{\Psi}=\and{\frac{d}{d\theta}}{\theta=0}\Psi(\hat{u}_{0}+\theta\tilde{u}_{0})$,
satisfy

$\begin{array}[]{rcl}d\tilde{x}_{0}&=&\bigg{[}g_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{x}_{0}(t)+\displaystyle\int\dfrac{\partial
g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\bigg{]}dt+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l}(x_{0}(t))\tilde{x}_{0}(t)dW_{0}^{l}(t),\\\
\tilde{x}_{0}(0)&=&0;\\\ \end{array}$
$\begin{array}[]{rcl}\dfrac{\partial\tilde{m}_{x_{0}}}{\partial
t}&=&-A_{1}^{*}\tilde{m}_{x_{0}}(x,t)-\text{\rm
div}\bigg{\\{}\Big{[}G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial G_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}(x,t)\Big{]}m_{x_{0}}(x,t)\\\
&&+G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{m}_{x_{0}}(x,t)\bigg{\\}},\\\
\tilde{m}_{x_{0}}(x,0)&=&0;\\\ \end{array}$
$\begin{array}[]{rcl}-\partial_{t}\tilde{\Psi}&=&\bigg{[}H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial H_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}(x,t)-A_{1}\tilde{\Psi}(x,t)\bigg{]}dt-
K_{\tilde{\Psi}}(x,t)dW_{0}(t),\\\
\tilde{\Psi}(x,T)&=&h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\displaystyle\int\dfrac{\partial
h_{1}}{\partial
m}(x,x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi.\\\
\end{array}$

Introduce the adjoint processes $p(t)$, ${q}(x,t)$ and ${r}(x,t)$ as stated in
the lemma statement and consider the following differentials

$\begin{array}[]{rcl}&&d(p^{*}\tilde{x}_{0})\\\
&=&-\tilde{x}_{0}^{*}(t)\bigg{[}f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+\displaystyle\int{r}(x,t)H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))dx\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)dx\bigg{]}dt\\\
&&+p^{*}(t)\bigg{[}\displaystyle\int\dfrac{\partial g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\bigg{]}dt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
$\begin{array}[]{rcl}&&d\displaystyle\int{q}(x,t)\tilde{m}_{x_{0}}(x,t)dx\\\
&=&\displaystyle\int{q}(x,t)\bigg{[}-\text{\rm
div}\Big{[}[G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}]m_{x_{0}}(x,t)\Big{]}\bigg{]}dxdt\\\
&&-\displaystyle\int\Big{[}p(t)\dfrac{\partial g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(\xi,t),\hat{u}_{0}(t))(x)+\displaystyle\int{r}(\xi,t)\dfrac{\partial
H_{1}}{\partial m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\\\
&&+\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\Big{]}\tilde{m}_{x_{0}}(x,t)dxdt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
$\begin{array}[]{rcl}&&d\displaystyle\int{r}(x,t)\tilde{\Psi}(x,t)dx\\\
&=&\displaystyle\int\bigg{[}-\text{\rm
div}\Big{[}G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\Big{]}\bigg{]}\tilde{\Psi}(x,t)dxdt\\\
&&-\displaystyle\int
r(x,t)\bigg{[}H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial H_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\bigg{]}dxdt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$

Using the results above, we have

$\begin{array}[]{rcl}&&d\displaystyle\Big{\\{}p^{*}\tilde{x}_{0}+\int{q}(x,t)\tilde{m}_{x_{0}}(x,t)dx-\int{r}(x,t)\tilde{\Psi}(x,t)dx\Big{\\}}\\\
&=&\bigg{[}\displaystyle-\tilde{x}_{0}^{*}(t)f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+p^{*}(t)g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\\\
&&-\displaystyle\int\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\tilde{m}_{x_{0}}(x,t)dx\bigg{]}dt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$

Integrating and taking expectation on both sides gives

$\begin{array}[]{rl}&\mathbb{E}\bigg{[}(h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx)\tilde{x}_{0}(T)dx\\\
&\displaystyle+\int\Big{[}\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(x,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial
m}(\xi,x_{0}(T),m_{x_{0}}(x,T))(x)d\xi\Big{]}\tilde{m}_{x_{0}}(x,T)dx\\\
&\displaystyle-\int
r(x,T)\Big{[}h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\displaystyle\int\dfrac{\partial
h_{1}}{\partial
m}(x,x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi\Big{]}dx\bigg{]}\\\
=&\displaystyle\mathbb{E}\int_{0}^{T}\bigg{\\{}\displaystyle-\tilde{x}_{0}^{*}(t)f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+p^{*}(t)g_{0,\hat{u}_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\\\
&-\displaystyle\int\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\tilde{m}_{x_{0}}(x,t)dx\bigg{\\}}dt.\\\
\end{array}$

Finally we consider (12) and obtain

$\begin{array}[]{rcl}0&=&\mathbb{E}\displaystyle\int_{0}^{T}\Big{\\{}f_{0,\hat{u}_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})+p^{*}g_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})\Big{\\}}\tilde{u}_{0}dt.\end{array}$

Since $\tilde{u}_{0}$ is arbitrary, the control is optimal for the dominating
player only if

$\begin{split}f_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})+p^{*}g_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})=0,\text{a.e.
t.}\end{split}$

Again, we are considering a minimization problem with the first order
condition given above. Since the unique existence of a minimizer,
$\hat{u}_{0}$ is assumed, we conclude that $\hat{u}_{0}$ satisfies the infimum

$f_{0}(x_{0},m_{x_{0}},\hat{u}_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},\hat{u}_{0})=\inf_{u_{0}}\Big{\\{}f_{0}(x_{0},m_{x_{0}},u_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},u_{0})\Big{\\}}.\\\ $

∎

Let
$G_{0}(x,m_{x_{0}},p)=g_{0}(x_{0},m_{x_{0}},\hat{u}_{0}(x_{0},m_{x_{0}},p))$.
We then conclude the main result in this section.

###### Theorem 2.7

The necessary condition for Problems 2.1, 2.2 and 2.3 is provided by the
following six equations

$\begin{array}[]{l}\left\\{\begin{array}[]{rll}dx_{0}&=&G_{0}(x_{0}(t),m_{x_{0}(t)}(t),p(t))dt+\sigma_{0}(x_{0}(t))dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \\\ \dfrac{\partial m_{x_{0}}}{\partial
t}&=&-A_{1}^{*}m_{x_{0}}(x,t)-{\text{div}}\Big{(}G_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}m_{x_{0}}(x,t)\Big{)},\\\
m_{x_{0}}(x,0)&=&\omega(x),\\\ \\\
-\partial_{t}\Psi&=&\Big{(}H_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)&=&h_{1}(x,x_{0}(T),m_{x_{0}}(x,T)).\\\
\end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rll}-dp&=&\bigg{[}H_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),p(t))+\displaystyle\int{r}(x,t)H_{x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))dx\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D{q}^{*}(x,t)m_{x_{0}(t)}(x,t)dx\bigg{]}dt\\\
&&-\sum_{l=1}^{d_{0}}K_{p}^{l}(t)dW_{0}^{l}(t)+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l*}(x_{0}(t))K_{p}^{l}(t)dt,\\\
p(T)&=&h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx;\\\
\\\ -\partial_{t}{q}&=&\bigg{[}-A_{1}{q}(x,t)+\dfrac{\partial H_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(\xi,t),p(t))(x)\\\
&&+D{q}(x,t)G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+\displaystyle\int D{q}(\xi,t)\dfrac{\partial G_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)m_{x_{0}}(\xi,t)d\xi\\\
&&+\displaystyle\int{r}(\xi,t)\dfrac{\partial H_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\bigg{]}dt-
K_{q}(x,t)dW_{0}(t),\\\ {q}(x,T)&=&\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(\xi,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial m}(\xi,x_{0}(T),m_{x_{0}}(\xi,T))(x)d\xi;\\\ \\\
\dfrac{\partial{r}}{\partial t}&=&-A_{1}^{*}{r}(x,t)-\text{\rm
div}\bigg{[}{r}(x,t)H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\bigg{]},\\\
{r}(x,0)&=&0.\end{array}\right.\end{array}$

##  3 Linear Quadratic Case

In this section we present a special case of the problem in the Linear
Quadratic setting in which both necessary and sufficient condition could be
established. Suppose that the state evolutions of the processes
$x_{0}(t),x_{1}(t)$ are described by

$\left\\{\begin{array}[]{rcl}dx_{0}&=&\Big{(}A_{0}x_{0}(t)+B_{0}z(t)+C_{0}u_{0}(x_{0}(t),t)\Big{)}dt+\sigma_{0}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\
dx_{1}&=&\Big{(}A_{1}x_{1}(t)+B_{1}z(t)+C_{1}u_{1}(x_{1}(t),t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
x_{1}(0)&=&\xi_{1}.\\\ \end{array}\right.$

To simplify, the matrices $A_{0},C_{0},B_{0},\sigma_{0}$;
$A_{1},C_{1},B_{1},D,\sigma_{1}$ are assumed to be constant though the case
with time dependent and deterministic function are similar. For if the
dominating player did not exist, it is customary to consider $z(t)$ as
deterministic, and the equilibrium condition is

$z(t)=\mathbb{E}x_{1}(t).$ (13)

We can first find an optimal stochastic control for the representative agent
given $z$, then solve for the fixed point equation (13). However, one cannot
assume $z(t)$ to be deterministic when the dominating player exists, which
induces a two-layer problem. Since the dominating player can directly
influence the mean field term in the present setting, $z(t)$ should be adapted
to the filtration $\mathcal{F}_{t}^{0}$. The equilibrium condition hence is

$z(t)=\mathbb{E}^{\mathcal{F}_{t}^{0}}x_{1}(t)=\int\xi
m_{x_{0}(t)}(\xi,t)d\xi.$ (14)

We define control problems for both the dominating player and the
representative agent and fist solve for the control problem of the
representative agent as if both $x_{0}(t)$ and $z(t)$ as exogenous. The next
problem is to solve the equilibrium condition (14) as a fixed point property.
Finally we solve for the control problem of the dominating player, but now
$z(t)$ is regarded as endogenous.

For any vector $v$ and matrix $M$ with appropriate dimensions, we write the
inner product $v^{*}Mv$ as $|v|^{2}_{M}$ for simplicity. Problems 2.1, 2.2 and
2.3 can now be rewritten in the present Linear Quadratic framework as follows:

###### Problem 3.1

Control of the Representative Agent
Given the process $x_{0}$ and $\kappa$, find a control
$u_{1}\in\mathcal{A}_{1}$ which minimizes the cost functional

$\begin{split}J_{1}(u_{1},x_{0},\kappa)&=\mathbb{E}\bigg{[}\int_{0}^{T}(|x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1}|_{Q_{1}}^{2}+u^{*}_{1}(t)R_{1}u_{1}(t))dt\\\
&\qquad\qquad+|x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}|_{\bar{Q}_{1}}^{2}\bigg{]}.\end{split}$

###### Problem 3.2

Equilibrium Condition
Let $x_{1}:=x_{1,\kappa}$ be the trajectory of the representative agent with
the optimal control found in Problem 3.1. Find the process $z(t)$ such that
the fixed point property is satisfied

$z(t)=\mathbb{E}^{\mathcal{F}_{t}^{0}}x_{1,z}(t),$

###### Problem 3.3

Control of the Dominating Player
Find a control $u_{0}\in\mathcal{A}_{0}$ which minimizes the cost functional

$\begin{split}J_{0}(u_{0})&=\mathbb{E}\bigg{[}\int_{0}^{T}(|x_{0}(t)-E_{0}z(t)-\zeta_{0}|_{Q_{0}}^{2}+u_{0}^{*}(t)R_{0}u_{0}(t))dt\\\
&\qquad\qquad+|x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta}_{0}|_{\bar{Q}_{0}}^{2}\bigg{]},\end{split}$

where $z$ is the solution given in Problem 3.2.

For simplicity, $E_{0},F,R_{0},R_{1},E_{1},\zeta_{0},\zeta_{1},Q_{0},Q_{1}$
are constant matrices and vectors; $Q_{0},Q_{1},R_{0},R_{1}$ are positive
symmetric and invertible. Note that Problems 3.1 and 3.3 are strictly convex
quadratic and coercive, we can write the stochastic principle.

###### Lemma 3.4

(Control of Representative Agent)
Problem 3.1 is uniquely solvable and the optimal control $\hat{u}_{1}(t)$ is
$-R_{1}^{-1}C_{1}^{*}n(t)$, where $n$ satisfies the adjoint process

$\begin{split}\left\\{\begin{array}[]{rlc}-dn&=\Big{(}A_{1}^{*}n(t)+Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt-
Z_{n,0}(t)dW_{0}(t)-Z_{n,1}(t)dW_{1}(t),\\\
n(T)&=\bar{Q}_{1}\Big{(}x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\end{array}\right.\end{split}$

###### Proof

Consider a perturbation of the optimal control
$\hat{u}_{1}+\theta\tilde{u}_{1}$, where $\tilde{u}_{1}$ is adapted to the
filtration $\mathcal{G}_{t}$. The original state $x_{1}$ becomes
$x_{1}+\theta\tilde{x}_{1}$ with

$\begin{split}\left\\{\begin{array}[]{rlc}d\tilde{x}_{1}&=\Big{(}A_{1}\tilde{x}_{1}(t)+C_{1}\tilde{u}_{1}(t)\Big{)}dt,\\\
\tilde{x}_{1}(0)&=0.\end{array}\right.\end{split}$

the optimality of $\hat{u}_{1}$ is expressed by the following Euler condition

$\begin{split}0&=\and{\dfrac{d}{d\theta}}{\theta=0}J_{1}(\hat{u}_{1}+\theta\tilde{u}_{1},x_{0},\kappa)\\\
&=\mathbb{E}\bigg{[}\int_{0}^{T}[\tilde{x}_{1}^{*}(t)Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})+\tilde{u}_{1}^{*}(t)R_{1}\hat{u}_{1}(t)]dt\\\
&\qquad\qquad+\tilde{x}_{1}^{*}(T)\bar{Q}_{1}(x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1})\bigg{]}\\\
\end{split}$

On the other hand, we have

$\begin{split}d(n^{*}\tilde{x}_{1})&=\Big{(}n^{*}(t)C_{1}\tilde{u}_{1}(t)-\tilde{x}_{1}^{*}(t)Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt\\\
&\qquad+\tilde{x}_{1}(t)Z_{n,0}(t)dW_{0}(t)+\tilde{x}_{1}(t)Z_{n,1}(t)dW_{1}(t).\end{split}$

Integrate both side and take expectation, combining with the Euler condition,
it becomes

$\mathbb{E}\int_{0}^{T}(n^{*}(t)C_{1}+\hat{u}_{1}^{*}(t)R_{1})\tilde{u}_{1}(t)dt=0,\quad\forall\tilde{u}_{1}(\cdot)\text{
adpated to }\mathcal{G},$

which implies, together with an application of tower property,

$\hat{u}_{1}(t)=-R_{1}^{-1}C_{1}^{*}n(t).$ (15)

Therefore the stochastic maximum principle for the representative agent is
expressed by the system

$\left\\{\begin{array}[]{rlc}dx_{1}&=\Big{(}A_{1}x_{1}(t)+B_{1}\kappa(t)-C_{1}R_{1}^{-1}C_{1}^{*}n(t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
x_{1}(0)&=\xi_{1}.\\\
-dn&=\Big{(}A_{1}^{*}n(t)+Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt-
Z_{n,0}(t)dW_{0}(t)-Z_{n,1}(t)dW_{1}(t),\\\
n(T)&=\bar{Q}_{1}\Big{(}x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\end{array}\right.$
(16)

and hence for every exogenous pair $(x_{0},\kappa)$, this system defines a
unique pair $(x_{1},n)$. By the convexity and coerciveness of the cost
functional, $\hat{u}_{1}$ is uniquely defined and the sufficient condition is
automatically satisfied. ∎

###### Remark

The optimal control has the representation
$-R_{1}^{-1}C_{1}^{*}n(t)=-R_{1}^{-1}C_{1}^{*}(P_{t}x_{1}(t)+g(t))$, where $P$
satisfies the symmetric Riccati equation

$\left\\{\begin{array}[]{rl}&\dot{P_{t}}+P_{t}A_{1}+A_{1}^{*}P_{t}-P_{t}C_{1}R_{1}^{-1}C_{1}^{*}P_{t}+Q_{1}=0,\\\
&P_{T}=\bar{Q}_{1};\\\ \end{array}\right.\\\ $

and $g$ satisfies the BSDE

$\left\\{\begin{array}[]{rcl}-dg&=&\Big{(}(A_{1}-C_{1}R_{1}^{-1}C_{1}^{*}P_{t})^{*}g(t)+(P_{t}B_{1}-Q_{1}E_{1})+(P_{t}D-Q_{1}F)x_{0}(t)\kappa(t)-Q_{1}\zeta_{1}\Big{)}dt\\\
&&\qquad-Z_{g,0}(t)dW_{0}(t)-Z_{g,1}(t)dW_{1}(t),\\\
g(T)&=&\bar{Q}_{1}\Big{(}-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\\\
\end{array}\right.$

To obtained the equilibrium condition stated in Problem (3.2), we simply take
expectation conditional on $\mathcal{F}_{t}^{0}$ on both sides of Equation
(16). By requiring $\mathbb{E}^{\mathcal{F}^{0}_{t}}x_{1}(t)=z(t)$, and
replace $\kappa(t)$ by $z(t)$, we have the following system, which is
analogical to the SHJB-FP (11) in Section 2.

$\left\\{\begin{array}[]{rlc}dz&=\Big{(}(A_{1}+B_{1})z(t)-C_{1}R_{1}^{-1}C_{1}^{*}m(t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
z_{1}(0)&=\mathbb{E}[\xi_{1}].\\\
-dm&=\Big{(}A_{1}^{*}m(t)+Q_{1}(I-E_{1})z(t)-Q_{1}Fx_{0}(t)-Q_{1}\zeta_{1}\Big{)}dt-
Z_{m}(t)dW_{0}(t),\\\
m(T)&=\bar{Q}_{1}(I-\bar{E}_{1})z(T)-\bar{Q}_{1}\bar{F}x_{0}(T)-\bar{Q}_{1}\bar{\zeta}_{1}.\end{array}\right.$
(17)

We next proceed on the problem of the dominating player.

###### Lemma 3.5

(Control of the Dominating Player)
Problem 3.3 is uniquely solvable and the optimal control $\hat{u}_{0}(t)$ is
$-R_{0}^{-1}C_{0}^{*}p(t)$, where $p$ satisfies

$\left\\{\begin{array}[]{rcl}-dp&=&\Big{(}A_{0}^{*}{p}(t)+D^{*}q(t)-(Q_{1}F)^{*}r(t)+Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt-
Z_{p}(t)dW_{0}(t),\\\
{p}(T)&=&\bar{F}^{*}\bar{Q}_{1}r(T)+\bar{Q}_{0}(x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta}_{0}).\\\
-dq&=&\Big{(}(A_{1}+B_{1})^{*}q(t)+B_{0}^{*}{p}(t)+(I-E_{1})^{*}Q_{1}^{*}r(t)-E_{0}^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt-
Z_{q}(t)dW_{0}(t),\\\
q(T)&=&(I-\bar{E}_{1})^{*}\bar{Q}_{1}r(T)-\bar{E}_{0}^{*}\bar{Q}_{0}(x_{0}(t)-\bar{E}_{0}z(t)-\bar{\zeta}_{0}).\\\
dr&=&\Big{(}A_{1}r(t)-C_{1}R_{1}^{-1}C_{1}^{*}q(t)\Big{)}dt,\\\
{r}(0)&=&0.\end{array}\right.$ (18)

###### Proof

Consider $\hat{u}_{0}+\theta\tilde{u}_{0}$ the perturbation of the optimal
control, where $\tilde{u}_{0}$ is adapted to the filtration
$\mathcal{F}_{t}^{0}$. The original states $x_{0}$, $z$, $m$ become
$x_{0}+\theta\tilde{x}_{0}$, $z+\theta\tilde{z}$, $m+\theta\tilde{m}$ with

$\left\\{\begin{array}[]{rcl}d\tilde{x}_{0}&=&\Big{(}A_{0}\tilde{x}_{0}(t)+B_{0}\tilde{z}(t)+C_{0}\tilde{u}_{0}(t)\Big{)}dt,\\\
\tilde{x}(0)&=&0.\\\
d\tilde{z}&=&\Big{(}(A_{1}+B_{1})\tilde{z}(t)-C_{1}R_{1}^{-1}C_{1}^{*}\tilde{m}(t)+D\tilde{x}_{0}(t)\Big{)}dt,\\\
\tilde{z}(0)&=&0.\\\
-d\tilde{m}&=&\Big{(}A_{1}^{*}\tilde{m}(t)+Q_{1}(I-E_{1})\tilde{z}(t)-Q_{1}F\tilde{x}_{0}(t)\Big{)}dt-
Z_{\tilde{m}}dW_{0}(t)\\\
\tilde{m}(T)&=&\bar{Q}_{1}(I-\bar{E}_{1})\tilde{z}(T)-\bar{Q}_{1}\bar{F}\tilde{x}_{0}(T).\\\
\end{array}\right.$

The corresponding maximum principle for $\hat{u}_{0}$ is

$\begin{split}0&=\and{\dfrac{d}{d\theta}}{\theta=0}J_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})\\\
&=\mathbb{E}\bigg{[}\int_{0}^{T}[(\tilde{x}_{0}(t)-E_{0}\tilde{z}(t))^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})+\hat{u}_{0}^{*}R_{0}\tilde{u}_{0}]dt\\\
&\qquad\qquad+(\tilde{x}_{0}(T)-\bar{E}_{0}\tilde{z}(T))^{*}Q_{0}(x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta_{0}})\bigg{]}\end{split}$
(19)

On the other hand, we can easily check that

$\begin{split}d({p}^{*}\tilde{x}_{0}+q^{*}\tilde{z}-r^{*}\tilde{m})=\Big{(}{p}^{*}(t)C_{0}\tilde{u}_{0}(t)-(\tilde{x}_{0}(t)-E_{0}\tilde{z}(t))^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt+\\{\dots\\}dW_{0}(t).\end{split}$

Integrating and also taking expectation on both sides of the last equation,
together with an application of (19), we deduce that

$\begin{split}\mathbb{E}\int_{0}^{T}({p}^{*}C_{0}+\hat{u}_{0}^{*}R_{0})\tilde{u}_{0}dt=0,\quad\forall\tilde{u}_{0}\text{
adapted to }\mathcal{F}_{t}^{0},\end{split}$

which implies the desired result by again the application of tower property,

$\hat{u}_{0}(t)=-R_{0}^{-1}C_{0}^{*}p(t)(t).$ (20)

∎

Summarizing the results we obtained so far, we present the main theorem in
this section.

###### Theorem 3.6

The necessary and sufficient conditions for the unique existence of the
solution to Problems 3.1, 3.2 and 3.3 are described by the following six
equations in matrix form

$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}d\begin{pmatrix}x_{0}\\\ r\\\
z\end{pmatrix}&=&\left\\{\begin{pmatrix}A_{0}&0&B_{0}\\\ 0&A_{1}&0\\\
D&0&A_{1}+B_{1}\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\ r(t)\\\
z(t)\end{pmatrix}\right.\\\
&&\left.-\begin{pmatrix}C_{0}R_{0}^{-1}C_{0}^{*}&0&0\\\
0&C_{1}R_{1}^{-1}C_{1}^{*}&0\\\
0&0&C_{1}R_{1}^{-1}C_{1}^{*}\end{pmatrix}\begin{pmatrix}p(t)\\\ q(t)\\\
m(t)\end{pmatrix}\right\\}dt+\begin{pmatrix}\sigma_{0}\\\ 0\\\
0\end{pmatrix}dW_{0},\\\ \begin{pmatrix}x_{0}(0)\\\ r(0)\\\
z(0)\end{pmatrix}&=&\begin{pmatrix}\xi_{0}\\\ 0\\\
\mathbb{E}[\xi_{1}]\end{pmatrix}.\\\ \end{array}\right.\\\ \\\
&\left\\{\begin{array}[]{rcl}-d\begin{pmatrix}p\\\ q\\\
m\end{pmatrix}&=&\left\\{\begin{pmatrix}A_{0}^{*}&D^{*}&0\\\
B_{0}^{*}&A_{1}^{*}+B_{1}^{*}&0\\\
0&0&A_{1}^{*}\end{pmatrix}\begin{pmatrix}p(t)\\\ q(t)\\\
m(t)\end{pmatrix}\right.\\\
&&\left.+\begin{pmatrix}Q_{0}&-(Q_{1}F)^{*}&-Q_{0}E_{0}\\\
-(Q_{0}E_{0})^{*}&(Q_{1}(I-E_{1}))^{*}&E_{0}^{*}Q_{0}E_{0}\\\
-Q_{1}F&0&Q_{1}(I-E_{1})\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\ r(t)\\\
z(t)\end{pmatrix}+\begin{pmatrix}-Q_{0}\zeta_{0}\\\ E_{0}^{*}Q_{0}\zeta_{0}\\\
-Q_{1}\zeta_{1}\end{pmatrix}\right\\}dt-\begin{pmatrix}Z_{p}(t)\\\ Z_{q}(t)\\\
Z_{m}(t)\end{pmatrix}dW_{0}(t),\\\ \begin{pmatrix}p(T)\\\ q(T)\\\
m(T)\end{pmatrix}&=&\begin{pmatrix}\bar{Q}_{0}&-(\bar{Q}_{1}\bar{F})^{*}&-\bar{Q}_{0}\bar{E}_{0}\\\
-\bar{Q}_{0}\bar{E}_{0})^{*}&(\bar{Q}_{1}(I-\bar{E}_{1}))^{*}&\bar{E}_{0}^{*}\bar{Q}_{0}\bar{E}_{0}\\\
\bar{Q}_{1}F&0&\bar{Q}_{1}(I-\bar{E}_{1})\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\
r(t)\\\ z(t)\end{pmatrix}+\begin{pmatrix}-\bar{Q}_{0}\bar{\zeta}_{0}\\\
\bar{E}_{0}^{*}\bar{Q}_{0}\bar{\zeta}_{0}\\\
-\bar{Q}_{1}\bar{\zeta}_{1}\end{pmatrix}.\\\ \end{array}\right.\end{array}$
(21)

###### Remark

One can easily compare these six equations with those stated in Theorem 2.7.
We obtain the same results by applying the general theory, however, it is more
convenient to acquire these six equations directly under the Linear Quadratic
setting, which also illuminates the power of using the principle of
separation. On comparison with the intermediary result obtained in HN2 . The
latter work did not take account of the third adjoint equation $r$ since it
fails to consider the impact on $m$ with respect to the change of the control
of the dominating player.

##  4 Fixed Point Problem

In this section, we provide a sufficient condition, which solely depends on
the coefficients of the mean field game system, for the unique existence of
the solution to Problems 3.1, 3.2 and 3.3 by means of tackling a non-symmetric
Riccati equation. To facilitate our argument, we define

${\bf x}:=\begin{pmatrix}x_{0}\\\ r\\\ z\end{pmatrix};\quad{\bf
p}:=\begin{pmatrix}p\\\ q\\\ m\end{pmatrix};\quad{\bf
Z}:=\begin{pmatrix}Z_{p}\\\ Z_{q}\\\ Z_{m}\end{pmatrix}.$

Hence we can write (21) as

$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}d{\bf
x}&=&\Big{(}(\mathcal{A}+\mathcal{B}){\bf x}(t)-\mathcal{C}{\bf
p}(t)\Big{)}dt+\sigma dW_{0}(t),\\\ {\bf x}(0)&=&\xi.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-d{\bf p}&=&\Big{(}\mathcal{A}^{*}{\bf
p}(t)+(\mathcal{Q}+\mathcal{S}){\bf x}(t)+{\bf k}\Big{)}dt-{\bf
Z}(t)dW_{0}(t),\\\ {\bf p}(T)&=&(\bar{\mathcal{Q}}+\bar{\mathcal{S}}){\bf
x}(T)+\bar{\bf k},\\\ \end{array}\right.\end{array}$ (22)

where

$\begin{array}[]{l}\mathcal{A}:=\begin{pmatrix}A_{0}&B_{0}&0\\\
D&A_{1}+B_{1}&0\\\
0&0&A_{1}\end{pmatrix};\quad\mathcal{B}:=\begin{pmatrix}0&-B_{0}&B_{0}\\\
-D&-B_{1}&0\\\
D&0&B_{1}\end{pmatrix};\quad\mathcal{C}:=\begin{pmatrix}C_{0}R_{0}^{-1}C_{0}^{*}&0&0\\\
0&C_{1}R_{1}^{-1}C_{1}^{*}&0\\\ 0&0&C_{1}R_{1}^{-1}C_{1}^{*}\end{pmatrix};\\\
{\sigma}:=\begin{pmatrix}\sigma_{0}\\\ 0\\\
0\end{pmatrix};\quad\xi:=\begin{pmatrix}\xi_{0}\\\ 0\\\
0\end{pmatrix};\quad\mathcal{Q}+\mathcal{S}:=\begin{pmatrix}Q_{0}&-F^{*}Q_{1}&-Q_{0}E_{0}\\\
-E_{0}^{*}Q_{0}&\quad(I-E_{1})^{*}Q_{1}^{*}&E_{0}^{*}Q_{0}E_{0}\\\
-Q_{1}F&0&\quad Q_{1}(I-E_{1})\end{pmatrix};\quad{\bf
k}:=\begin{pmatrix}-Q_{0}\zeta_{0}\\\ E_{0}^{*}Q_{0}\zeta_{0}\\\
-Q_{1}\zeta_{1}\end{pmatrix};\\\
\bar{\mathcal{Q}}+\bar{\mathcal{S}}:=\begin{pmatrix}\bar{Q}_{0}&-(\bar{Q}_{1}\bar{F})^{*}&-\bar{Q}_{0}\bar{E}_{0}\\\
-\bar{Q}_{0}\bar{E}_{0})^{*}&(\bar{Q}_{1}(I-\bar{E}_{1}))^{*}&\bar{E}_{0}^{*}\bar{Q}_{0}\bar{E}_{0}\\\
\bar{Q}_{1}F&0&\bar{Q}_{1}(I-\bar{E}_{1})\end{pmatrix};\quad\bar{\bf
k}:=\begin{pmatrix}-\bar{Q}_{0}\bar{\zeta}_{0}\\\
\bar{E}_{0}^{*}\bar{Q}_{0}\bar{\zeta}_{0}\\\
-\bar{Q}_{1}\bar{\zeta}_{1}\end{pmatrix},\end{array}$

where $\mathcal{Q}$ and $\bar{\mathcal{Q}}$ are positive matrices. Consider
the following non-symmetric Riccati equation

$\left\\{\begin{array}[]{l}\dot{\Gamma}+\mathcal{A}^{*}\Gamma_{t}+\Gamma_{t}(\mathcal{A}+\mathcal{B})-\Gamma_{t}\mathcal{C}\Gamma_{t}+(\mathcal{Q}+\mathcal{S})=0,\\\
\Gamma_{T}=\bar{\mathcal{Q}};\\\ \end{array}\right.$ (23)

and the backward ODE

$\left\\{\begin{array}[]{rcl}-d{\bf
g}&=&\Big{(}(\mathcal{A}^{*}-\Gamma_{t}\mathcal{B}){\bf g}(t)+{\bf
k}\Big{)}dt,\\\ {\bf g}(T)&=&\bar{\bf k}.\end{array}\right.$ (24)

It is easy to check that ${\bf p}(t)=\Gamma_{t}{\bf x}(t)+{\bf g}(t)$. With
respect to this affine form, the forward backward equation (22) admits a
unique solution if and only if (23) admits a unique solution. In accordance
with Theorem 2.4.3 in Ma and Young FBSDE  or AB , we have the following
proposition.

###### Proposition 4.1

Suppose the following forward-backward ordinary differential equations

$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}\dfrac{dX}{dt}&=&(\mathcal{A}+\mathcal{B})X(t)-\mathcal{C}Y(t)\\\
X(0)&=&0.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-\dfrac{dY}{dt}&=&\mathcal{A}^{*}Y(t)+(\mathcal{Q}+\mathcal{S})X(t),\\\
Y(T)&=&(\bar{\mathcal{Q}}+\bar{\mathcal{S}})X(T).\\\
\end{array}\right.\end{array}$

admits a unique solution for any $t_{0}\in[0,T]$. Then there is a unique
solution of (23).

Our next theorem concludes the results in this section.

###### Theorem 4.2

Let $\phi(s,t)$ be the fundamental solution to $\mathcal{A}$. Suppose that

$\Big{(}1+\sqrt{T}\|\phi\|_{T}\cdot\|\mathcal{B}\mathcal{Q}^{-\frac{1}{2}}\|\Big{)}\Big{(}1+N(S)\Big{)}<2,$

where $\|\cdot\|$ stands for usual Euclidean norm. Then there exists a unique
solution of equation (21), and hence a unique (mean field) equilibrium exists.
Here,

$\|\phi\|_{T}:=\sup_{0\leq t\leq
T}\sqrt{\|\phi^{*}(T,t)\bar{\mathcal{Q}}^{\frac{1}{2}}\|^{2}+\int_{t}^{T}\|\phi^{*}(s,t)\mathcal{Q}^{\frac{1}{2}}\|^{2}ds}$

and

$N(S)=\max\\{\|\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}\bar{\mathcal{Q}}^{-\frac{1}{2}}\|,\|\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}\mathcal{Q}^{-\frac{1}{2}}\|\\}$

###### Proof

Let $x,y$ be elements in the Hilbert Space
$\mathcal{H}^{2}([0,T];\mathbb{R}^{n_{0}+n_{1}+n_{1}})$ endowed with the inner
product

$\langle
x,y\rangle_{\mathcal{H}}=x^{*}(T)\bar{\mathcal{Q}}y(T)+\int_{0}^{T}x^{*}(s)\mathcal{Q}y(s)ds$

Furthermore,
$\|\cdot\|_{\mathcal{H}}:=|\langle\cdot,\cdot\rangle|_{\mathcal{H}}^{\frac{1}{2}}$
stands for the induced norm under this inner product. We consider the forward
backward ordinary differential equation

$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}\dfrac{dX}{dt}&=&\mathcal{A}X(t)-\mathcal{C}Y(t)+\mathcal{B}x(t)\\\
X(0)&=&0.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-\dfrac{dY}{dt}&=&\mathcal{A}^{*}Y(t)+\mathcal{Q}X(t)+\mathcal{S}x(t),\\\
Y(T)&=&\bar{\mathcal{Q}}X(T)+\bar{\mathcal{S}}x(T).\\\
\end{array}\right.\end{array}$ (25)

Observe that both $\mathcal{C}$ and $\mathcal{Q}$ are positive definite,
Equation (25) corresponds a well-defined (deterministic) control problem.
Hence, $x\mapsto X$ is well defined in $\mathcal{H}^{2}$. It suffices to show
that this mapping is indeed a contraction. Consider the inner product

$\dfrac{d}{dt}(X^{*}Y)=-Y^{*}(t)\mathcal{C}Y(t)+Y^{*}(t)\mathcal{B}x(t)-X^{*}(t)\mathcal{Q}X(t)-X^{*}(t)\mathcal{S}x(t).$

Taking integration on $[0,T]$ yields

$X^{*}(T)\bar{\mathcal{Q}}X(T)+\int_{0}^{T}X^{*}(t)\mathcal{Q}X(t)dt=X(T)^{*}\bar{\mathcal{S}}x(T)+\int_{0}^{T}-Y^{*}(t)\mathcal{C}Y(t)+Y^{*}(t)\mathcal{B}x(t)-X^{*}(t)\mathcal{S}x(t)dt.$

By the positivity of $\mathcal{C}$, and Cauchy-Schwarz inequality, we have

$\begin{array}[]{rcl}\|X\|_{\mathcal{H}}^{2}&\leq&\|\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}\bar{\mathcal{Q}}^{-\frac{1}{2}}\|\cdot\Big{(}X^{*}(T)\bar{\mathcal{Q}}^{\frac{1}{2}}\cdot\bar{\mathcal{Q}}^{\frac{1}{2}}x(T)\Big{)}\\\
&&\quad+\|\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}\mathcal{Q}^{-\frac{1}{2}}\|\cdot\displaystyle\int_{0}^{T}X^{*}(t)\mathcal{Q}^{\frac{1}{2}}\cdot\mathcal{Q}^{\frac{1}{2}}x(t)dt+\displaystyle\int_{0}^{T}Y^{*}(t)\mathcal{B}x(t)dt\\\
&\leq&N(S)\Big{(}\|X\|_{\mathcal{H}}\cdot\|x\|_{\mathcal{H}}\Big{)}+\displaystyle\int_{0}^{T}Y^{*}(t)\mathcal{B}x(t)dt.\\\
\end{array}$ (26)

On the other hand, for $t\in[0,T]$, we have

$\begin{split}Y(t)&=\phi^{*}(T,t)\Big{(}\bar{\mathcal{Q}}X(T)+\bar{\mathcal{S}}x(T)\Big{)}+\int_{t}^{T}\phi^{*}(s,t)\Big{(}\mathcal{Q}X(s)+\mathcal{S}x(s)\Big{)}ds\\\
&=\phi^{*}(T,t)\bar{\mathcal{Q}}^{\frac{1}{2}}\Big{(}\bar{\mathcal{Q}}^{\frac{1}{2}}X(T)+\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}x(T)\Big{)}+\int_{t}^{T}\phi^{*}(s,t)\mathcal{Q}^{\frac{1}{2}}\Big{(}\mathcal{Q}^{\frac{1}{2}}X(s)+\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}x(s)\Big{)}ds\\\
\end{split}$

which implies

$\sup_{t\in[0,T]}\|Y_{t}\|\leq\|\phi\|_{T}\Big{(}\|X\|_{\mathcal{H}}+N(S)\|x\|_{\mathcal{H}}\Big{)}.$
(27)

Combining Equations (26) and (27) yields

$\begin{array}[]{rcl}\|X\|_{\mathcal{H}}^{2}&\leq&N(S)\Big{(}\|X\|_{\mathcal{H}}\cdot\|x\|_{\mathcal{H}}\Big{)}+\sqrt{T}\|\phi\|_{T}\Big{(}\|X\|_{\mathcal{H}}+N(S)\|x\|_{\mathcal{H}}\Big{)}\|\mathcal{B}Q^{\frac{1}{2}}\|\|x\|_{\mathcal{H}}\end{array}$

which shows that $x\mapsto X$ is a contraction if

$\Big{(}1+\sqrt{T}\|\phi\|_{T}\cdot\|\mathcal{B}\mathcal{Q}^{-\frac{1}{2}}\|\Big{)}\Big{(}1+N(S)\Big{)}<2.$

∎

##  5 Conclusion

In this paper, by adopting adjoint equation approach, we provide the general
theory and discuss the necessary condition for optimal controls for both the
dominating player and the representative agent, and study the corresponding
fixed point problem in relation to the equilibrium condition. A convenient
necessary and sufficient condition has been provided under the Linear
Quadratic setting; in particular, a illuminative sufficient condition, which
only involves the coefficient of the mean field game system, for the unique
existence of the equilibrium control has been given. Finally, proof of the
convergence result of finite player game to mean field counterpart is provided
in Appendix. Applications of the present model in connection with central bank
lending and systematic risk in financial context will be provided in the
future work.

##  6 Acknowledgements

The first author-Alain Bensoussan acknowledges the financial support of the
Hong Kong RGC GRF 500113 and the National Science Foundation under grant DMS
1303775. The second author-Michael Chau acknowledges the financial support
from the Chinese University of Hong Kong, and the present work constitutes a
part of his work for his postgraduate dissertation. The third author-Phillip
Yam acknowledges the financial support from The Hong Kong RGC GRF 404012 with
the project title: Advanced Topics In Multivariate Risk Management In Finance
And Insurance, The Chinese University of Hong Kong Direct Grants 2010/2011
Project ID: 2060422 and 2011/2012 Project ID: 2060444. Phillip Yam also
expresses his sincere gratitude to the hospitality of both Hausdorff Center
for Mathematics and Hausdorff Research Institute for Mathematics of the
University of Bonn during the preparation of the present work.

##  Appendix A Appendix

###  A.1 $\epsilon$-Nash Equilibrium

We now establish that the solutions of Problems 2.1 and 2.2 is an
$\epsilon$-Nash Equilibrium. Suppose that there are $N$ representative agents
behaving in similar manner, so that the state of the dominating player and the
$i$-th agent satisfies the following SDE respectively:

$\begin{array}[]{l}\left\\{\begin{array}[]{rcl}dy_{0}&=&g_{0}\Big{(}y_{0}(t),\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{y_{1}^{j}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}y_{0}(t)\Big{)}dW_{0}(t),\\\
y_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rcl}dy_{1}^{i}&=&g_{1}\Big{(}y_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(t)},u_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}y_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
y_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.\end{array}$ (28)

where $\delta_{y}$ is Dirac measure with a unit mass at $y$. We call Equation
(28) the empirical system. The corresponding objective functional for the
$i$-th agent is:

$\begin{array}[]{rcl}\mathcal{J}^{N,i}({\bf
u})=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}y_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(t)},u^{i}(t)\Big{)}dt+h_{1}\Big{(}y_{1}^{i}(T),y_{0}(T),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(T)}\Big{)}\bigg{]},\end{array}$

where ${\bf u}=(u_{1}^{1},u_{1}^{2},\dots,u_{1}^{N})$. We expect that when
$N\rightarrow\infty$, the hypothetical approximation is described by (1), that
is:

$\begin{array}[]{l}\left\\{\begin{array}[]{rcl}dx_{0}&=&g_{0}\Big{(}x_{0}(t),m_{x_{0}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rcl}dx_{1}^{i}&=&g_{1}\Big{(}x_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
x_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.\end{array}$ (29)

We call Equation (29) the mean field system. The corresponding limiting
objective functional for the $i$-th player is

$\begin{array}[]{rcl}\mathcal{J}^{i}(u_{1}^{i})=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}x_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{i}(t)\Big{)}dt+h_{1}\Big{(}x_{1}^{i}(T),x_{0}(T),m_{x_{0}(T)}\Big{)}\bigg{]}.\end{array}$
(30)

Using Corollary 2.5, the necessary condition for optimality is described by
the SHJB-FP coupled equation (11). To proceed, we assume that the optimal
control $\hat{\bf u}=(\hat{u}_{1}^{1},\hat{u}_{1}^{2},\dots,\hat{u}_{1}^{N})$
exists. To avoid ambiguity, denote $\hat{x}_{1}^{i}$ and $\hat{y}_{1}^{i}$ the
states dynamics of $x_{1}^{i}$ and $y_{1}^{i}$ corresponding to the optimal
control $\hat{u}_{1}^{i}$. The mean field term $m_{x_{0}(t)}$, is the
probability measure of the optimal trajectory $\hat{x}_{1}^{i}$ at time $t$,
conditioning on $\mathcal{F}^{0}_{t}$. Under this construction, being
conditional on $\mathcal{F}^{0}_{t}$, $\\{\hat{x}_{1}^{i}\\}_{i}$ are
identical and independent processes; while $\\{\hat{y}_{1}^{i}\\}_{i}$ are
dependent on each other through the empirical distribution. For simplicity,
for two density functions $m$ and $m^{\prime}$, we write
$W_{2}(md\lambda,m^{\prime}d\lambda)=W_{2}(m,m^{\prime})$.

###### Lemma A.1

Suppose the assumptions (A.1-A.3) hold. If $m_{x_{0}(t)}$ is chosen to be the
density function of $\hat{x}_{1}^{i}$ conditional on $\mathcal{F}_{0}^{t}$,
then

$\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\Big{]}=\mathcal{O}(\dfrac{1}{N}).\\\
$

###### Proof

Observe that for any $t\in[0,T]$

$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}g_{0}\Big{(}y_{0}(s),\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},u_{0}(s)\Big{)}-g_{0}\Big{(}x_{0}(s),m_{x_{0}(s)},u_{0}(s)\Big{)}\bigg{|}^{2}ds\\\
&&\qquad\qquad+\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}\sigma_{0}\Big{(}y_{0}(s)\Big{)}-\sigma_{0}\Big{(}x_{0}(s)\Big{)}\bigg{|}^{2}ds\bigg{\\}},\end{array}$

and

$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}g_{1}\Big{(}\hat{y}_{1}^{i}(s),y_{0}(s),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\hat{u}_{1}^{i}(s)\Big{)}-g_{1}\Big{(}\hat{x}_{1}^{i}(s),x_{0}(s),m_{x_{0}(s)},\hat{u}_{1}^{i}(s)\Big{)}\bigg{|}^{2}ds\\\
&&\qquad\qquad+\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}\sigma_{1}\Big{(}\hat{y}_{1}^{i}(s)\Big{)}-\sigma_{1}\Big{(}\hat{x}_{1}^{i}(s)\Big{)}\bigg{|}^{2}ds\bigg{\\}},\end{array}$

By the Lipschitz assumptions, we have

$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}ds\bigg{\\}}\\\
&&\qquad+C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}+\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},m_{x_{0}(t)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{1}^{i}(s)-x_{1}^{i}(s)\Big{|}^{2}ds\bigg{\\}}\\\
&\leq&C\bigg{\\{}\mathbb{E}\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}\Big{|}y_{0}(u)-x_{0}(u)\Big{|}^{2}+\displaystyle\sup_{u\leq
s}\Big{|}\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)\Big{|}^{2}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds\bigg{\\}},\\\
\end{array}$ (31)

where $C>0$ is a constant, changing line by line, depends only on $T$ and $K$.
By definition, for any Dirac measures $\delta_{y}$ on $\mathbb{R}^{n_{1}}$ and
density function $m$, we have

$W_{2}^{2}(\delta_{y},m)=\int_{\mathbb{R}^{n_{1}}}|y-x|^{2}dm(x).$

Also observe that the joint measure
$\frac{1}{N}\sum_{j=1}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{x}_{1}^{j}(s))}$ on
${\mathbb{R}^{n_{1}}\times{\mathbb{R}^{n_{1}}}}$ has respective marginals
$\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)}$ and
$\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}$ on
${\mathbb{R}^{n_{1}}}$. Using the definition of Wasserstein metric, we
evaluate

$\begin{array}[]{rcl}\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\bigg{]}&\leq&\mathbb{E}\bigg{[}\displaystyle\int_{\mathbb{R}^{n_{1}}\times{\mathbb{R}^{n_{1}}}}|y-x|^{2}d\bigg{(}\dfrac{1}{N}\sum_{j=1}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{x}_{1}^{j}(s))}(y,x)\bigg{)}\bigg{]}\\\
&\leq&\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\mathbb{E}\Big{|}\hat{y}_{1}^{j}(s)-\hat{x}_{1}^{j}(s)\Big{|}^{2}\\\
&=&\mathbb{E}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2},\end{array}$
(32)

where the last equality results from the fact that
$\\{\hat{y}^{j}-\hat{x}^{j}\\}_{j=1}^{N}$ are symmetric. Similarly, we also
have

$\begin{array}[]{rcl}\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\bigg{]}&\leq&\mathbb{E}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}.\end{array}$

Combining with (31) and applying Gronwall’s inequality, we have

$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&Ce^{Ct}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{t}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds\bigg{]}.\\\
\end{array}$ (33)

By definition of the Wasserstein metric, we have

$\begin{array}[]{rcl}&&W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&=&\displaystyle\inf_{\Gamma}\displaystyle\int_{\mathbb{R}^{n_{1}}}\displaystyle\int_{\mathbb{R}^{n_{1}}}|x-y|^{2}d\Gamma_{\Big{(}\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}}(x,y)\\\
&\leq&\displaystyle\int_{\mathbb{R}^{n_{1}}}\displaystyle\int_{\mathbb{R}^{n_{1}}}|x|^{2}-2x\cdot
y+|y|^{2}d\Big{(}\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}(x)dm_{x_{0}(s)}(y)\\\
&=&\Big{|}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)\Big{|}^{2}-2\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)\Big{)}\cdot\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)+\mathbb{E}^{\mathcal{F}^{0}_{s}}|\hat{x}_{1}^{j}(s)|^{2}\\\
&=&\Big{|}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}\\\
\end{array}$

Hence,

$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}\\\
&\leq&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+2\displaystyle\sum_{j<k}^{N}[\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)]\cdot[\hat{x}_{1}^{k}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)]\bigg{]}\\\
\end{array}$

Recall that given $\mathcal{F}^{0}_{t}$, $\\{\hat{x}_{1}^{j}\\}_{j}$ are
identically and independently distributed, we thus get

$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}\\\
&\leq&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+2\displaystyle\sum_{j<k}^{N}[\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)]\cdot[\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)]\bigg{]}\\\
&=&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+0\bigg{]}\\\
&=&\dfrac{1}{N}\mathbb{E}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}.\\\
\end{array}$

Similar estimate applies on the second term in Equation (33). Put $t=T$, we
finally have

$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&\dfrac{Ce^{CT}}{N}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}ds\bigg{]}\\\
&\leq&\dfrac{4Ce^{CT}}{N}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{|}\hat{x}_{1}^{j}(s)\Big{|}^{2}ds\bigg{]}\end{array}$
(34)

With the linear growth assumptions, for any $t\in[0,T]$, we easily get the
estimates

$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq t}|x_{0}(t)|^{2}\leq
C\mathbb{E}\bigg{\\{}|\xi_{0}|^{2}+\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}|x_{0}(u)|^{2}+\mathbb{E}^{\mathcal{F}_{s}^{0}}|\hat{x}_{1}^{j}(s)|^{2}+|u_{0}(s)|^{2}ds\bigg{\\}}\end{array}$

and

$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{x}_{1}^{j}(t)|^{2}\leq
C\mathbb{E}\bigg{\\{}|\xi_{1}^{j}|^{2}+\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}|\hat{x}_{1}^{j}(u)|^{2}+\displaystyle\sup_{u\leq
s}|x_{0}(u)|^{2}+\mathbb{E}^{\mathcal{F}_{s}^{0}}|\hat{x}_{1}^{j}(s)|^{2}+|\hat{u}_{1}^{j}(s)|^{2}ds\bigg{\\}}\end{array}$

Applying the Tower property of expectation and the Gronwall’s inequality on
the sum of the two inequalities above yields

$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|x_{0}(t)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{x}_{1}^{j}(t)|^{2}\leq
Ce^{Ct}\mathbb{E}\bigg{\\{}|\xi_{0}|^{2}+|\xi_{1}^{j}|^{2}+\displaystyle\int_{0}^{t}|\hat{u}_{1}^{j}(s)|^{2}+|u_{0}(s)|^{2}ds\bigg{\\}}<\infty\end{array}$

We have the order for the estimate (34)

$\mathbb{E}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}=\mathcal{O}(\dfrac{1}{N}).$ (35)

∎

We also have approximation for the cost functionals.

###### Lemma A.2

$\begin{array}[]{rcl}&&\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}_{1}^{i})=O(\dfrac{1}{\sqrt{N}}).\\\ \end{array}$

###### Proof

With the quadratic assumptions (4), we have

$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}\hat{y}_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},\hat{u}^{i}(t)\Big{)}-f_{1}\Big{(}\hat{x}_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},\hat{u}_{1}^{i}(t)\Big{)}dt\\\
&&\qquad\qquad+h_{1}\Big{(}\hat{y}_{1}^{i}(T),y_{0}(T),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)}\Big{)}-h_{1}\Big{(}\hat{x}_{1}^{i}(T),\hat{x}_{0}(T),m_{x_{0}(T)}\Big{)}\bigg{]}\\\
&\leq&C\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|+|\hat{x}_{1}^{i}(t)|+|y_{0}(t)|+|x_{0}(t)|+\Big{(}\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(t)|^{2}}{N-1}\Big{)}^{\frac{1}{2}}+\Big{(}\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(t)|^{2}\Big{)}^{\frac{1}{2}}+2|\hat{u}^{i}(t)|\Big{]}\\\
&&\qquad\qquad\qquad\cdot\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|+|y_{0}(t)-x_{0}(t)|+W_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\\\
&&\qquad+\Big{[}1+|\hat{y}_{1}^{i}(T)|+|\hat{x}_{1}^{i}(T)|+|y_{0}(T)|+|x_{0}(T)|+\Big{(}\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(T)|^{2}}{N-1}\Big{)}^{\frac{1}{2}}+\Big{(}\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(T)|^{2}\Big{)}^{\frac{1}{2}}\Big{]}\\\
&&\qquad\qquad\qquad\cdot\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|+|y_{0}(T)-x_{0}(T)|+W_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}.\\\
\end{array}$

An application of Hölder’s inequality, and the symmetry on
$\\{\hat{x}_{1}^{j}\\}_{j}$ gives

$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(t)|^{2}}{N-1}+\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}+\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(T)|^{2}}{N-1}+\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(T)|^{2}\Big{]}\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}}\\\
&=&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}\Big{]}\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}}\\\
\end{array}$

By the linear growth assumptions on $g_{0}$, $\sigma_{0}$, $g_{1}$ and
$\sigma_{1}$, it is easy to show that

$\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt$

and

$\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}\Big{]}$

are bounded (independent of $N$). We finally arrive at the estimates

$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}},\\\
\end{array}$

which goes to $0$ as $N\rightarrow\infty$, as shown in Lemma A.1. Hence

$|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}_{1}^{i})|=O(\dfrac{1}{\sqrt{N}}).$

∎

In the previous lemmas, we assumed that all players adopt their corresponding
mean field optimal controls. By symmetry, the convergences of state dynamics
and the cost functionals are then established. To show that the mean field
optimal controls ${\bf u}$ indeed constitute a $\epsilon$-Nash equilibrium on
the empirical system, without loss of generality, we assume that the first
player did not obey the mean field optimal control. In particular, let
$u_{1}^{1}$ be an arbitrary control in $\mathcal{A}_{1}$, define ${\bf
u}:=(u_{1}^{1},\hat{u}_{1}^{2},\dots,\hat{u}_{1}^{N})$. We then have the
following empirical and mean field SDEs for the dominating player, the $1$-st
player and the $i$-th player ($i>1$) respectively:

$\left\\{\begin{array}[]{rcl}dy_{0}&=&g_{0}\Big{(}y_{0}(t),\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(t)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(t)}\Big{)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}y_{0}(t)\Big{)}dW_{0}(t),\\\
y_{0}(0)&=&\xi_{0}.\\\
dx_{0}&=&g_{0}\Big{(}x_{0}(t),m_{x_{0}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.$ (36)
$\left\\{\begin{array}[]{rcl}dy_{1}^{1}&=&g_{1}\Big{(}y_{1}^{1}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(t)},u_{1}^{1}(t)\Big{)}dt+\sigma_{1}\Big{(}y_{1}^{1}(t)\Big{)}dW_{1}^{1}(t),\\\
y_{1}^{1}(0)&=&\xi_{1}^{1}.\\\
dx_{1}^{1}&=&g_{1}\Big{(}x_{1}^{1}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{1}(t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}^{1}(t)\Big{)}dW_{1}^{1}(t),\\\
x_{1}^{1}(0)&=&\xi_{1}^{1}.\\\ \end{array}\right.$ (37)
$\left\\{\begin{array}[]{rcl}d\hat{y}_{1}^{i}&=&g_{1}\Big{(}\hat{y}_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(t)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)}\Big{)},\hat{u}_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}\hat{y}_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
\hat{y}_{1}^{i}(0)&=&\xi_{1}^{i}.\\\
d\hat{x}_{1}^{i}&=&g_{1}\Big{(}\hat{x}_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},\hat{u}_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}\hat{x}_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
\hat{x}_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.$ (38)

We claim that if $m_{x_{0}}$ is the density function of $\hat{x}_{1}^{i}$
conditioning on $\mathcal{F}^{0}$, then we have the convergence
$y_{0}\rightarrow x_{0}$, $y_{1}^{1}\rightarrow x_{1}^{1}$ and
$\hat{y}_{1}^{i}\rightarrow\hat{x}_{1}^{i}$ in the sense of the following
lemma

###### Lemma A.3

$\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{1}^{1}(u)-x_{1}^{1}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\Big{]}\rightarrow 0,\quad\text{
as }N\rightarrow\infty.\\\ $

###### Proof

We first show the convergence of the dominating player and the $i$-th player.
Similar to the proof of Lemma A.1, we first have

$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}\bigg{\\}}ds\\\
&&\qquad+C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}+\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}\bigg{\\}}ds\\\
&\leq&C\mathbb{E}\displaystyle\int_{0}^{t}\bigg{[}\displaystyle\sup_{u\leq
s}\Big{|}y_{0}(u)-x_{0}(u)\Big{|}^{2}+\displaystyle\sup_{u\leq
s}\Big{|}\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)\Big{|}^{2}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}(\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)})\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}ds\\\
\end{array}$ (39)

By the same argument used in Equation (32) in Lemma A.1, we have

$\mathbb{E}\Big{[}W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\Big{]}+\mathbb{E}\Big{[}W_{2}^{2}(\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)})\Big{]}\leq
2\mathbb{E}|\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}|^{2}.$

Hence, by applying Gronwall’s inequality on Equation (39), we have

$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&Ce^{Ct}\mathbb{E}\displaystyle\int_{0}^{t}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}ds\\\
\end{array}$ (40)

For the first term in (40), consider the following joint measure on
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{1}}$

$\begin{split}\mu(x,y)&=\dfrac{1}{N}\displaystyle\sum_{j=2}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{y}_{1}^{j}(s))}(x,y)+\dfrac{1}{N(N-1)}\displaystyle\sum_{j=2}^{N}\delta_{(y_{1}^{1}(s),\hat{y}_{1}^{j}(s))}(x,y),\\\
\end{split}$

which has respective marginals

$\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}\quad\text{and}\quad\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}.$

By the definition of Wasserstein metric,

$\begin{array}[]{rcl}&&\mathbb{E}\Big{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}\Big{]}\\\
&\leq&\mathbb{E}\Big{[}\displaystyle\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{1}}}|x-y|^{2}d\mu(x,y)\Big{]}\\\
&=&\mathbb{E}\Big{[}\dfrac{1}{N(N-1)}\displaystyle\sum_{j=2}^{N}|y_{1}^{1}(s)-\hat{y}_{1}^{j}(s)|^{2}\Big{]}\\\
&=&\dfrac{1}{N}\mathbb{E}|y_{1}^{1}(s)-\hat{y}_{1}^{2}(s)|^{2},\end{array}$

where the last equality results from symmetry on $\\{y_{1}^{j}\\}_{j}$,
clearly goes to $0$ as $N\rightarrow\infty$. Similar argument applies for the
third term in (40). For the convergence of the second and the forth term, we
refer to the argument in the last part of Lemma A.1 and the results follow.

For the convergence of the $1$-st player, the procedure are similar and we do
not provide here. ∎

We conclude from the similar procedures to show the convergence of the cost
functional. In particular, we have

$|\mathcal{J}^{N,1}({\bf
u})-\mathcal{J}^{1}(u_{1}^{1})|=O(\dfrac{1}{\sqrt{N}}).$

###### Theorem A.4

$\hat{\bf u}$ is an $\epsilon$-Nash equilibrium.

###### Proof

Summarizing all the obtained results in this section, we can conclude

$\begin{array}[]{rcl}&&|\mathcal{J}^{N,1}(\hat{\bf
u})-\mathcal{J}^{1}(\hat{u}_{1}^{1})|=O(\dfrac{1}{\sqrt{N}});\\\
&&|\mathcal{J}^{N,1}({\bf
u})-\mathcal{J}^{1}(u_{1}^{1})|=O(\dfrac{1}{\sqrt{N}}).\\\ \end{array}$

Since $\hat{u}_{1}^{1}$ is optimal control, we have
$\mathcal{J}^{1}(\hat{u}_{1}^{1})\leq\mathcal{J}^{1}(u_{1}^{1})$. We deduce

$\mathcal{J}^{N,i}(\hat{\bf u})\leq\mathcal{J}^{N,1}({\bf
u})+O(\dfrac{1}{\sqrt{N}}).$

Hence, $\hat{\bf u}$ is an $\epsilon$-Nash equilibrium. ∎

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