# Equilibrium pricing of securities in the co-presence of
cooperative and non-cooperative populations 111Forthcoming in ESAIM: Control,
Optimisation and Calculus of Variations.

Masaaki Fujii222Quantitative Finance Course, Graduate School of Economics, The
University of Tokyo.

( The first version: 27 September, 2022
This version: 21 June, 2023. )

###### Abstract

In this work, we develop an equilibrium model for price formation of
securities in a market composed of two populations of different types: the
first one consists of cooperative agents, while the other one consists of non-
cooperative agents. The trading of every cooperative member is assumed to be
coordinated by a central planner. In the large population limit, the problem
for the central planner is shown to be a conditional extended mean-field
control. In addition to the convexity assumptions, if the relative size of the
cooperative population is small enough, then we are able to show the existence
of a unique equilibrium for both the finite-agent and the mean-field models.
The strong convergence to the mean-field model is also proved under the same
conditions.

Keywords : market clearing, mean-field games, extended mean-field control,
controlled-FBSDEs

##  1 Introduction

In this paper, we study the problem of equilibrium price formation by
considering, as an example, a simple model of a securities market. The prices
of securities have always been the center of interest of many people for
centuries. They inspire people to consider their properties not only for
speculative reasons but also for true academic interests, such as how they are
determined, how they behave in a given environment, how they are influenced by
preferences and the level of risk-averseness of investors, and so on. In fact,
since the appearance of modern securities markets, the huge amount of price
data they provide has induced the significant developments of related academic
areas, such as financial mathematics, financial economics, statistical
inference etc. However, despite its importance, we are still quite far from
the complete understanding of a price process even when we restrict our
attention to a standard liquid stock. The main reason behind this difficulty
is that any price process is inevitably affected by complicated interactions
among a large number of market participants, which includes various financial
firms, individual investors, governments and even supernational institutions.

The recent progress in the mean field game (MFG) theory has started to shed
new light on the long-standing problem of multi-agent games. The MFG theory
was pioneered by the seminal works of Lasry & Lions [43, 44, 45] and Huang et
al. [38, 39, 40], which characterizes a Nash equilibrium by a coupled system
of Hamilton-Jacobi-Bellman and Kolmogorov equations. A probabilistic approach
to the MFG theory based on forward backward stochastic differential equations
(FBSDEs) of McKean-Vlasov type was established by Carmona & Delarue [12, 13].
The two volumes of [14, 15] by the same authors provide readers a detailed
account of the topic. The method using the concept of relaxed controls (see,
for example, [41, 42]), which can significantly weaken the regularity
assumptions, has also been intensively studied. For interested readers, there
are an excellent lecture note [11] and many monographs such as [10, 33, 34] on
various topics regarding the MFG theory.

Along side of its theoretical developments, MFGs have successfully found vast
applications in various areas. In particular, energy and financial markets
have been popular targets for analysis. As is well known, any price in these
free competitive markets is determined so that it balances the supply and
demand of the asset. Unfortunately, this market clearing condition does not
fit well to the concept of Nash equilibrium. In fact, if we shift a control of
one agent away from her equilibrium strategy while keeping the others
unchanged, then the market clearing condition would be broken down. Since the
MFG theory has been developed primarily for the analysis of Nash equilibrium,
a standard approach in the literature assumes a specific structure of the
price process without imposing the market clearing condition. A popular way is
to hypothesize that the price process is decomposed into two parts; one is a
so-called fundamental price whose dynamics is exogenously given and the other
is a price-impact term whose size is affected by actions of agents in a
specific manner. Another way is to suppose that a price is given by a certain
function of total demand or supply of the asset. One can find many interesting
examples on optimal executions [26, 27, 21], games in a smart grid [6, 17],
price formation in an electricity market [23, 24], entry and exit games [4],
trading with different beliefs [16], and trading of exhaustible commodities
[37], just to name a few. There are also macroeconomic applications of MFGs,
for examples, on growth, inequality and unemployment, and so on [3, 5, 9, 31].

In the above examples, the structure of a price process is assumed exogenously
and the emphasis of research is not on the price process itself. In this
paper, on the other hand, our primary interest is in the construction of a
price process endogenously from the market clearing condition. There has also
been recent progress in the field of MFGs regarding this direction of
research. Gomes & Saúde [35] present a deterministic model of electricity
price formation. Different mathematical approaches and numerical calculation
techniques on the same model are developed by Ashrafyan et al. [7, 8]. Its
generalization with a random supply function is provided by Gomes et al. [32].
Evangelista et al. [22] use the actual high-frequency data to give a promising
numerical results for the listed stocks in several exchanges. Shrivats et al.
[54] solve the price formation problem of solar renewable energy certificate
(SREC), and Firoozi et al. [25] deal with a principal-agent problem in the
associated emission market. Fujii & Takahashi [28] solve a stochastic mean-
field price model of securities and prove that the market is cleared in the
large population limit. In [29], the same authors provide the FBSDE
characterization of the market clearing price in the finite-agent market, and
then prove its strong convergence to the mean-field model of [28]. Its
extension to the presence of a major agent is studied in [30].

In this paper, we build upon the previous works [28, 29, 30] and develop a new
model of a securities market composed of two populations of different types:
the first one consists of cooperative agents, while the other one consists of
non-cooperative agents. Every agent is supposed to be a registered financial
firm trying to minimize her cost functional by the continuous trading at the
common exchange. What is new in this paper is the existence of the cooperative
population, which can be a large number of subsidiaries of a giant financial
group or a temporary alliance of many financial firms for a certain purpose,
such as liquidation of assets of a defaulted financial firm. In addition to
the pure academic interests, understanding the market with cooperative agents
is of critical importance for financial regulators as well as policy makers.
The trading of every cooperative member is assumed to be coordinated by a
central planner. We show that, in the mean field limit, this cooperation makes
the problem of the central planner an extended mean-field control [2] with
common noise, where the law of the controls of the cooperative members enters
her cost functional. Here, let us refer to some works on extended mean-field
control problems: Graber [36] studies a linear-quadratic model for production
of a exhaustible resource; Acciaio et al. [2] present the necessary and
sufficient conditions of Pontryagin’s maximum principle; Nie & Yan [47] study
the problem under the partial observation; Motte & Pham [46] investigate a
mean-field Markov decision problem; Djete et al. [18, 19] and Djete [20]
provide a dynamic programing principle as well as the limit theory by making
use of the relaxed control technique and its generalization.

To the author’s knowledge, this is the first work studying the equilibrium
price formation in the co-presence of cooperative and non-cooperative agents.
Using a general convex cost function for the cooperative agents, we show that
there exists a unique market clearing equilibrium for a given duration if the
relative size of the cooperative population is small enough. After solving the
problem in the finite-agent market, we construct the corresponding MFG model
and finally prove the strong convergence when the large population limit is
taken. We want to emphasize that, although we have chosen a model of a
securities market, the developed methods (with appropriate modifications) will
also be useful to describe other markets with cooperative producers and/or
buyers, for example, the crude oil market with OPEC a cooperative association
of exporters. In addition to the financial economics perspective, the current
work also contributes to the MFG theory in two ways. Firstly, in addition to
the sufficiency condition, we show the existence of a unique optimal solution
to an extended mean-field control problem with common noise. The existence
result using the adjoint equations for this type of problem is new except
those obtained in the linear-quadratic settings [2, 36]. Moreover, due to the
co-presence of non-cooperative agents, the problem has a non-standard form of
controlled McKean-Vlasov FBSDEs instead of controlled McKean-Vlasov SDEs.
Next, although the general convergence results for extended mean-field control
problems are already studied by [19, 20], our proof is directly built upon the
result for the standard (not extended) mean-field control problem by Carmona &
Delarue [13] and the monotonicity technique by Fujii & Takahashi [29]. In
return for our strong convexity assumptions, the proof is done in a
straightforward manner.

The organization of the paper is as follows. After explaining some notations
in Section 2, we solve the finite-agent market model in Section 3. The
corresponding mean-field model is built and solved in Section 4. Finally, in
Section 5, we prove the strong convergence to the mean-field limit. Section 6
concludes the paper.

##  2 Notations

We first introduce $(1+N_{1}+N_{2})$ complete probability spaces.

  * •

$(\Omega^{0},{\cal F}^{0},\mathbb{P}^{0})$ is a complete probability space
with a complete and right-continuous filtration $\mathbb{F}^{0}:=({\cal
F}_{t}^{0})_{t\geq 0}$ generated by $d_{0}$-dimensional standard Brownian
motion $W^{0}:=(W_{t}^{0})_{t\geq 0}$.

  * •

For each $1\leq i\leq N_{1}$, $(\overline{\Omega}^{1,i},\overline{{\cal
F}}^{1,i},\overline{\mathbb{P}}^{1,i})$ is a complete probability space with a
complete and right-continuous filtration
$\overline{\mathbb{F}}^{1,i}:=(\overline{{\cal F}}_{t}^{1,i})_{t\geq 0}$
generated by $d_{1}$-dimensional standard Brownian motion
$W^{i}:=(W^{i}_{t})_{t\geq 0}$ and a $\sigma$-algebra $\sigma(\xi^{i})$
belonging to $\overline{{\cal F}}^{1,i}_{0}$, generated by a square integrable
$\mathbb{R}^{n}$-valued random variable $\xi^{i}$. The distribution of
$\xi^{i}$ is the same for every $i$.

  * •

For each $1\leq j\leq N_{2}$, $(\overline{\Omega}^{2,j},\overline{{\cal
F}}^{2,j},\overline{\mathbb{P}}^{2,j})$ is a complete probability space with a
complete and right-continuous filtration
$\overline{\mathbb{F}}^{2,j}:=(\overline{{\cal F}}_{t}^{2,j})_{t\geq 0}$
generated by $d_{2}$-dimensional standard Brownian motion
$B^{j}:=(B_{t}^{j})_{t\geq 0}$ and a $\sigma$-algebra $\sigma(\eta^{j})$
belonging to $\overline{{\cal F}}_{0}^{2,j}$, generated by a square integrable
$\mathbb{R}^{n}$-valued random variable $\eta^{j}$. The distribution of
$\eta^{j}$ is the same for every $j$.

We then introduce probability spaces defined by the Cartesian products of
those defined above.

  * •

$(\overline{\Omega},\overline{{\cal F}},\overline{\mathbb{P}})$ is a
probability space defined by the product
$\overline{\Omega}:=\overline{\Omega}^{1,1}\times\ldots\times\overline{\Omega}^{1,N_{1}}\times\overline{\Omega}^{2,1}\times\ldots\times\overline{\Omega}^{2,N_{2}}$
with $(\overline{{\cal F}},\overline{\mathbb{P}})$ the completion of
$(\otimes_{i=1}^{N_{1}}\overline{{\cal
F}}^{1,i}\otimes_{j=1}^{N_{2}}\overline{{\cal
F}}^{2,j},~{}\otimes_{i=1}^{N_{1}}\overline{\mathbb{P}}^{1,i}\otimes_{j=1}^{N_{2}}\overline{\mathbb{P}}^{2,j})$.
$\overline{\mathbb{F}}:=(\overline{{\cal F}}_{t})_{t\geq 0}$ denotes the
complete and right-continuous augmentation of
$(\otimes_{i=1}^{N_{1}}\overline{{\cal
F}}^{1,i}_{t}\otimes_{j=1}^{N_{2}}\overline{{\cal F}}^{2,j}_{t})_{t\geq 0}$.

  * •

$(\Omega,{\cal F},\mathbb{P})$ is a probability space defined by the product
$\Omega:=\Omega^{0}\times\overline{\Omega}$ with $({\cal F},\mathbb{P})$ the
completion of $({\cal F}^{0}\otimes\overline{{\cal
F}},~{}\mathbb{P}^{0}\otimes\overline{\mathbb{P}})$. $\mathbb{F}:=({\cal
F}_{t})_{t\geq 0}$ denotes the complete and right-continuous augmentation of
$({\cal F}_{t}^{0}\otimes\overline{{\cal F}}_{t})_{t\geq 0}$. The expectation
with respect to $\mathbb{P}$ is denoted by $\mathbb{E}[\cdot]$, and a generic
element of $\Omega$ is denoted by $\omega=(\omega^{0},\overline{\omega})$ with
$\omega^{0}\in\Omega^{0}$ and $\overline{\omega}\in\overline{\Omega}$.

For notational simplicity, we do not distinguish a random variable $X$ defined
on a marginal probability space, such as $(\overline{\Omega},\overline{{\cal
F}},\overline{\mathbb{P}})$, with its natural extension to $(\Omega,{\cal
F},\mathbb{P})$. Without loss of generality, we may always suppose that the
probability spaces are the completions of those countably generated by working
on the Borel sets of appropriate Polish spaces.

Throughout the work, $T~{}(>0)$ is a given time horizon, the symbol $L$
denotes a given positive constant, and the symbol $C$ a general positive
constant which may change line by line. For any measurable space
$(\Omega,{\cal G})$ with filtration $\mathbb{G}:=({\cal G}_{t})_{t\geq 0}$, we
use the following notations for frequently encountered spaces:

  * •

${\cal S}^{n}_{+}$ denotes the space of $n\times n$ strictly positive definite
symmetric matrices.

  * •

$\mathbb{L}^{2}({\cal G};\mathbb{R}^{d})$ denotes the set of
$\mathbb{R}^{d}$-valued ${\cal G}$-measurable random variables $X$ satisfying

$\begin{split}\|X\|_{\mathbb{L}^{2}}:=\mathbb{E}[|X|^{2}]^{\frac{1}{2}}<\infty.\end{split}$

  * •

$\mathbb{S}^{2}(\mathbb{G};\mathbb{R}^{d})$ is the set of
$\mathbb{R}^{d}$-valued $\mathbb{G}$-adapted continuous processes $X$
satisfying

$\small\begin{split}\|X\|_{\mathbb{S}^{2}}:=\mathbb{E}\bigl{[}\sup_{t\in[0,T]}|X_{t}|^{2}\bigr{]}^{\frac{1}{2}}<\infty.\end{split}$

  * •

$\mathbb{H}^{2}(\mathbb{G};\mathbb{R}^{d})$ is the set of
$\mathbb{R}^{d}$-valued $\mathbb{G}$-progressively measurable processes $Z$
satisfying

$\small\begin{split}\|Z\|_{\mathbb{H}^{2}}:=\mathbb{E}\Bigl{[}\Bigl{(}\int_{0}^{T}|Z_{t}|^{2}dt\Bigr{)}\Bigr{]}^{\frac{1}{2}}<\infty.\end{split}$

  * •

${\cal L}(X)$ denotes the law of a random variable $X$.

  * •

${\cal P}(\mathbb{R}^{d})$ is the set of probability measures on
$(\mathbb{R}^{d},{\cal B}(\mathbb{R}^{d}))$.

  * •

${\cal P}_{p}(\mathbb{R}^{d})$ with $p\geq 1$ is the subset of ${\cal
P}(\mathbb{R}^{d})$ with finite $p$th moment, i.e. the set of $\mu\in{\cal
P}(\mathbb{R}^{d})$ satisfying

$\small\begin{split}M_{p}(\mu):=\Bigl{(}\int_{\mathbb{R}^{d}}|x|^{p}\mu(dx)\Bigr{)}^{\frac{1}{p}}<\infty.\end{split}$

We always assign ${\cal P}_{p}(\mathbb{R}^{d})$ the $p$-Wasserstein distance
$W_{p}$, which makes ${\cal P}_{p}(\mathbb{R}^{d})$ a Polish space. It is
defined by, for any $\mu,\nu\in{\cal P}_{p}(\mathbb{R}^{d})$,

$\small\begin{split}W_{p}(\mu,\nu):=\inf_{\pi\in\Pi_{p}(\mu,\nu)}\Bigl{[}\Bigl{(}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{p}\pi(dx,dy)\Bigr{)}^{\frac{1}{p}}\Bigr{]},\end{split}$

whre $\Pi_{p}(\mu,\nu)\subset{\cal P}_{p}(\mathbb{R}^{d}\times\mathbb{R}^{d})$
denotes the set of probability measures with marginals $\mu$ and $\nu$.

  * •

For $k=1,2$, the empirical mean of $N_{k}$ variables $(x^{i})_{i=1}^{N_{k}}$
is denoted by $\mathfrak{m}_{k}(x):=\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}x^{i}.$

We often omit the arguments such as $(\mathbb{G},\mathbb{R}^{d})$ when they
are clear from the context.

##  3 Price formation in a finite-agent market

In this section, we consider the price formation problem in a finite-agent
market composed of two populations: the first one consists of $N_{1}$
cooperative agents, and the second one consists of $N_{2}$ non-cooperative
agents. We often use $i$ (respectively, $j$) to index an agent in the first
(respectively, second) population. Each agent represents a registered
financial firm trading at a common securities exchange. She is supposed to
have many individual clients who cannot directly access to the exchange, and
she provides financial services by using her own inventory to answer the sale
and purchase requests from those clients. This is the so-called over-the-
counter (OTC) market. Therefore, every agent faces the stochastic order flows
from her OTC clients in addition to idiosyncratic as well as common market
shocks. Under such an environment, every agent carries out the optimal
continuous trading at the exchange to minimize her cost functional. For the
members of the first population, we assume that every trading is coordinated
by a central planner.

We suppose that there are $n$ tradable securities in the market, whose prices
are denoted by $(\varpi_{t}^{k})_{k=1}^{n}$. The goal of the paper is to
determine the price process $\varpi:=(\varpi_{t})_{t\in[0,T]}$ endogenously so
that it clears the market, i.e. matches the supply and demand of securities at
the exchange. Unless otherwise stated, we will work on the filtered
probability space $(\Omega,{\cal F},\mathbb{F},\mathbb{P})$ defined in the
last section. By construction, $(\xi^{i})_{i=1}^{N_{1}}$ and also
$(\eta^{j})_{j=1}^{N_{2}}$ are independently and identically distributed
(i.i.d.). Let us introduce the following processes to describe the various
types of information:

  * •

$c^{0}:=(c_{t}^{0})_{t\in[0,T]}$ belongs to
$\mathbb{H}^{2}(\mathbb{F}^{0};\mathbb{R}^{m})$ with
$\sup_{t\in[0,T]}\mathbb{E}[|c_{t}^{0}|^{2}]<\infty$, which is used to
represent common market information including the cash flows from the
securities.

  * •

For $1\leq i\leq N_{1}$, $c^{i}:=(c_{t}^{i})_{t\in[0,T]}$ belongs to
$\mathbb{H}^{2}(\overline{\mathbb{F}}^{1,i};\mathbb{R}^{m})$ with
$\sup_{t\in[0,T]}\mathbb{E}[|c_{t}^{i}|^{2}]<\infty$. They are used to
represent idiosyncratic shocks for the first population, and
$(c^{i})_{i=1}^{N_{1}}$ are i.i.d.

  * •

For $1\leq j\leq N_{2}$,
$\mathfrak{c}^{j}:=(\mathfrak{c}_{t}^{j})_{t\in[0,T]}$ belongs to
$\mathbb{H}^{2}(\overline{\mathbb{F}}^{2,j};\mathbb{R}^{m})$ with
$\sup_{t\in[0,T]}\mathbb{E}[|\mathfrak{c}_{t}^{j}|^{2}]<\infty$. They are used
to represent idiosyncratic shocks for the second population, and
$(\mathfrak{c}^{j})_{j=1}^{N_{2}}$ are i.i.d.

Here, the dimension $m\in\mathbb{N}$ is commonly chosen just for notational
simplicity.

Before going to study the details of the optimization problem faced by each
population separately, let us summarize the important features of each problem
here.

  * •

The optimization problem for the second population $1\leq j\leq N_{2}$:

    * –

Each agent $j$ tries to minimize her cost functional by continuous trading at
the exchange (see $(\ref{non-c-problem})$).

    * –

Her cost functional (see $(\ref{2nd-cost-functional})$) depends on her trading
speed, her inventory size, the price of securities as well as the shocks
induced by $(c^{0},\mathfrak{c}^{j})$.

    * –

She is just a price taker (see Definition 3.1) and hence treats the price of
securities $\varpi$ as an externally given stochastic process.

    * –

Each agent $1\leq j\leq N_{2}$ solves her own problem independently. In
particular, she does not care about the actions of the other agents.

  * •

The optimization problem for the first population $1\leq i\leq N_{1}$:

    * –

The cost functional (see $(\ref{1st-cost-functional})$) of each agent $i$
depends on her trading speed, her inventory size, the empirical distribution
of the inventories for the first population, the price of securities as wall
as the shocks induced by $(c^{0},c^{i})$.

    * –

They try to minimize the sum of the cost functionals of the whole population
$1\leq i\leq N_{1}$ by continuous trading at the exchange, which is fully
coordinated by a central planner (see $(\ref{central-problem})$).

    * –

The first population as a whole has non-negligible market share and hence the
actions of the central planner can influence the price of securities $\varpi$
(see $(\ref{vp-expression-1})$), which in turn affects the actions of the
second population.

    * –

Assuming the agents in the second population rationally react to the price
change, the central planner solves the cost minimization problem with those
reactions taken into account.

It is immediate to see that the problem of each agent in the second population
is a usual stand-alone optimization. Hence it makes perfect sense to start our
investigation from solving this problem. The optimization problem for the
central planner of the first population becomes much more complicated. We will
see that the market clearing condition (see Definition 3.2) connects her
trading actions and the price impacts. Through the reactions to the price
change, the central planner must control not only the actions of the first
population but also those of the second population indirectly. This results in
stochastic control of a large system of FBSDEs instead of an SDE.

###  3.1 Problem for the non-cooperative agents

####  3.1.1 Problem description

We first study the problem for the non-cooperative agents in the second
population. Here, we basically follow the technique developed in [28, 29]. Let
us introduce three $\mathbb{F}^{0}$-progressively measurable bounded
processes, $b:=(b_{t})_{t\in[0,T]}$, $\rho_{2}:=(\rho_{2}(t))_{t\in[0,T]}$ and
$\Lambda=(\Lambda_{t})_{t\in[0,T]}$, where $b_{t}$ and $\rho_{2}(t)$ are non-
negative, and $\Lambda_{t}$ is ${\cal S}^{n}_{+}$-valued. We introduce the
following measurable functions;
$l_{2}:[0,T]\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}^{n}$,
$(\sigma_{2}^{0},\sigma_{2}):[0,T]\times(\mathbb{R}^{m})^{2}\rightarrow(\mathbb{R}^{n\times
d_{0}},\mathbb{R}^{n\times d_{2}})$,
$\overline{f}_{2}:[0,T]\times\mathbb{R}^{n}\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}$
and $g_{2}:\mathbb{R}^{n}\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}$. For
each
$t\in[0,T],x,\varpi,\alpha\in\mathbb{R}^{n},c^{0},\mathfrak{c}\in\mathbb{R}^{m}$,
we also set

$\small\begin{split}f_{2}(t,x,\varpi,\alpha,c^{0},\mathfrak{c}):=\langle\alpha,\varpi\rangle+\frac{1}{2}\langle\alpha,\Lambda_{t}\alpha\rangle-
b_{t}\langle
x,\varpi\rangle+\overline{f}_{2}(t,x,c^{0},\mathfrak{c}).\end{split}$

Here, $\langle\cdot,\cdot\rangle$ denotes an inner product of $n$-dimensional
vectors. For given stochastic processes
$(c^{0}_{t},(\mathfrak{c}^{j}_{t})_{j=1}^{N_{2}},$
$b_{t},\Lambda_{t})_{t\in[0,T]}$, we use the following abbreviations to
simplify the notations: for $1\leq j\leq N_{2}$ and
$(t,x,\varpi,\alpha)\in[0,T]\times(\mathbb{R}^{n})^{3}$,

$\small\begin{split}&l_{2}^{j}(t):=l_{2}(t,c_{t}^{0},\mathfrak{c}_{t}^{j}),\quad(\sigma_{2}^{j,0}(t),\sigma_{2}^{j}(t)):=(\sigma_{2}^{0},\sigma_{2})(t,c_{t}^{0},\mathfrak{c}_{t}^{j}),\\\
&\overline{f}_{2}^{j}(t,x):=\overline{f}_{2}(t,x,c_{t}^{0},\mathfrak{c}_{t}^{j}),\quad
g_{2}^{j}(x):=g_{2}(x,c_{T}^{0},\mathfrak{c}_{T}^{j}),\\\
&f_{2}^{j}(t,x,\varpi,\alpha):=\langle\alpha,\varpi\rangle+\frac{1}{2}\langle\alpha,\Lambda_{t}\alpha\rangle-
b_{t}\langle x,\varpi\rangle+\overline{f}_{2}^{j}(t,x).\end{split}$ ( 3.1)

Before providing various conditions on these functions, let us describe the
financial problem for each non-cooperative agent indexed by
$j=1,\ldots,N_{2}$. The following concept is important.

######  Definition 3.1 (price taker).

An agent is called a price taker if she behaves under the assumption that
there is no price impact from her trading. In other words, she always accepts
the market price as it is without trying to control it.

The following assumption is used throughout this work.

######  Assumption 3.1.

Every non-cooperative agent is a price taker.

This is a natural assumption if the size of population $N_{2}$ is large enough
so that the market share of every individual agent is negligibly small. Under
Assumption 3.1, every non-cooperative agent considers the price process
$\varpi\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ of the $n$ securities as
an exogenously given input. We suppose that the problem for each non-
cooperative agent $1\leq j\leq N_{2}$ is to minimize the cost functional

$\inf_{\alpha^{j}\in\mathbb{A}_{2}}J^{j}(\alpha^{j})$ ( 3.2)

under the dynamic constraint

$\small\begin{split}dx_{t}^{j}&=(\alpha_{t}^{j}+\rho_{2}(t)\varpi_{t}+l_{2}^{j}(t))dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\quad
x_{0}^{j}=\eta^{j}.\end{split}$ ( 3.3)

Here, the cost functional is defined by

$\small\begin{split}J^{j}(\alpha^{j}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f_{2}^{j}(t,x_{t}^{j},\varpi_{t},\alpha_{t}^{j})dt+g_{2}^{j}(x_{T}^{j})\Bigr{]},\end{split}$
( 3.4)

and $\mathbb{A}_{2}:=\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ is the space
of admissible controls.

Let us provide an economic interpretation of each term;
$x^{j}=(x_{t}^{j})_{t\in[0,T]}$ is an $\mathbb{R}^{n}$-valued process giving
the time evolution of the position size of the $n$ securities for the $j$th
agent, and $\eta^{j}$ denotes its initial value. Any negative component of
$(x_{t}^{j})\in\mathbb{R}^{n}$ represents a short position at time $t$.
$\alpha^{j}=(\alpha^{j}_{t})_{t\in[0,T]}\in\mathbb{A}_{2}$ is a control of the
$j$th agent denoting her trading rate of the securities. Here, each component
of $\alpha_{t}^{j}dt~{}(\in\mathbb{R}^{n})$ gives the amount of the
corresponding security bought (or sold if negative) at the exchange within a
time interval $[t,t+dt]$. In addition to the direct trading via the exchange,
the level of her inventory is affected by the stochastic order flows from her
OTC clients, which are described by $(\rho_{2}(t)\varpi_{t}+l_{2}^{j}(t))dt$
and $(\sigma_{2}^{j,0}(t)dW_{t}^{0},\sigma_{2}^{j}(t)dB_{t}^{j})$. In
particular, the term $\rho_{2}(t)\varpi_{t}$ represents a reaction of her
clients to the price level. Since it is naturally expected that the OTC
clients do not have any information about the agent’s securities position, we
assume the terms $(\rho_{2},l_{2}^{j},\sigma_{2}^{j,0},\sigma_{2}^{j})$, which
describe the OTC order flows, to be independent from $x^{j}$. $f_{2}^{j}$ and
$g_{2}^{j}$ denote the running and terminal cost functions, respectively;
$\langle\alpha,\varpi\rangle$ in $f_{2}^{j}$ represents the direct trading
cost of the securities of price $\varpi$ with trading rate $\alpha$;
$\frac{1}{2}\langle\alpha,\Lambda_{t}\alpha\rangle$ an internal trading cost
or a trading fee to be paid to the exchange; $-b_{t}\langle x,\varpi\rangle$
the reduction of the cost based on the mark-to-market value of the inventory
with a stochastic discount factor $b_{t}$; $\overline{f}_{2}^{j}$ the other
contributions to the running cost depending on the inventory level,
informations and cash flows $(c^{0},\mathfrak{c}^{j})$. The terminal cost
function $g_{2}^{j}$ represents the penalty on the open position at the
closing time $T$.

######  Remark 3.1.

From the problem setup and $(\ref{j-convention})$, we see that all the
heterogeneities among the non-cooperative agents come from i.i.d. inputs only;
idiosyncratic informations $(\mathfrak{c}^{j})_{j=1}^{N_{2}}$, initial
conditions $(\eta^{j})_{j=1}^{N_{2}}$, and the Brownian motions
$(B^{j})_{j=1}^{N_{2}}$.

######  Remark 3.2.

One may wonder why we do not allow $\varpi_{T}$-dependence in $g_{2}^{j}$ as
in the work [28]. This is due to the existence of the first population. Since
they have non-negligible market share, their last-minute trading at $\\{T\\}$
can influence $\varpi_{T}$ despite the fact that it is on the Lebesgue-null
set. In other words, if there exists $\varpi_{T}$ dependence in the terminal
cost function, the first population can drastically change it without changing
their securities position at all. This easily makes the system ill-defined.
Because of this reason, we assume that the terminal cost functions of the both
population (see $(\ref{2nd-cost-functional})$ and $(\ref{1st-cost-
functional})$ below) are independent from $\varpi_{T}$. 333The other
possibility is to forbid the last-minute trading as in [30]. This is not a big
restriction from the modeling perspective, since the cost functionals can
still depend on the time integral of $(\varpi_{t})_{t\in[0,T]}$.

######  Remark 3.3.

The definition of the admissible space
$\mathbb{A}_{2}=\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ used above implies
that each agent is supposed to know, in addition to the common information
$\mathbb{F}^{0}$, all the idiosyncratic information $\overline{\mathbb{F}}$ of
the other agents. (The perfect information is to be assumed also for the first
population.) Ideally, we would like to restrict the information set available
to each agent $j$ to the filtration
$\mathbb{F}^{0}\otimes\overline{\mathbb{F}}^{2,j}$. Unfortunately however, to
the best of the author’s knowledge, there is no existing work to successfully
achieve the market clearing equilibrium in the finite agent market without
assuming the perfect information. Interestingly, though, we shall observe that
the realistic information setup is recovered in the large population limit.
See the discussion in Remark 4.2 and the one following Theorem 5.1.

####  3.1.2 Solving the individual problem for $j=1,\ldots,N_{2}$

In order to solve the above problem, let us introduce the following
conditions.

######  Assumption 3.2.

(i) $\Lambda=(\Lambda_{t})_{t\in[0,T]}$ is an $\mathbb{F}^{0}$-progressively
measurable ${\cal S}_{+}^{n}$-valued process such that both $\Lambda_{t}$ and
$\overline{\Lambda}_{t}~{}(:=\Lambda_{t}^{-1})$ are bounded by $L$ for every
$t\in[0,T]$.
(ii) For every $(t,c^{0},\mathfrak{c})\in[0,T]\times(\mathbb{R}^{m})^{2}$,
$|l_{2}(t,c^{0},\mathfrak{c})|+|\sigma_{2}^{0}(t,c^{0},\mathfrak{c})|+|\sigma_{2}(t,c^{0},\mathfrak{c})|\leq
L(1+|c^{0}|+|\mathfrak{c}|).$
(iii) For every
$(t,x,c^{0},\mathfrak{c})\in[0,T]\times\mathbb{R}^{n}\times(\mathbb{R}^{m})^{2}$,
$|\overline{f}_{2}(t,x,c^{0},\mathfrak{c})|+|g_{2}(x,c^{0},\mathfrak{c})|\leq
L(1+|x|^{2}+|c^{0}|^{2}+|\mathfrak{c}|^{2}).$
(iv) For every $(t,c^{0},\mathfrak{c})\in[0,T]\times(\mathbb{R}^{m})^{2}$,
$\overline{f}_{2}$ and $g_{2}$ are once continuously differentiable in $x$,
and their derivatives have the affine form in $x$:

$\small\begin{split}&\partial_{x}\overline{f}_{2}(t,x,c^{0},\mathfrak{c})=c_{f}(t,c^{0},\mathfrak{c})x+h_{f}(t,c^{0},\mathfrak{c}),\quad\partial_{x}g_{2}(x,c^{0},\mathfrak{c})=c_{g}(c^{0},\mathfrak{c})x+h_{g}(c^{0},\mathfrak{c}).\end{split}$

Here, $(c_{f},h_{f}):[0,T]\times(\mathbb{R}^{m})^{2}\rightarrow({\cal
S}^{n}_{+},~{}\mathbb{R}^{n})$ and
$(c_{g},h_{g}):(\mathbb{R}^{m})^{2}\rightarrow({\cal
S}^{n}_{+},~{}\mathbb{R}^{n})$ are measurable functions that satisfy, with
some positive constants $\gamma_{2}^{f},\gamma_{2}^{g}>0$,

$\small\begin{split}&|h_{f}(t,c^{0},\mathfrak{c})|+|h_{g}(c^{0},\mathfrak{c})|\leq
L(1+|c^{0}|+|\mathfrak{c}|),\quad|c_{f}(t,c^{0},\mathfrak{c})|+|c_{g}(c^{0},\mathfrak{c})|\leq
L,\\\
&\langle\theta,c_{f}(t,c^{0},\mathfrak{c})\theta\rangle\geq\gamma_{2}^{f}|\theta|^{2},~{}\langle\theta,c_{g}(c^{0},\mathfrak{c})\theta\rangle\geq\gamma_{2}^{g}|\theta|^{2},~{}\forall\theta\in\mathbb{R}^{n}.\end{split}$

(v) $b:=(b_{t})_{t\in[0,T]}$ and $\rho_{2}:=(\rho_{2}(t))_{t\in[0,T]}$ are
$\mathbb{F}^{0}$-progressively measurable bounded processes satisfying

$\small\begin{split}0\leq b_{t}\leq
L,\quad\frac{b_{t}^{2}}{2\gamma_{2}^{f}}\leq\rho_{2}(t)\leq L,\quad\forall
t\in[0,T].\end{split}$

######  Remark 3.4.

The benefit (i.e. negative cost) due to the continuous payoff (respectively,
lump-sum payoff at $T$) from the securities can be included in $h_{f}$
(respectively, $h_{g}$). The convexity of cost functions (i.e. risk averseness
of agents) is determined by $c_{f}(t,c_{t}^{0},\mathfrak{c}_{t}^{j})$ and
$c_{g}(c_{T}^{0},\mathfrak{c}_{T}^{j})$, which are stochastic and also
heterogeneous among the agents by the factors
$(\mathfrak{c}^{j})_{j=1}^{N_{2}}$. The affine structures in (iv) will be used
in later sections to guarantee the convexity of Hamiltonian for the
“cooperative” agents.

With the adjoint variable $y\in\mathbb{R}^{n}$, the Hamiltonian for
$(\ref{non-c-problem})$ of the $j$th agent is given by

$\small\begin{split}H_{2}^{j}(t,x,y,\varpi,\alpha):=\langle
y,\alpha+\rho_{2}(t)\varpi+l_{2}^{j}(t)\rangle+\langle\alpha,\varpi\rangle+\frac{1}{2}\langle\alpha,\Lambda_{t}\alpha\rangle-
b_{t}\langle x,\varpi\rangle+\overline{f}_{2}^{j}(t,x).\end{split}$

For each $\omega\in\Omega$, this gives a map from
$[0,T]\times(\mathbb{R}^{n})^{4}$ to $\mathbb{R}$. Under Assumption 3.2, for a
given adjoint variable $y$, $H_{2}^{j}$ is jointly and strictly convex in
$(x,\alpha)$. Its unique minimizer $\widehat{\alpha}$ is given by
$\widehat{\alpha}=-\overline{\Lambda}_{t}(y+\varpi).$ Hence, for a given
$\varpi\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$, the system of the state
and adjoint equations from the maximum principle is given by

$\small\begin{split}&dx_{t}^{j}=(-\overline{\Lambda}_{t}(y_{t}^{j}+\varpi_{t})+\rho_{2}(t)\varpi_{t}+l_{2}^{j}(t))dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
&dy_{t}^{j}=-(\partial_{x}\overline{f}_{2}^{j}(t,x_{t}^{j})-b_{t}\varpi_{t})dt+z_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{z}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}z_{t}^{j,k}dB_{t}^{k}\end{split}$
( 3.5)

with $x_{0}^{j}=\eta^{j}$ and $y_{T}^{j}=\partial_{x}g_{2}^{j}(x_{T}^{j})$,
for each agent $1\leq j\leq N_{2}$.

######  Theorem 3.1.

Let $\varpi\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ be given. Under
Assumption 3.2, for each $j=1,\ldots,N_{2}$, there exists a unique square
integrable solution with
$(x^{j},y^{j})\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})$
to the FBSDE $(\ref{fbsde-non-C})$. 444Obviously, every integrand of Brownian
motions is square integrable, such as
$z^{j,0}\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n\times d_{0}})$. Since all
we need is this square integrability, we drop the details to save the space.
Moreover, the problem $(\ref{non-c-problem})$ has a unique optimal solution
$\widehat{\alpha}^{j}:=(\widehat{\alpha}_{t}^{j})_{t\in[0,T]}$ given by
$\widehat{\alpha}_{t}^{j}=-\overline{\Lambda}_{t}(y_{t}^{j}+\varpi_{t})$.

###### Proof.

This is a simple adaptation of [29, Theorem 3.1]. By the convexity and the
differentiability of the Hamiltonian, it is standard to check the sufficiency
of the Pontryagin’s maximum principle. The uniqueness of the optimal solution
follows from the strict convexity of $H_{2}^{j}$ in $\alpha$. Thus it only
remains to prove the existence of the unique solution to the FBSDE
$(\ref{fbsde-non-C})$. For each $(t,\omega)\in[0,T]\times\Omega$, let us
define the maps from state variables
$u:=(x^{j},y^{j})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ by

$\small\begin{split}&B_{x^{j}}(t,u):=-\overline{\Lambda}_{t}(y^{j}+\varpi_{t})+\rho_{2}(t)\varpi_{t}+l_{2}^{j}(t),\\\
&F_{y^{j}}(t,u):=-\partial_{x}\overline{f}_{2}^{j}(t,x^{j})+b_{t}\varpi_{t},\quad
G_{y^{j}}(u):=\partial_{x}g_{2}^{j}(x^{j}).\end{split}$

Under Assumption 3.2, they are uniformly Lipschitz continuous in $u$, and
$(B_{x^{j}}(t,u),F_{y^{j}}(t,u),G_{y^{j}}(u))$ satisfy the square
integrability for a given $u$. Hence it suffices to check the Peng-Wu’s
monotonicity [49, (H2.3)]. For two inputs $(u,\acute{u})$, let us set $\Delta
x^{j}:=x^{j}-\acute{x}^{j},\Delta y^{j}:=y^{j}-\acute{y}^{j}$, $\Delta
B_{x^{j}}(t):=B_{x^{j}}(t,u)-B_{x^{j}}(t,\acute{u})$, $\Delta
F_{y^{j}}(t):=F_{y^{j}}(t,u)-F_{y^{j}}(t,\acute{u})$ and $\Delta
G_{y^{j}}:=G_{y^{j}}(u)-G_{y^{j}}(\acute{u})$. Then, from condition (iv), it
is straightforward to see $\langle\Delta B_{x^{j}}(t),\Delta
y^{j}\rangle+\langle\Delta F_{y^{j}}(t),\Delta
x^{j}\rangle\leq-\gamma_{2}^{f}|\Delta x^{j}|^{2}$ and $\langle\Delta
G_{y^{j}},\Delta x^{j}\rangle\geq\gamma_{2}^{g}|\Delta x^{j}|^{2}$. Thus the
existence of a unique solution to $(\ref{fbsde-non-C})$ follows from [49,
Theorem 2.6] with $\mu_{1}=\gamma_{2}^{g}$, $\beta_{1}=\gamma_{2}^{f}$ with
the identity matrix $G=I_{n\times n}$. ∎

####  3.1.3 Market clearing equilibrium with given order flows from the first
population

Let us denote by $\beta^{i}:=(\beta_{t}^{i})_{t\in[0,T]}$ the trading rate of
each agent $i=1,\ldots,N_{1}$ in the first population at the securities
exchange.

######  Definition 3.2.

We say that the market clearing condition is satisfied if the next equality
holds:

$\small\begin{split}\sum_{j=1}^{N_{2}}\alpha_{t}^{j}+\sum_{i=1}^{N_{1}}\beta_{t}^{i}=0,\quad
dt\otimes d\mathbb{P}\text{-a.e.}\end{split}$

######  Definition 3.3.

With given order flows
$(\beta^{i}\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n}),i=1,\ldots,N_{1})$
from the first population, if there exists a price process
$\varpi\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ and the set of optimal
solutions $(\widehat{\alpha}^{j})_{j=1}^{N_{2}}$ to the problem $(\ref{non-c-
problem})$ that satisfies the market clearing condition, then we say that the
market clearing equilibrium exists with the equilibrium price process $\varpi$
for the given order flows $(\beta^{i})_{i=1}^{N_{1}}$.

Since we know
$\widehat{\alpha}_{t}^{j}=-\overline{\Lambda}_{t}(y_{t}^{j}+\varpi_{t})$ from
Theorem 3.1, the market clearing condition implies that the price process
$\varpi_{t}$ must be equal to $\varpi_{t}(y_{t},\beta_{t})$ defined by

$\small\varpi_{t}(y,\beta):=-\mathfrak{m}_{2}(y)+\delta_{\mathfrak{n}}\Lambda_{t}\mathfrak{m}_{1}(\beta),$
( 3.6)

where

$\small\delta_{\mathfrak{n}}:=N_{1}/N_{2}$

denotes the ratio of the two populations. Recall that $\mathfrak{m}_{2}(y)$
and $\mathfrak{m}_{1}(\beta)$ give the empirical means of
$(y^{j})_{j=1}^{N_{2}}$ and $(\beta^{i})_{i=1}^{N_{1}}$, respectively. As in
[28, 29], the relation $(\ref{vp-expression-1})$ leads us to consider the
FBSDEs;

$\small\begin{split}&dx_{t}^{j}=\bigl{(}-\overline{\Lambda}_{t}(y_{t}^{j}+\varpi_{t}(y_{t},\beta_{t}))+\rho_{2}(t)\varpi_{t}(y_{t},\beta_{t})+l_{2}^{j}(t)\bigr{)}dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
&dy_{t}^{j}=-(\partial_{x}\overline{f}_{2}^{j}(t,x_{t}^{j})-b_{t}\varpi_{t}(y_{t},\beta_{t}))dt+z_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{z}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}z_{t}^{j,k}dB_{t}^{k}\end{split}$
( 3.7)

with $x_{0}^{j}=\eta^{j}$ and $y_{T}^{j}=\partial_{x}g_{2}^{j}(x_{T}^{j})$,
$1\leq j\leq N_{2}$. It is a coupled system by the interaction in
$\varpi_{t}(y,\beta)$. The relevance of $(\ref{fbsde-eq-non-C})$ may be
understood from the next theorem.

######  Theorem 3.2.

Let $(\beta^{i})_{i=1}^{N_{1}}$ with
$\beta^{i}\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ be given. Under
Assumption 3.2, there exists a unique square integrable solution with
$(x,y)=((x^{j})_{j=1}^{N_{2},},(y^{j})_{j=1}^{N_{2}})\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}$
to the system of FBSDEs $(\ref{fbsde-eq-non-C})$. Moreover, there exists a
unique market clearing equilibrium with the equilibrium price process
$(\varpi_{t}=\varpi_{t}(y_{t},\beta_{t}))_{t\in[0,T]}$ for the given order
flows $(\beta^{i})_{i=1}^{N_{1}}$.

###### Proof.

This is an adaptation of [29, Theorems 3.2 and 3.3]. Firstly, the existence of
the solution to $(\ref{fbsde-eq-non-C})$ is a necessary condition so that
$(\widehat{\alpha}^{j})_{j=1}^{N_{2}}$ clears the market with the given order
flows $(\beta^{i})_{i=1}^{N_{1}}$. Conversely, if there exists a square
integrable solution to the system $(\ref{fbsde-eq-non-C})$, then one can check
that the price process $(\varpi_{t}(y_{t},\beta_{t}))_{t\geq 0}$ defined by
its solution $y$ actually clears the market when it is used as an input price
process $\varpi$ for the problem $(\ref{non-c-problem})$. In fact, due to the
uniqueness of the solution to $(\ref{fbsde-non-C})$ by Theorem 3.1, the
backward solutions $y^{j}$ of the two equations coincide.

Hence it suffices to show that there exists a unique square integrable
solution to $(\ref{fbsde-eq-non-C})$. As before, this is done by Peng-Wu’s
continuation method. For each $(t,\omega)$, we define the maps from
$u:=((x^{j})_{j=1}^{N_{2}},(y^{j})_{j=1}^{N_{2}})\in(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}$
to $\mathbb{R}^{n}$ by

$\small\begin{split}&B_{x^{j}}(t,u):=-\overline{\Lambda}_{t}(y^{j}+\varpi_{t}(y,\beta_{t}))+\rho_{2}(t)\varpi_{t}(y,\beta_{t})+l_{2}^{j}(t),\\\
&F_{y^{j}}(t,u):=-\partial_{x}\overline{f}_{2}^{j}(t,x^{j})+b_{t}\varpi_{t}(y,\beta_{t}),\quad
G_{y^{j}}(u):=\partial_{x}g_{2}^{j}(x^{j})\end{split}$

with $1\leq j\leq N_{2}$. By Assumption 3.2, they are uniformly Lipschitz
continuous with respect to $u$. For each $u$, it is easy to confirm the square
integrability;
$B_{x^{j}}(\cdot,u),F_{x^{j}}(\cdot,u)\in\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$,
and $G_{y^{j}}(u)\in\mathbb{L}^{2}({\cal F}_{T};\mathbb{R}^{n})$. Now, it only
remains to confirm the Peng-Wu’s monotonicity conditions. For two inputs
$(u,\acute{u})$, let us set $\Delta x^{j}:=x^{j}-\acute{x}^{j}$, $\Delta
y^{j}=y^{j}-\acute{y}^{j}$, $\Delta
B_{x^{j}}(t):=B_{x^{j}}(t,u)-B_{x^{j}}(t,\acute{u})$, $\Delta
F_{y^{j}}(t):=F_{y^{j}}(t,u)-F_{y^{j}}(t,\acute{u})$ and $\Delta
G_{y^{j}}:=G_{y^{j}}(u)-G_{y^{j}}(\acute{u})$. We have

$\small\begin{split}&\sum_{j=1}^{N_{2}}\langle\Delta B_{x^{j}}(t),\Delta
y^{j}\rangle+\sum_{j=1}^{N_{2}}\langle\Delta F_{y^{j}}(t),\Delta
x^{j}\rangle\\\ &=-\sum_{j=1}^{N_{2}}\langle\overline{\Lambda}_{t}(\Delta
y^{j}-\mathfrak{m}_{2}(\Delta y))+\rho_{2}(t)\mathfrak{m}_{2}(\Delta y),\Delta
y^{j}\rangle-\sum_{j=1}^{N_{2}}\langle
c^{f}(t,c_{t}^{0},\mathfrak{c}_{t}^{j})\Delta
x^{j}+b_{t}\mathfrak{m}_{2}(\Delta y),\Delta x^{j}\rangle\\\ &\leq-
N_{2}\rho_{2}(t)|\mathfrak{m}_{2}(\Delta
y)|^{2}-\gamma_{2}^{f}\sum_{j=1}^{N_{2}}|\Delta
x^{j}|^{2}+b_{t}\sum_{j=1}^{N_{2}}|\langle\Delta x^{j},\mathfrak{m}_{2}(\Delta
y)\rangle|\leq-\frac{\gamma_{2}^{f}}{2}\sum_{j=1}^{N_{2}}|\Delta
x^{j}|^{2}.\end{split}$

In the first inequality, we have used the condition (iv) and the fact that
$\sum_{j=1}^{N_{2}}\langle\overline{\Lambda}_{t}\Delta y^{j},\Delta
y^{j}\rangle\geq N_{2}\langle\overline{\Lambda}_{t}\mathfrak{m}_{2}(\Delta
y),\mathfrak{m}_{2}(\Delta y)\rangle$. The Young’s inequality and the
assumption (v) give the last one. From the condition (iv), we also have
$\sum_{j=1}^{N_{2}}\langle\Delta G_{y^{j}},\Delta
x^{j}\rangle\geq\gamma_{2}^{g}\sum_{j=1}^{N_{2}}|\Delta x^{j}|^{2}.$ Hence, we
can apply [49, Theorem 2.6] with
$(\beta_{1},\mu_{1})=(\gamma^{f}_{2}/2,\gamma_{2}^{g})$ and the identity
matrix $G$. ∎

######  Remark 3.5.

It is interesting to observe an economic role played by the term
$\rho_{2}(t)\varpi_{t}$. Since $\rho_{2}$ is positive, it increases the inflow
of securities to the agents’ inventory when the price $\varpi$ rises. This is
equivalent to the increase of the sales from the OTC clients. Since agents
(i.e. financial firms) are trying to maintain the optimal inventory level,
this reduces the demand of the securities among the financial firms.
Assumption 3.2 (v) guarantees that this is stronger than the opposite effect
caused by the mark-to-market term $-b_{t}\langle x,\varpi\rangle$. By making
the demand for securities react in an opposite way to the price level, the
term $\rho_{2}(t)\varpi_{t}$ seems to prevent market bubbles/crashes from
happening and allow the equilibrium to exist for general duration $T$.

###  3.2 Problem for the cooperative agents

####  3.2.1 Problem description

We now study the problem for the first population that consists of cooperative
agents. In contrast to the previous case, the agents are no longer price
takers. Although the market share of each agent may be negligibly small, their
coordinated actions in total can influence the behavior of the price process.
In order to describe their problem in a concrete manner, let us first
introduce the following measurable functions;
$l_{1}:[0,T]\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}^{n}$,
$(\sigma_{1}^{0},\sigma_{1}):[0,T]\times(\mathbb{R}^{m})^{2}\rightarrow(\mathbb{R}^{n\times
d_{0}},\mathbb{R}^{n\times d_{1}})$,
$f_{1}:[0,T]\times\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times
A_{1}\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}$, and
$g_{1}:\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times(\mathbb{R}^{m})^{2}\rightarrow\mathbb{R}$. Here,
$A_{1}$ is a closed convex subset of $\mathbb{R}^{n}$. We also introduce
$\mathbb{R}$-valued $\mathbb{F}^{0}$-progressively measurable process
$\rho_{1}:=(\rho_{1}(t))_{t\in[0,T]}$. As in $(\ref{j-convention})$, for given
stochastic processes $(c^{0}_{t},c_{t}^{i})_{t\in[0,T]}$, we often use the
following abbreviations

$\small\begin{split}&l_{1}^{i}(t):=l_{1}(t,c_{t}^{0},c_{t}^{i}),\quad(\sigma_{1}^{i,0}(t),\sigma_{1}^{i}(t)):=(\sigma_{1}^{0},\sigma_{1})(t,c_{t}^{0},c_{t}^{i}),\\\
&f_{1}^{i}(t,x,\mu,\varpi,\beta):=f_{1}(t,x,\mu,\varpi,\beta,c_{t}^{0},c_{t}^{i}),\quad
g_{1}^{i}(x,\mu):=g_{1}(x,\mu,c_{T}^{0},c_{T}^{i})\end{split}$ ( 3.8)

for $(t,x,\mu,\varpi,\beta)\in[0,T]\times\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times A_{1}$ and $1\leq i\leq
N_{1}$. We also use, for $1\leq j\leq N_{2}$,

$\small\begin{split}c_{f}^{j}(t):=c_{f}^{j}(t,c_{t}^{0},\mathfrak{c}_{t}^{j}),\quad
c_{g}^{j}:=c_{g}(c_{T}^{0},\mathfrak{c}_{T}^{j}).\end{split}$

Let us denote by $\beta:=(\beta^{i})_{i=1}^{N_{1}}$ with
$\beta^{i}:=(\beta_{t}^{i})_{t\in[0,T]}$ the trading rate of the cooperative
agents at the exchange. $X^{i}=(X_{t}^{i})_{t\in[0,T]}$ is an
$\mathbb{R}^{n}$-valued process describing the evolution of the position size
of the $n$ securities of the $i$th agent. We suppose that it obeys the
dynamics given by

$\small\begin{split}&dX_{t}^{i}=(\beta_{t}^{i}+\rho_{1}(t)\varpi_{t}(y_{t},\beta_{t})+l_{1}^{i}(t))dt+\sigma_{1}^{i,0}(t)dW_{t}^{0}+\sigma_{1}^{i}(t)dW_{t}^{i},\quad
X_{0}^{i}=\xi^{i},\end{split}$

$1\leq i\leq N_{1}$. Here, the term $\varpi_{t}(y_{t},\beta_{t})$ is defined
by $(\ref{vp-expression-1})$ with $y=(y^{j})_{j=1}^{N_{2}}$ being the solution
to the system of FBSDEs $(\ref{fbsde-eq-non-C})$. The economic interpretation
is similar to the previous case; each component of
$\beta_{t}^{i}dt~{}(\in\mathbb{R}^{n})$ gives the amount of the corresponding
security bought (or sold if negative) by the $i$th agent at the exchange
within a time interval $[t,t+dt]$;
$(\rho_{1}(t)\varpi_{t}(y_{t},\beta_{t})+l_{1}^{i}(t))dt$ and
$(\sigma_{1}^{i,0}(t)dW_{t}^{0},\sigma_{1}^{i}(t)dW_{t}^{i})$ give the
stochastic order flows from her OTC clients. We suppose that the cost
functional for each agent $i$ is given by

$\small\begin{split}J_{1}^{i}(\beta):=\mathbb{E}\Bigl{[}\int_{0}^{T}f_{1}^{i}(t,X_{t}^{i},\nu_{t}^{N_{1}},\varpi_{t}(y_{t},\beta_{t}),\beta_{t}^{i})dt+g_{1}^{i}(X_{T}^{i},\nu_{T}^{N_{1}})\Bigr{]},\end{split}$
( 3.9)

where $\nu^{N_{1}}_{t}$ is the time-$t$ empirical distribution of the
securities position among the cooperative agents, i.e.
$\nu_{t}^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X_{t}^{i}}$.
$f_{1}^{i}$ and $g_{1}^{i}$ denote the running and terminal costs,
respectively. Since the agents are cooperative, it is natural to suppose that
each of them takes care of not only her own inventory but also its
distribution across the members. This is represented by the dependence on the
empirical distribution $\nu^{N_{1}}$ in the cost functions. The running cost
$f_{1}^{i}$ is allowed to have general convex dependence on $\varpi$ to
describe the agent’s preference on the price level.

We suppose that all the trading actions in the first population are
coordinated by the central planner, whose problem is defined by

$\small\begin{split}\inf_{\beta^{i}\in\mathbb{A}_{1},~{}i=1,\ldots,N_{1}}\sum_{i=1}^{N_{1}}J_{1}^{i}(\beta).\end{split}$
( 3.10)

Here, $\mathbb{A}_{1}:=\mathbb{H}^{2}(\mathbb{F};A_{1})$ is the admissible
space for each $\beta^{i}$. Thus, at every time $t\in[0,T]$, each control
$\beta_{t}^{i}$ has to take values in a given closed convex set $A_{1}$. The
central planner is responsible to control every $\beta^{i},i=1,\ldots,N_{1}$
within $\mathbb{A}_{1}$ to minimize the total cost of the whole population.
Due to the reactions from the non-cooperative agents, she needs to deal with
complicated price impacts from her own trading. Notice that, in addition to
the direct impact from $\mathfrak{m}_{1}(\beta)$-term in $\varpi_{t}(\cdot)$,
she also receives implicit feedbacks from $\mathfrak{m}_{2}(y)$-term via
$(\ref{fbsde-eq-non-C})$. In fact, the coupling with $(\ref{fbsde-eq-non-C})$
makes the problem for the central planner an optimization with respect to the
system of controlled-FBSDEs instead of controlled-SDEs. See the works by J.
Yong [55, 56] for general discussions on the problems of controlled-FBSDEs.

######  Remark 3.6.

Similarly to the second population, we see that all the heterogeneities among
the cooperative agents come from i.i.d. inputs only; idiosyncratic
informations $(c^{i})_{i=1}^{N_{1}}$, initial conditions
$(\xi^{i})_{i=1}^{N_{1}}$, and the Brownian motions $(W^{i})_{i=1}^{N_{1}}$.

Before giving the set of main assumptions, let us mention about the concept of
differentiability of functions defined on the space of probability measures.
In this work, we adopt the notion of $L$-differentiability used in [14, 15],
which was first introduced by P.L.Lions in his lecture at College de France.
Here, the differentiation is based on the lifting of function ${\cal
P}_{2}(\mathbb{R}^{n})\ni\mu\mapsto u(\mu)\in\mathbb{R}$ to a function
$\widetilde{u}$ defined on the Hilbert space $\mathbb{L}^{2}(\Omega,{\cal
F},\mathbb{P};\mathbb{R}^{n})$ by $\widetilde{u}(X):=u({\cal L}(X))$ with
$X\in\mathbb{L}^{2}(\Omega,{\cal F},\mathbb{P};\mathbb{R}^{n})$. Here, the
probability space $(\Omega,{\cal F},\mathbb{P})$ is chosen arbitrarily with a
Polish set $\Omega$ and an atomless probability measure $\mathbb{P}$ to
support the random variable $X$ with ${\cal L}(X)=\mu$. The $L$-derivative of
$u(\mu)$ is defined as the Frechet derivative of the lifted function
$\widetilde{u}(X)$.

######  Definition 3.4 (Definition 5.22 in [14]).

A function $u$ on ${\cal P}_{2}(\mathbb{R}^{n})$ is said to be
$L$-differentiable at $\mu_{0}\in{\cal P}_{2}(\mathbb{R}^{n})$ if there exists
a random variable $X$ with law $\mu_{0}$ such that the lifted function
$\widetilde{u}$ is Frechet differentiable at $X$.

We distinguish the Frechet derivative by using the symbol $D$. By [14,
Proposition 5.24], if $u$ is $L$-differentiable at $\mu_{0}$, the
$\widetilde{u}$ is Frechet differentiable at any $\acute{X}$ with ${\cal
L}(\acute{X})=\mu_{0}$ and the law of the pair
$(\acute{X},D\widetilde{u}(\acute{X}))$ is independent of the choice of the
random variable $\acute{X}$. Thus, the $L$-derivative may be denoted by
$\partial_{\mu}u(\mu_{0})(\cdot):\mathbb{R}^{n}\ni
x\mapsto\partial_{\mu}u(\mu_{0})(x)\in\mathbb{R}^{n}$, which is uniquely
defined $\mu_{0}$-a.e. on $\mathbb{R}^{n}$. By its definition, the
$L$-derivative satisfies

$\small\begin{split}u(\mu)=u(\mu_{0})+\mathbb{E}\bigl{[}\langle
X-X_{0},\partial_{\mu}u(\mu_{0})(X_{0})\rangle\bigr{]}+o(\|X-X_{0}\|_{\mathbb{L}^{2}}),\end{split}$

for any random variables $X$ and $X_{0}$ with ${\cal L}(X)=\mu$ and ${\cal
L}(X_{0})=\mu_{0}$. For example, if the function $u$ is of the form
$u(\mu):=\int_{\mathbb{R}^{n}}h(x)\mu(dx)$ for some function
$h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, we have
$\widetilde{u}(X)=\mathbb{E}[h(X)]$ with ${\cal L}(X)=\mu$. If the function
$h$ is differentiable, then the definition of $L$-derivative implies that
$\partial_{\mu}u(\mu)(\cdot)=\partial_{x}h(\cdot)$. See [14, 15] for details
and many more examples.

In our analysis, in particular, functions on empirical distributions play a
special role. Suppose that a function $u:{\cal
P}_{2}(\mathbb{R}^{n})\rightarrow\mathbb{R}$ is given. By considering the map
$(\mathbb{R}^{n})^{N}\ni(x^{i})_{i=1}^{N}\mapsto
u\Bigr{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{x^{i}}\Bigr{)}\in\mathbb{R}$, $u$
can also be seen as a function on $(\mathbb{R}^{n})^{N}$. If $u:{\cal
P}_{2}(\mathbb{R}^{n})\rightarrow\mathbb{R}$ is $L$-differentiable, [14,
Proposition 5.35] tells that the function $u$, when seen as a function on
$(\mathbb{R}^{n})^{N}$, is differentiable for each $x^{i}$ and satisfies

$\small\begin{split}\partial_{x^{i}}u(x^{1},\ldots,x^{N})=\frac{1}{N}\partial_{\mu}u\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{x^{i}}\Bigr{)}(x^{i}).\end{split}$
( 3.11)

Here, with slight abuse of notations, we used the same symbol $u$ to denote
the two maps.

Now, our assumptions on the cooperative agents are given as follows.

######  Assumption 3.3.

(0) The population sizes $(N_{1},N_{2})$ satisfy $\delta_{\mathfrak{n}}\leq
L$, where $\delta_{\mathfrak{n}}:=N_{1}/N_{2}$.
(i) For every $(t,c^{0},c)\in[0,T]\times(\mathbb{R}^{m})^{2}$,
$|l_{1}(t,c^{0},c)|+|\sigma_{1}^{0}(t,c^{0},c)|+|\sigma_{1}(t,c^{0},c)|\leq
L(1+|c^{0}|+|c|).$
(ii) For every
$(t,x,\mu,\varpi,\beta,c^{0},c)\in[0,T]\times\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times
A_{1}\times(\mathbb{R}^{m})^{2}$, the cost functions satisfy

$\small\begin{split}&|f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)|\leq
L(1+|x|^{2}+M_{2}(\mu)^{2}+|\varpi|^{2}+|\beta|^{2}+|c^{0}|^{2}+|c|^{2}),\\\
&|g_{1}(x,\mu,c^{0},c)|\leq
L(1+|x|^{2}+M_{2}(\mu)^{2}+|c^{0}|^{2}+|c|^{2}).\end{split}$

and, with another inputs
$(\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta})$, they also satisfy the
local Lipschitz continuity;

$\small\begin{split}&|f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)-f_{1}(t,\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta},c^{0},c)|\\\
&\quad\leq
L\bigl{(}1+|x|+|\acute{x}|+M_{2}(\mu)+M_{2}(\acute{\mu})+|\varpi|+|\acute{\varpi}|+|c^{0}|+|c|\bigr{)}(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})+|\varpi-\acute{\varpi}|+|\beta-\acute{\beta}|),\\\
&|g_{1}(x,\mu,c^{0},c)-g_{1}(\acute{x},\acute{\mu},c^{0},c)|\leq
L(1+|x|+|\acute{x}|+M_{2}(\mu)+M_{2}(\acute{\mu})+|c^{0}|+|c|)(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})).\end{split}$

(iii) The cost functions $f_{1}$ and $g_{1}$ are once continuously
differentiable in $(x,\mu,\varpi,\beta)$ and $(x,\mu)$ respectively with
uniformly Lipschitz continuous derivatives in the sense that

$\small\begin{split}&|[(\partial_{x},\partial_{\varpi},\partial_{\beta})f_{1}](t,x,\mu,\varpi,\beta,c^{0},c)-[(\partial_{x},\partial_{\varpi},\partial_{\beta})f_{1}](t,\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta},c^{0},c)|\leq
L(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})+|\varpi-\acute{\varpi}|+|\beta-\acute{\beta}|),\\\
&|\partial_{\mu}f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)(v)-\partial_{\mu}f_{1}(t,\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta},c^{0},c)(\acute{v})|\leq
L(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})+|\varpi-\acute{\varpi}|+|\beta-\acute{\beta}|+|v-\acute{v}|),\\\
&|\partial_{x}g_{1}(x,\mu,c^{0},c)-\partial_{x}g_{1}(\acute{x},\acute{\mu},c^{0},c)|\leq
L(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})),\\\
&|\partial_{\mu}g_{1}(x,\mu,c^{0},c)(v)-\partial_{\mu}g_{1}(\acute{x},\acute{\mu},c^{0},c)(\acute{v})|\leq
L(|x-\acute{x}|+W_{2}(\mu,\acute{\mu})+|v-\acute{v}|),\end{split}$

hold for every $(t,c^{0},c)\in[0,T]\times(\mathbb{R}^{m})^{2}$ and
$(x,\mu,\varpi,\beta,v),(\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta},\acute{v})\in\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times A_{1}\times\mathbb{R}^{n}$.
Moreover, the derivatives satisfy the linear growth property:

$\small\begin{split}&|[(\partial_{x},\partial_{\varpi},\partial_{\beta})f_{1}](t,x,\mu,\varpi,\beta,c^{0},c)|+|\partial_{\mu}f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)(v)|\leq
L(1+|x|+M_{2}(\mu)+|\varpi|+|\beta|+|v|+|c^{0}|+|c|),\\\
&|\partial_{x}g_{1}(x,\mu,c^{0},c)|+|\partial_{\mu}g_{1}(x,\mu,c^{0},c)(v)|\leq
L(1+|x|+M_{2}(\mu)+|v|+|c^{0}|+|c|).\end{split}$

(iv) For any $(t,c^{0},c)\in[0,T]\times(\mathbb{R}^{m})^{2}$, $f_{1}$ is
jointly convex in $(x,\mu,\varpi,\beta)$ in the sense that

$\small\begin{split}&f_{1}(t,\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta},c^{0},c)-f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)-\bigl{\langle}\partial_{(x,\varpi,\beta)}f_{1}(t,x,\mu,\varpi,\beta,c^{0},c),(\acute{x}-x,\acute{\varpi}-\varpi,\acute{\beta}-\beta)\bigr{\rangle}\\\
&-\mathbb{E}\bigl{[}\langle\partial_{\mu}f_{1}(t,x,\mu,\varpi,\beta,c^{0},c)(X),\acute{X}-X\rangle\bigr{]}\geq\frac{\gamma_{1}^{f}}{2}|\acute{x}-x|^{2}+\frac{\lambda_{\beta}}{2}|\acute{\beta}-\beta|^{2}+\frac{\lambda_{\varpi}}{2}|\acute{\varpi}-\varpi|^{2}\end{split}$

holds for any pair of inputs
$(x,\mu,\varpi,\beta),(\acute{x},\acute{\mu},\acute{\varpi},\acute{\beta})\in\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times A_{1}$ and random variables
$X,\acute{X}$ with ${\cal L}(X)=\mu,{\cal L}(\acute{X})=\acute{\mu}$. Here,
$\gamma_{1}^{f},\lambda_{\beta},\lambda_{\varpi}>0$ are some positive
constants, and $\partial_{(x,\varpi,\beta)}f_{1}$ stands for the gradient of
$f_{1}$ in the joint variables $(x,\varpi,\beta)$.
(v) For any $(c^{0},c)\in(\mathbb{R}^{m})^{2}$, $g_{1}$ is jointly convex in
$(x,\mu)$ with some positive constant $\gamma_{1}^{g}>0$,

$\small\begin{split}&g_{1}(\acute{x},\acute{\mu},c^{0},c)-g_{1}(x,\mu,c^{0},c)-\langle\partial_{x}g_{1}(x,\mu,c^{0},c),\acute{x}-x\rangle-\mathbb{E}\bigl{[}\langle\partial_{\mu}g_{1}(x,\mu,c^{0},c)(X),\acute{X}-X\rangle\bigr{]}\geq\frac{\gamma_{1}^{g}}{2}|\acute{x}-x|^{2}\end{split}$

holds for any pair of inputs
$(x,\mu),(\acute{x},\acute{\mu})\in\mathbb{R}^{n}\times{\cal
P}_{2}(\mathbb{R}^{n})$ and random variables $X,\acute{X}$ with ${\cal
L}(X)=\mu,{\cal L}(\acute{X})=\acute{\mu}$.
(vi) $\rho_{1}:=(\rho_{1}(t))_{t\in[0,T]}$ is an $\mathbb{R}$-valued
$\mathbb{F}^{0}$-progressively measurable bounded process with
$|\rho_{1}(t)|\leq L,~{}\forall t\in[0,T]$. Moreover, $\rho_{2}$ satisfies

$\small\begin{split}\max\bigl{(}\frac{1}{L},\frac{2b_{t}^{2}}{\gamma_{2}^{f}}\bigr{)}\leq\rho_{2}(t)\leq
L,\quad\forall t\in[0,T].\end{split}$

######  Remark 3.7.

Note that, $f_{1}$ can have much more general form than $f_{2}$. In
particular, $f_{1}$ having a linear quadratic form in $(x,\varpi,\beta)$ with
appropriate convexity is a special example satisfying the above assumptions.

####  3.2.2 Adjoint equations

Let us first introduce the notations $X,P\in(\mathbb{R}^{n})^{N_{1}},\beta\in
A_{1}^{N_{1}}$ and $x,y,p,r\in(\mathbb{R}^{n})^{N_{2}}$ in the following way;
$X:=(X^{i})_{i=1}^{N_{1}}$, $P:=(P^{i})_{i=1}^{N_{1}}$,
$\beta:=(\beta^{i})_{i=1}^{N_{1}}$, $x:=(x^{j})_{j=1}^{N_{2}}$,
$y:=(y^{j})_{j=1}^{N_{2}}$, $p:=(p^{j})_{j=1}^{N_{2}}$ and
$r:=(r^{j})_{j=1}^{N_{2}}$.

With adjoint variables $(P,p,r)$, the Hamiltonian $H$ for the problem
$(\ref{central-problem})$ is defined by

$\small\begin{split}&H(t,X,x,y,P,p,r,\beta)\\\
&:=\sum_{i=1}^{N_{1}}\bigl{\\{}\langle
P^{i},\beta^{i}+\rho_{1}(t)\varpi_{t}(y,\beta)+l_{1}^{i}(t)\rangle+f_{1}^{i}(t,X^{i},\nu^{N_{1}},\varpi_{t}(y,\beta),\beta^{i})\bigr{\\}}\\\
&\quad+\sum_{j=1}^{N_{2}}\bigl{\\{}\langle
p^{j},-\overline{\Lambda}_{t}(y^{j}+\varpi_{t}(y,\beta))+\rho_{2}(t)\varpi_{t}(y,\beta)+l_{2}^{j}(t)\rangle+\langle
r^{j},-\partial_{x}\overline{f}_{2}^{j}(t,x^{j})+b_{t}\varpi_{t}(y,\beta)\rangle\bigr{\\}},\end{split}$
( 3.12)

which gives, for each $(t,\omega)$, a map from
$(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}\times
A_{1}^{N_{1}}$ to $\mathbb{R}$. Here, $\varpi_{t}(y,\beta)$ is defined by
$(\ref{vp-expression-1})$ and
$\nu^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X^{i}}$. With slight
abuse of notation, let us also write the above Hamiltonian as
$H(t,X,x,y,P,p,r,\varpi,\beta)$ when we treat $\varpi\in\mathbb{R}^{n}$ as a
separate argument. From the number of arguments involved in the expression,
one may understand which definition is used without confusion. Note that, in
the both definitions, the dependence on $\nu^{N_{1}}$ is treated as a function
of $X=(X^{i})_{i=1}^{N_{1}}$. For each $u:=(X,x,y,P,p,r)$, we wan to find a
minimizer:

$\small\begin{split}\widehat{\beta}(t,u):={\rm
argmin}\bigl{(}H(t,X,x,y,P,p,r,\beta):\beta=(\beta^{i})_{i=1}^{N_{1}}\in
A_{1}^{N_{1}}\bigr{)}.\end{split}$

We often use the following derivatives: with $\varpi:=\varpi_{t}(y,\beta)$ and
$f_{1}^{i}(t,u,\beta^{i}):=f_{1}^{i}(t,X^{i},\nu^{N_{1}},\varpi,\beta^{i})$,

$\small\begin{split}&\partial_{\beta^{i}}H(t,u,\beta)=\partial_{\beta^{i}}H(t,u,\varpi,\beta)+\frac{\Lambda_{t}}{N_{2}}\partial_{\varpi}H(t,u,\varpi,\beta),\\\
&=P^{i}+\partial_{\beta^{i}}f_{1}^{i}(t,u,\beta^{i})+\frac{\Lambda_{t}}{N_{2}}\sum_{k=1}^{N_{1}}\partial_{\varpi}f_{1}^{k}(t,u,\beta^{k})+\delta_{\mathfrak{n}}\rho_{1}(t)\Lambda_{t}\mathfrak{m}_{1}(P)+(-I+\rho_{2}(t)\Lambda_{t})\mathfrak{m}_{2}(p)+b_{t}\Lambda_{t}\mathfrak{m}_{2}(r),\\\
&\partial_{X^{i}}H(t,u,\beta)=\partial_{x}f_{1}^{i}(t,u,\beta^{i})+\frac{1}{N_{1}}\sum_{k=1}^{N_{1}}\partial_{\mu}f_{1}^{k}(t,u,\beta^{k})(X^{i}),\\\
&\partial_{x^{j}}H(t,u,\beta)=-c_{f}^{j}(t)r^{j},\\\
&\partial_{y^{j}}H(t,u,\beta)=-\frac{1}{N_{2}}\sum_{k=1}^{N_{1}}\partial_{\varpi}f_{1}^{k}(t,u,\beta^{k})-\delta_{\mathfrak{n}}\rho_{1}(t)\mathfrak{m}_{1}(P)-b_{t}\mathfrak{m}_{2}(r)+\overline{\Lambda}_{t}(\mathfrak{m}_{2}(p)-p^{j})-\rho_{2}(t)\mathfrak{m}_{2}(p).\end{split}$
( 3.13)

Here, we have used $(\ref{L-d-empirical})$ to get
$\partial_{X^{i}}H(t,u,\beta)$.

######  Lemma 3.1.

Under Assumptions 3.2-3.3, for each $(t,\omega)$ and a given set of adjoint
variables
$(P,p,r)\in(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}$,
the Hamiltonian $H$ defined by $(\ref{H-1-finite})$ is jointly convex in
$(X,x,y,\beta)\in(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}\times
A_{1}^{N_{1}}$ and strictly so in $(X,\beta)$. In particular, for any
$t\in[0,T]$ and input $u:=(X,x,y,P,p,r)$, there exists a unique minimizer
$(\widehat{\beta}_{t}^{i}=\widehat{\beta}^{i}(t,u))_{i=1}^{N_{1}}$ with
$\widehat{\beta}_{t}^{i}\in A_{1}$ that satisfies the following linear growth
and the Lipschitz continuity properties:

$\small\begin{split}&\sum_{i=1}^{N_{1}}|\widehat{\beta}^{i}(t,u)|^{2}\leq
C\sum_{i=1}^{N_{1}}\bigr{[}1+|X^{i}|^{2}+|P^{i}|^{2}+|\mathfrak{m}_{2}(y)|^{2}+|\mathfrak{m}_{2}(p)|^{2}+|\mathfrak{m}_{2}(r)|^{2}+|c^{0}_{t}|^{2}+|c^{i}_{t}|^{2}\bigr{]},\\\
&\sum_{i=1}^{N_{1}}|\widehat{\beta}^{i}(t,u)-\widehat{\beta}^{i}(t,\acute{u})|^{2}\leq
C\sum_{i=1}^{N_{1}}\bigl{[}|X^{i}-\acute{X}^{i}|^{2}+|P^{i}-\acute{P}^{i}|^{2}+|\mathfrak{m}_{2}(y-\acute{y})|^{2}+|\mathfrak{m}_{2}(p-\acute{p})|^{2}+|\mathfrak{m}_{2}(r-\acute{r})|^{2}\bigr{]},\end{split}$

where
$\acute{u}:=(\acute{X},\acute{x},\acute{y},\acute{P},\acute{p},\acute{r})$ is
another input and $C$ is some positive constant independent of the population
sizes. Moreover, the minimizer $\widehat{\beta}(t,u)$ is coupled to the second
population in a symmetric way; it is independent from $x$ and depends on
$y,p,r$ only through their empirical means.

###### Proof.

First, we check the relevant convexity of $H$. For a given $(P,p,r)$, we
consider two arbitrary sets of state variables
$(X,x,y),(\acute{X},\acute{x},\acute{y})$ and controls $\beta,\acute{\beta}$.
With $u:=(X,x,y,P,p,r)$ and
$\acute{u}:=(\acute{X},\acute{x},\acute{y},P,p,r)$, we compare $H(t,u,\beta)$
and $H(t,\acute{u},\acute{\beta})$. For notational simplicity, let us set
$\nu^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X^{i}},\acute{\nu}^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{\acute{X}^{i}}$,
$\varpi:=\varpi_{t}(y,\beta),\acute{\varpi}:=\varpi_{t}(\acute{y},\acute{\beta})$,
$\Delta X^{i}:=\acute{X}^{i}-X^{i}$,
$\Delta\beta^{i}:=\acute{\beta}^{i}-\beta^{i}$,
$\Delta\varpi:=\acute{\varpi}-\varpi=-\mathfrak{m}_{2}(\Delta
y)+\delta_{\mathfrak{n}}\Lambda_{t}\mathfrak{m}_{1}(\Delta\beta)$, etc. We
also put
$f_{1}^{i}(t,u,\beta^{i}):=f_{1}^{i}(t,X^{i},\nu^{N_{1}},\varpi,\beta^{i}),f_{1}^{i}(t,\acute{u},\acute{\beta}^{i}):=f_{1}^{i}(t,\acute{X}^{i},\acute{\nu}^{N_{1}},\acute{\varpi},\acute{\beta}^{i})$.
Since the linear terms in $H$ cancel, we obtain by direct calculation using
$(\ref{eq-H1-beta})$,

$\small\begin{split}&H(t,\acute{u},\acute{\beta})-H(t,u,\beta)\\\
&-\sum_{i=1}^{N_{1}}\bigl{[}\langle\partial_{\beta^{i}}H(t,u,\beta),\Delta\beta^{i}\rangle+\langle\partial_{X^{i}}H(t,u,\beta),\Delta
X^{i}\rangle\bigr{]}-\sum_{j=1}^{N_{2}}\bigl{[}\langle\partial_{x^{j}}H(t,u,\beta),\Delta
x^{j}\rangle+\langle\partial_{y^{j}}H(t,u,\beta),\Delta
y^{j}\rangle\bigr{]}\\\
&=\sum_{i=1}^{N_{1}}\bigl{[}f_{1}^{i}(t,\acute{u},\acute{\beta}^{i})-f_{1}^{i}(t,u,\beta^{i})-\langle\partial_{\beta}f_{1}^{i}(t,u,\beta^{i}),\Delta\beta^{i}\rangle-\langle\partial_{\varpi}f_{1}^{i}(t,u,\beta^{i}),\Delta\varpi\rangle\\\
&-\langle\partial_{x}f_{1}^{i}(t,u,\beta^{i}),\Delta
X^{i}\rangle-\mathbb{E}^{\theta}[\langle\partial_{\mu}f_{1}^{i}(t,u,\beta^{i})(X^{\theta}),\Delta
X^{\theta}\rangle]\bigr{]}\geq\frac{1}{2}\sum_{i=1}^{N_{1}}\bigl{(}\gamma_{1}^{f}|\Delta
X^{i}|^{2}+\lambda_{\beta}|\Delta\beta^{i}|^{2}+\lambda_{\varpi}|\Delta\varpi|^{2}\bigr{)},\end{split}$
( 3.14)

which gives the desired convexity. Since $A_{1}$ is convex and closed, it
guarantees the existence of the unique minimizer. Here, we have introduced a
random variable $\theta$ on a probability space $(\Omega^{\theta},{\cal
F}^{\theta},\mathbb{P}^{\theta})$, which is distributed uniformly on the set
$\\{1,2,\ldots,N_{1}\\}$. We use $\mathbb{E}^{\theta}[\cdot]$ to denote the
expectation with respect to $\mathbb{P}^{\theta}$. In particular,

$\small\begin{split}\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\sum_{k=1}^{N_{1}}\langle\partial_{\mu}f_{1}^{k}(t,u,\beta^{k})(X^{i}),\Delta
X^{i}\rangle=\sum_{k=1}^{N_{1}}\mathbb{E}^{\theta}\bigl{[}\langle\partial_{\mu}f_{1}^{k}(t,u,\beta^{k})(X^{\theta}),\Delta
X^{\theta}\rangle\bigr{]}.\end{split}$ ( 3.15)

Since the distribution of the random variables $X^{\theta},\acute{X}^{\theta}$
are equal to $\nu^{N_{1}}$ and $\acute{\nu}^{N_{1}}$ respectively, Assumption
3.3 (iv) gives the inequality $(\ref{H1-joint-convexity})$.

We now prove the latter claims. The convexity $(\ref{H1-joint-convexity})$
implies that, for any fixed $(\beta_{0}^{i}\in A_{1})_{i=1}^{N_{1}}$,

$\small\begin{split}H(t,u,\beta_{0})\geq H(t,u,\widehat{\beta}(t,u))\geq
H(t,u,\beta_{0})+\sum_{i=1}^{N_{1}}\langle\widehat{\beta}_{t}^{i}-\beta_{0}^{i},\partial_{\beta^{i}}H(t,u,\beta_{0})\rangle+\frac{\lambda_{\beta}}{2}\sum_{i=1}^{N_{1}}|\widehat{\beta}_{t}^{i}-\beta_{0}^{i}|^{2},\end{split}$

which gives, by Young’s inequality,
$\sum_{i=1}^{N_{1}}|\widehat{\beta}_{t}^{i}-\beta_{0}^{i}|^{2}\leq
C\sum_{i=1}^{N_{1}}|\partial_{\beta^{i}}H(t,u,\beta_{0})|^{2}$, where $C$
depends on $\lambda_{\beta}$. Hence $(\ref{eq-H1-beta})$, the conditions (0)
and (iii) give the desired growth property for $\widehat{\beta}$. Finally, let
us put $\widehat{\beta}_{t}^{i}=\widehat{\beta}^{i}(t,u)$ and
$\widehat{\beta}_{t}^{\prime,i}=\widehat{\beta}^{i}(t,\acute{u})$ the
minimizers of $H$ with two inputs $u=(X,x,y,P,p,r)$ and
$\acute{u}=(\acute{X},\acute{x},\acute{y},\acute{P},\acute{p},\acute{r})$,
respectively. Using the optimality condition, we have
$\sum_{i=1}^{N_{1}}\langle\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i},\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}_{t}^{\prime})-\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t})\rangle\leq
0.$ By rearrangement,

$\small\begin{split}\sum_{i=1}^{N_{1}}\langle\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i},\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t}^{\prime})-\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t})\rangle\leq\sum_{i=1}^{N_{1}}\langle\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i},\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t}^{\prime})-\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}_{t}^{\prime})\rangle.\end{split}$

The convexity of the left hand side in $\beta$ yields
$\lambda_{\beta}\sum_{i=1}^{N_{1}}|\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i}|^{2}\leq\sum_{i=1}^{N_{1}}\langle\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i},\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t}^{\prime})-\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}_{t}^{\prime})\rangle$
and hence, an application of Young’s inequality gives

$\small\begin{split}\sum_{i=1}^{N_{1}}|\widehat{\beta}_{t}^{\prime,i}-\widehat{\beta}_{t}^{i}|^{2}\leq
C\sum_{i=1}^{N_{1}}|\partial_{\beta^{i}}H(t,u,\widehat{\beta}_{t}^{\prime})-\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}_{t}^{\prime})|^{2}.\end{split}$

From $(\ref{eq-H1-beta})$, the conditions (0) and (iii) give the desired
Lipschitz continuity. The last statement on the coupling of
$\widehat{\beta}(t,u)$ is a direct consequence of the form of this Lipschitz
continuity. ∎

The system of coupled equations $1\leq i\leq N_{1},1\leq j\leq N_{2}$ that
characterizes the optimal solution to the problem $(\ref{central-problem})$ is
given as follows:

$\small\begin{split}dX_{t}^{i}&=(\widehat{\beta}_{t}^{i}+\rho_{1}(t)\varpi_{t}(y_{t},\widehat{\beta}_{t})+l_{1}^{i}(t))dt+\sigma_{1}^{i,0}(t)dW_{t}^{0}+\sigma^{i}_{1}(t)dW_{t}^{i},\\\
dx_{t}^{j}&=(-\overline{\Lambda}_{t}(y_{t}^{j}+\varpi_{t}(y_{t},\widehat{\beta}_{t}))+\rho_{2}(t)\varpi_{t}(y_{t},\widehat{\beta}_{t})+l_{2}^{j}(t))dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
dy_{t}^{j}&=-(\partial_{x}\overline{f}_{2}^{j}(t,x_{t}^{j})-b_{t}\varpi_{t}(y_{t},\widehat{\beta}_{t}))dt+z_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{z}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}z_{t}^{j,k}dB_{t}^{k},\\\
dP_{t}^{i}&=-\Bigl{\\{}\partial_{x}f_{1}^{i}(t,X_{t}^{i},\nu_{t}^{N_{1}},\varpi_{t}(y_{t},\widehat{\beta}_{t}),\widehat{\beta}_{t}^{i})+\frac{1}{N_{1}}\sum_{k=1}^{N_{1}}\partial_{\mu}f_{1}^{k}(t,X_{t}^{k},\nu_{t}^{N_{1}},\varpi_{t}(y_{t},\widehat{\beta}_{t}),\widehat{\beta}_{t}^{k})(X_{t}^{i})\Bigr{\\}}dt\\\
&\qquad+Q_{t}^{i,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}Q_{t}^{i,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}\mathfrak{Q}_{t}^{i,k}dB_{t}^{k},\\\
dp_{t}^{j}&=c_{f}^{j}(t)r_{t}^{j}dt+q_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{q}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}q_{t}^{j,k}dB_{t}^{k},\\\
dr^{j}_{t}&=-\Bigl{\\{}\overline{\Lambda}_{t}(\mathfrak{m}_{2}(p_{t})-p_{t}^{j})-\rho_{2}(t)\mathfrak{m}_{2}(p_{t})-\delta_{\mathfrak{n}}\rho_{1}(t)\mathfrak{m}_{1}(P_{t})-b_{t}\mathfrak{m}_{2}(r_{t})\\\
&\qquad-\frac{1}{N_{2}}\sum_{k=1}^{N_{1}}\partial_{\varpi}f_{1}^{k}(t,X_{t}^{k},\nu_{t}^{N_{1}},\varpi_{t}(y_{t},\widehat{\beta}_{t}),\widehat{\beta}_{t}^{k})\Bigr{\\}}dt\end{split}$
( 3.16)

with the boundary conditions; $X_{0}^{i}=\xi^{i}$, $x_{0}^{j}=\eta^{j}$,
$r_{0}^{j}=0$ and

$\small\begin{split}y_{T}^{j}=\partial_{x}g_{2}^{j}(x_{T}^{j}),\quad
P^{i}_{T}=\partial_{x}g_{1}^{i}(X_{T}^{i},\nu_{T}^{N_{1}})+\frac{1}{N_{1}}\sum_{k=1}^{N_{1}}\partial_{\mu}g_{1}^{k}(X_{T}^{k},\nu_{T}^{N_{1}})(X_{T}^{i}),\
\quad p_{T}^{j}=-c_{g}^{j}r_{T}^{j}.\end{split}$

Note that the two populations are coupled only through the empirical means
contained in $\varpi_{t}(y,\beta)$. The relevance of $(\ref{fbsde-full-
finite})$ is based on the next theorem.

######  Remark 3.8.

From $(\ref{eq-H1-beta})$, one can check that the drift terms of adjoint
variables $(P^{i},p^{j},r^{j})$ are given by
$-\partial_{X^{i}}H(t,u,\widehat{\beta})$,
$-\partial_{x^{j}}H(t,u,\widehat{\beta})$ and
$-\partial_{y^{j}}H(t,u,\widehat{\beta})$, respectively.

######  Theorem 3.3.

Let Assumption 3.2-3.3 be in force. Suppose that there exists a unique square
integrable solution to the system of equations $(\ref{fbsde-full-finite})$
with
$u:=(X,x,y,P,p,r)\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{1}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{1}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}$
and $\widehat{\beta}_{t}:=(\widehat{\beta}^{i}(t,u_{t}))_{i=1}^{N_{1}}$ the
minimizer of $H(t,u_{t},\cdot)$ for $dt\otimes d\mathbb{P}$-a.e. Then,
$\widehat{\beta}_{t},t\in[0,T]$ is the unique optimal solution to
$(\ref{central-problem})$.

###### Proof.

Let $u_{t}:=(X_{t},x_{t},y_{t},P_{t},p_{t},r_{t}),t\in[0,T]$ denote the
solution to $(\ref{fbsde-full-finite})$ with $\widehat{\beta}_{t}$ the
minimizer of $H(t,u_{t},\cdot)$, $dt\otimes d\mathbb{P}$-a.e. By Lemma 3.1 and
the properties of $(c^{0},c^{i})$ given at the beginning of Section 3,
$\widehat{\beta}$ is square integrable and hence admissible. We denote by
$(\acute{X}_{t},\acute{x}_{t},\acute{y}_{t}),t\in[0,T]$ the state processes
corresponding to another control
$\acute{\beta}=(\acute{\beta}^{i})_{i=1}^{N_{1}}\in\mathbb{A}_{1}^{N_{1}}$ and
set
$\acute{u}_{t}:=(\acute{X}_{t},\acute{x}_{t},\acute{y}_{t},P_{t},p_{t},r_{t}),t\in[0,T]$.
Let us put
$\nu^{N_{1}}_{t}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X_{t}^{i}},\acute{\nu}_{t}^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{\acute{X}_{t}^{i}}$,
$\acute{\varpi}_{t}:=\varpi_{t}(\acute{y}_{t},\acute{\beta}_{t})$,
$\varpi_{t}:=\varpi_{t}(y_{t},\widehat{\beta}_{t})$, $\Delta X:=\acute{X}-X$,
$\Delta x:=\acute{x}-x$, $\Delta\beta=\acute{\beta}-\widehat{\beta}$ and
similarly for the others. For notational ease, we also use
$f^{i}_{1}(t,u_{t},\widehat{\beta}^{i}_{t}):=f_{1}^{i}(t,X_{t}^{i},\nu_{t}^{N_{1}},\varpi_{t}(y_{t},\widehat{\beta}_{t}),\widehat{\beta}_{t}^{i})$
and
$f_{1}^{i}(t,\acute{u}_{t},\acute{\beta}_{t}^{i}):=f_{1}^{i}(t,\acute{X}_{t}^{i},\acute{\nu}_{t}^{N_{1}},\varpi_{t}(\acute{y}_{t},\acute{\beta}_{t}),\acute{\beta}_{t}^{i})$.

From Assumption 3.3 (v) and the technique used in $(\ref{theta-technique})$,
we have

$\small\begin{split}\sum_{i=1}^{N_{1}}\bigl{(}g_{1}^{i}(\acute{X}_{T}^{i},\acute{\nu}_{T}^{N_{1}})-g_{1}^{i}(X_{T}^{i},\nu_{T}^{N_{1}})\bigr{)}&\geq\sum_{i=1}^{N_{1}}\langle
P_{T}^{i},\Delta X_{T}^{i}\rangle=\sum_{i=1}^{N_{1}}\langle P_{T}^{i},\Delta
X_{T}^{i}\rangle+\sum_{j=1}^{N_{2}}\bigl{(}\langle p_{T}^{j},\Delta
x_{T}^{j}\rangle+\langle r_{T}^{j},\Delta
y_{T}^{j}\rangle\bigr{)}.\end{split}$

In the latter equality, we used the fact $\langle p_{T}^{j},\Delta
x_{T}^{j}\rangle+\langle r_{T}^{j},\Delta y_{T}^{j}\rangle=0$. By the initial
condition $\Delta X^{i}_{0}=\Delta x^{i}_{0}=r^{j}_{0}=0$ and the optimality
$\sum_{i}\langle\partial_{\beta^{i}}H(t,u_{t},\widehat{\beta}_{t}),\Delta\beta^{i}_{t}\rangle\geq
0$ $dt\otimes d\mathbb{P}$-a.e., an application of Itô formula yields

$\small\begin{split}\sum_{i=1}^{N_{1}}J_{1}^{i}(\acute{\beta})-\sum_{i=1}^{N_{1}}J_{1}^{i}(\widehat{\beta})&\geq\mathbb{E}\Bigl{[}\sum_{i=1}^{N_{1}}\langle
P_{T}^{i},\Delta X_{T}^{i}\rangle+\sum_{j=1}^{N_{2}}\bigl{(}\langle
p_{T}^{j},\Delta x_{T}^{j}\rangle+\langle r_{T}^{j},\Delta
y_{T}^{j}\rangle\bigr{)}+\int_{0}^{T}\sum_{i=1}^{N_{1}}\bigl{[}f_{1}^{i}(t,\acute{u}_{t},\acute{\beta}_{t}^{i})-f_{1}^{i}(t,u_{t},\widehat{\beta}_{t}^{i})\bigr{]}dt\Bigr{]}\\\
&\geq\mathbb{E}\int_{0}^{T}\bigl{[}H(t,\acute{u}_{t},\acute{\beta}_{t})-H(t,u_{t},\widehat{\beta}_{t})-\sum_{i=1}^{N_{1}}\bigl{(}\langle\partial_{X^{i}}H(t,u_{t},\widehat{\beta}_{t}),\Delta
X_{t}^{i}\rangle+\langle\partial_{\beta^{i}}H(t,u_{t},\widehat{\beta}_{t}),\Delta\beta_{t}^{i}\rangle\bigr{)}\\\
&\quad-\sum_{j=1}^{N_{2}}\bigl{(}\langle\partial_{x^{j}}H(t,u_{t},\widehat{\beta}_{t}),\Delta
x_{t}^{j}\rangle+\langle\partial_{y^{j}}H(t,u_{t},\widehat{\beta}_{t}),\Delta
y_{t}^{j}\rangle\bigr{)}\bigr{]}dt.\end{split}$

The conclusion follows from the joint convexity $(\ref{H1-joint-convexity})$
in Lemma 3.1. ∎

####  3.2.3 Existence of the optimal solution

The aim of this section is to prove the existence of a strong solution to
$(\ref{fbsde-full-finite})$, which then will be combined with Theorem 3.3 to
solve the optimization problem for the first population. We use the following
maps to study the monotonicity [49, (H2.3)]. For each $(t,\omega)$, we define
the maps on
$(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{1}}\times(\mathbb{R}^{n})^{N_{2}}\times(\mathbb{R}^{n})^{N_{2}}\ni
u:=(X,x,y,P,p,r)\mapsto\mathbb{R}^{n}$ by

$\small\begin{split}&B_{X^{i}}(t,u):=\widehat{\beta}^{i}+\rho_{1}(t)\varpi_{t}(y,\widehat{\beta})+l_{1}^{i}(t),\\\
&B_{x^{j}}(t,u):=-\overline{\Lambda}_{t}(y^{j}+\varpi_{t}(y,\widehat{\beta}))+\rho_{2}(t)\varpi_{t}(y,\widehat{\beta})+l_{2}^{j}(t),\\\
&B_{r^{j}}(t,u):=-\bigl{\\{}\overline{\Lambda}_{t}(\mathfrak{m}_{2}(p)-p^{j})-\rho_{2}(t)\mathfrak{m}_{2}(p)-\delta_{\mathfrak{n}}\rho_{1}(t)\mathfrak{m}_{1}(P)-b_{t}\mathfrak{m}_{2}(r)-\frac{1}{N_{2}}\sum_{k=1}^{N_{1}}\partial_{\varpi}f_{1}^{k}(t,X^{k},\nu^{N_{1}},\varpi_{t}(y,\widehat{\beta}),\widehat{\beta}^{k})\bigr{\\}},\\\
&F_{P^{i}}(t,u):=-\bigl{\\{}\partial_{x}f_{1}^{i}(t,X^{i},\nu^{N_{1}},\varpi_{t}(y,\widehat{\beta}),\widehat{\beta}^{i})+\frac{1}{N_{1}}\sum_{k=1}^{N_{1}}\partial_{\mu}f_{1}^{k}(t,X^{k},\nu^{N_{1}},\varpi_{t}(y,\widehat{\beta}),\widehat{\beta}^{k})(X^{i})\bigr{\\}},\\\
&F_{y^{j}}(t,u):=-(\partial_{x}\overline{f}_{2}^{j}(t,x^{j})-b_{t}\varpi_{t}(y,\widehat{\beta})),\quad
F_{p^{j}}(t,u):=c_{f}^{j}(t)r^{j},\end{split}$

and

$\small\begin{split}G_{P^{i}}(u):=\partial_{x}g_{1}^{i}(X^{i},\nu^{N_{1}})+\frac{1}{N_{1}}\sum_{k=1}^{N_{1}}\partial_{\mu}g_{1}^{k}(X^{k},\nu^{N_{1}})(X^{i}),\quad
G_{y^{j}}(u):=\partial_{x}g_{2}^{j}(x^{j}),\quad
G_{p^{j}}(u):=-c_{g}^{j}r^{j},\end{split}$

with $1\leq i\leq N_{1},1\leq j\leq N_{2}$. Here,
$\nu^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X^{i}}$ and
$(\widehat{\beta}^{i}:=\widehat{\beta}^{i}(t,u))_{i=1}^{N_{1}}$ is the
minimizer of the Hamiltonian $H(t,u,\cdot)$. Let
$u:=(X,x,y,P,p,r),\acute{u}:=(\acute{X},\acute{x},\acute{y},\acute{P},\acute{p},\acute{r})\in(\mathbb{R}^{n})^{2N_{1}+4N_{2}}$
be two arbitrary inputs. We use abbreviations, such as,
$\widehat{\beta}^{i}=\widehat{\beta}^{i}(t,u)$,
$\widehat{\beta}^{\prime,i}:=\widehat{\beta}^{i}(t,\acute{u})$,
$\varpi:=\varpi_{t}(y,\widehat{\beta})$,
$\acute{\varpi}:=\varpi_{t}(\acute{y},\widehat{\beta}^{\prime})$,
$f_{1}^{i}(t,u,\widehat{\beta}^{i}):=f_{1}^{i}(t,X^{i},\nu^{N_{1}},\varpi,\widehat{\beta}^{i})$,
$f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}):=f_{1}^{i}(t,\acute{X}^{i},\acute{\nu}^{N_{1}},\acute{\varpi},\widehat{\beta}^{\prime,i})$
and also put $\Delta X^{i}:=X^{i}-\acute{X}^{i}$, $\Delta
y^{j}:=y^{j}-\acute{y}^{j}$,
$\Delta\widehat{\beta}^{i}:=\widehat{\beta}^{i}-\widehat{\beta}^{\prime,i}$,
$\Delta\varpi:=\varpi-\acute{\varpi}$, and $\Delta
B_{X^{i}}(t):=B_{X^{i}}(t,u)-B_{X^{i}}(t,\acute{u})$, $\Delta
F_{y^{j}}(t,u):=F_{y^{j}}(t,u)-F_{y^{j}}(t,\acute{u})$, $\Delta
G_{P^{i}}:=G_{P^{i}}(u)-G_{P^{i}}(\acute{u})$, etc.

######  Lemma 3.2.

Under Assumptions 3.2-3.3, for any inputs $(u,\acute{u})$, the terminal
functions satisfy

$\small\begin{split}\sum_{i=1}^{N_{1}}\langle\Delta G_{P^{i}},\Delta
X^{i}\rangle+\sum_{j=1}^{N_{2}}\bigl{(}\langle\Delta G_{y^{j}},\Delta
x^{j}\rangle+\langle(-1)\Delta G_{p^{j}},\Delta
r^{j}\rangle\bigr{)}\geq\gamma_{1}^{g}\sum_{i=1}^{N_{1}}|\Delta
X^{i}|^{2}+\gamma_{2}^{g}\sum_{j=1}^{N_{2}}(|\Delta x^{j}|^{2}+|\Delta
r^{j}|^{2}).\end{split}$

###### Proof.

This is a direct result of Assumption 3.2 (iv) and Assumption 3.3 (v). ∎

Although the proof requires rather lengthy calculation, the following result
is crucial. It tells that the drift terms do not satisfy the Peng-Wu’s
monotonicity conditions. However, we shall show that the violation of the
monotonicity can be controlled by $\delta_{\mathfrak{n}}:=N_{1}/N_{2}$, the
ratio of the two population sizes.

######  Lemma 3.3.

Let Assumptions 3.2-3.3 hold. Then, for any $t\in[0,T]$ and inputs
$(u,\acute{u})$, there exists a positive constant $\delta_{0}^{F}$ such that,
for any $(N_{1},N_{2})$ satisfying
$\delta_{\mathfrak{n}}:=N_{1}/N_{2}\leq\delta_{0}^{F}$, the inequality

$\small\begin{split}{\cal
D}(t)\leq-\gamma^{f}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x^{j}|^{2}+|\Delta
r^{j}|^{2})\bigr{]}+\delta_{\mathfrak{n}}C\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y^{j}|^{2}+|\Delta
p^{j}|^{2})\bigr{]}\end{split}$

holds with
$\gamma^{f}:=\min\bigl{(}\frac{\gamma_{1}^{f}}{2},\frac{\gamma_{2}^{f}}{3}\bigr{)}$
and some positive constant $C$ independent of the population sizes. Here,

$\small\begin{split}{\cal D}(t)&:=\sum_{i=1}^{N_{1}}\langle\Delta
B_{X^{i}}(t),\Delta P^{i}\rangle+\sum_{j=1}^{N_{2}}\bigl{[}\langle\Delta
B_{x^{j}}(t),\Delta y^{j}\rangle+\langle(-1)\Delta B_{r^{j}}(t),\Delta
p^{j}\rangle\bigr{]}\\\ &~{}+\sum_{i=1}^{N_{1}}\langle\Delta
F_{P^{i}}(t),\Delta X^{i}\rangle+\sum_{j=1}^{N_{2}}\bigl{[}\langle\Delta
F_{y^{j}}(t),\Delta x^{j}\rangle+\langle(-1)\Delta F_{p^{j}}(t),\Delta
r^{j}\rangle\bigr{]}.\end{split}$

###### Proof.

Let us evaluate each term. From $(\ref{eq-H1-beta})$, we can express $\Delta
P^{i}$ as

$\small\begin{split}\Delta
P^{i}&=\partial_{\beta^{i}}H(t,u,\widehat{\beta})-\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}^{\prime})-(\partial_{\beta^{i}}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{\beta^{i}}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}))+(I-\rho_{2}(t)\Lambda_{t})\mathfrak{m}_{2}(\Delta
p)\\\ &-b_{t}\Lambda_{t}\mathfrak{m}_{2}(\Delta
r)-\delta_{\mathfrak{n}}\rho_{1}(t)\Lambda_{t}\mathfrak{m}_{1}(\Delta
P)-\frac{\Lambda_{t}}{N_{2}}\sum_{k=1}^{N_{1}}(\partial_{\varpi}f_{1}^{k}(t,u,\widehat{\beta}^{k})-\partial_{\varpi}f_{1}^{k}(t,\acute{u},\widehat{\beta}^{\prime,k})).\end{split}$

It gives, together with the optimality condition
$\sum_{i}\langle\partial_{\beta^{i}}H(t,u,\widehat{\beta})-\partial_{\beta^{i}}H(t,\acute{u},\widehat{\beta}^{\prime}),\Delta\widehat{\beta}^{i}\rangle\leq
0$,

$\small\begin{split}&\sum_{i=1}^{N_{1}}\langle\Delta B_{X^{i}}(t),\Delta
P^{i}\rangle=\sum_{i=1}^{N_{1}}\langle\Delta\widehat{\beta}^{i}+\rho_{1}(t)\Delta\varpi,\Delta
P^{i}\rangle\\\
&\leq-\sum_{i=1}^{N_{1}}\langle\partial_{\beta^{i}}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{\beta^{i}}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}),\Delta\widehat{\beta}^{i}\rangle-\sum_{i=1}^{N_{1}}\langle\partial_{\varpi}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{\varpi}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}),\Delta\varpi+\mathfrak{m}_{2}(\Delta
y)\rangle\\\
&\quad+N_{1}\langle(I-\rho_{2}(t)\Lambda_{t})\mathfrak{m}_{2}(\Delta
p)-b_{t}\Lambda_{t}\mathfrak{m}_{2}(\Delta
r),\mathfrak{m}_{1}(\Delta\widehat{\beta})\rangle-
N_{1}\rho_{1}(t)\langle\mathfrak{m}_{2}(\Delta y),\mathfrak{m}_{1}(\Delta
P)\rangle.\end{split}$

Since $\sum_{j}\langle\overline{\Lambda}_{t}(\Delta
y^{j}-\mathfrak{m}_{2}(\Delta y)),\Delta y^{j}\rangle\geq 0$ and
$\sum_{j}\langle\overline{\Lambda}_{t}(\Delta p^{j}-\mathfrak{m}_{2}(\Delta
p)),\Delta p^{j}\rangle\geq 0$, we have

$\small\begin{split}&\sum_{j=1}^{N_{2}}\langle B_{x^{j}}(t),\Delta
y^{j}\rangle\leq-N_{2}\rho_{2}(t)|\mathfrak{m}_{2}(\Delta
y)|^{2}+N_{1}\langle(\rho_{2}(t)\Lambda_{t}-I)\mathfrak{m}_{1}(\Delta\widehat{\beta}),\mathfrak{m}_{2}(\Delta
y)\rangle,\\\ &\sum_{j=1}^{N_{2}}\langle(-1)\Delta B_{r^{j}}(t),\Delta
p^{j}\rangle\leq-N_{2}\rho_{2}(t)|\mathfrak{m}_{2}(\Delta
p)|^{2}-N_{2}b_{t}\langle\mathfrak{m}_{2}(\Delta r),\mathfrak{m}_{2}(\Delta
p)\rangle-N_{1}\rho_{1}(t)\langle\mathfrak{m}_{1}(\Delta
P),\mathfrak{m}_{2}(\Delta p)\rangle\\\
&\qquad-\sum_{i=1}^{N_{1}}\langle\partial_{\varpi}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{\varpi}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}),\mathfrak{m}_{2}(\Delta
p)\rangle.\end{split}$

By the technique used for $(\ref{theta-technique})$ in Lemma 3.1, we get

$\small\begin{split}&\sum_{i=1}^{N_{1}}\langle\Delta F_{P^{i}}(t),\Delta
X^{i}\rangle=-\sum_{i=1}^{N_{1}}\langle\partial_{x}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{x}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}),\Delta
X^{i}\rangle\\\
&\qquad-\sum_{i=1}^{N_{1}}\mathbb{E}^{\theta}\bigl{[}\langle\partial_{\mu}f_{1}^{i}(t,u,\widehat{\beta}^{i})(X^{\theta})-\partial_{\mu}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i})(\acute{X}^{\theta}),\Delta
X^{\theta}\rangle\bigr{]}.\end{split}$

Finally, as a direct result from Assumption 3.2 (iv), we have

$\small\begin{split}\sum_{j=1}^{N_{2}}\langle\Delta F_{y^{j}}(t),\Delta
x^{j}\rangle+\sum_{j=1}^{N_{2}}\langle(-1)\Delta F_{p^{j}}(t),\Delta
r^{j}\rangle\leq-\gamma_{2}^{f}\sum_{j=1}^{N_{2}}(|\Delta x^{j}|^{2}+|\Delta
r^{j}|^{2})+N_{2}b_{t}\langle\Delta\varpi,\mathfrak{m}_{2}(\Delta
x)\rangle.\end{split}$

Summing all the terms and using Assumption 3.3 (iv), we get

$\small\begin{split}{\cal D}(t)\leq-\gamma_{1}^{f}\sum_{i=1}^{N_{1}}|\Delta
X^{i}|^{2}-\gamma_{2}^{f}\sum_{j=1}^{N_{2}}(|\Delta x^{j}|^{2}+|\Delta
r^{j}|^{2})-\sum_{i=1}^{N_{1}}(\lambda_{\beta}|\Delta\widehat{\beta}^{i}|^{2}+\lambda_{\varpi}|\Delta\varpi|^{2})+{\cal
R}(t),\end{split}$ ( 3.17)

where the residual term ${\cal R}(t)$ is defined by

$\small\begin{split}{\cal R}(t)&:=-N_{2}\rho_{2}(t)(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta
p)|^{2})-\sum_{i=1}^{N_{1}}\langle\partial_{\varpi}f_{1}^{i}(t,u,\widehat{\beta}^{i})-\partial_{\varpi}f_{1}^{i}(t,\acute{u},\widehat{\beta}^{\prime,i}),\mathfrak{m}_{2}(\Delta
y+\Delta p)\rangle\\\
&\quad+N_{1}\langle(1-\rho_{2}(t)\Lambda_{t})\mathfrak{m}_{2}(\Delta p-\Delta
y)-b_{t}\Lambda_{t}\mathfrak{m}_{2}(\Delta
r),\mathfrak{m}_{1}(\Delta\widehat{\beta})\rangle-
N_{1}\rho_{1}(t)\langle\mathfrak{m}_{1}(\Delta P),\mathfrak{m}_{2}(\Delta
y+\Delta p)\rangle\\\ &\quad-N_{2}b_{t}\langle\mathfrak{m}_{2}(\Delta
r),\mathfrak{m}_{2}(\Delta
p)\rangle+N_{2}b_{t}\langle\Delta\varpi,\mathfrak{m}_{2}(\Delta
x)\rangle.\end{split}$

We get an estimate on ${\cal R}(t)$ from the Lipschitz continuity of
$\partial_{\varpi}f_{1}^{i}$ and Young’s inequality. In particular,

$\small\begin{split}&N_{1}|\rho_{1}(t)\langle\mathfrak{m}_{1}(\Delta
P),\mathfrak{m}_{2}(\Delta y+\Delta p)\rangle|\leq
N_{2}\frac{\rho_{2}(t)}{2}(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta
p)|^{2})+\frac{N_{1}^{2}\rho_{1}(t)^{2}}{N_{2}\rho_{2}(t)}\Bigl{(}\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}|\Delta
P^{i}|^{2}\Bigr{)}\\\ &\qquad\leq
N_{2}\frac{\rho_{2}(t)}{2}(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta
p)|^{2})+C\frac{N_{1}}{N_{2}}\sum_{i=1}^{N_{1}}|\Delta P^{i}|^{2},\\\
&N_{2}b_{t}|\langle\Delta\varpi,\mathfrak{m}_{2}(\Delta
x)\rangle|\leq\frac{\gamma_{2}^{f}}{2}\sum_{j=1}^{N_{2}}|\Delta
x^{j}|^{2}+\frac{N_{2}b_{t}^{2}}{\gamma_{2}^{f}}|\mathfrak{m}_{2}(\Delta
y)|^{2}+C\frac{N_{1}}{N_{2}}\sum_{i=1}^{N_{1}}|\Delta\widehat{\beta}^{i}|^{2}\end{split}$

with some constant $C$. Similar calculations yield

$\small\begin{split}{\cal
R}(t)\leq&\sum_{i=1}^{N_{1}}\bigl{[}\frac{\gamma_{1}^{f}}{2}|\Delta
X^{i}|^{2}+\frac{\lambda_{\varpi}}{2}|\Delta\varpi|^{2}+\frac{\lambda_{\beta}}{2}|\Delta\widehat{\beta}^{i}|^{2}\bigr{]}+\frac{\gamma_{2}^{f}}{2}\sum_{j=1}^{N_{2}}(|\Delta
x^{j}|^{2}+|\Delta r^{j}|^{2})\\\ &+CN_{1}(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta p)|^{2}+|\mathfrak{m}_{2}(\Delta
r)|^{2})+C\frac{N_{1}}{N_{2}}\sum_{i=1}^{N_{1}}(|\Delta\widehat{\beta}^{i}|^{2}+|\Delta
P^{i}|^{2})\\\ &-N_{2}\frac{\rho_{2}(t)}{2}(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta
p)|^{2})+\frac{N_{2}b_{t}^{2}}{2\gamma_{2}^{f}}|\mathfrak{m}_{2}(\Delta
p)|^{2}+\frac{N_{2}b_{t}^{2}}{\gamma_{2}^{f}}|\mathfrak{m}_{2}(\Delta
y)|^{2}\\\ \leq&\sum_{i=1}^{N_{1}}\bigl{[}\frac{\gamma_{1}^{f}}{2}|\Delta
X^{i}|^{2}+\frac{\lambda_{\varpi}}{2}|\Delta\varpi|^{2}+\frac{\lambda_{\beta}}{2}|\Delta\widehat{\beta}^{i}|^{2}\bigr{]}+\frac{\gamma_{2}^{f}}{2}\sum_{j=1}^{N_{2}}(|\Delta
x^{j}|^{2}+|\Delta r^{j}|^{2})\\\ &+CN_{1}(|\mathfrak{m}_{2}(\Delta
y)|^{2}+|\mathfrak{m}_{2}(\Delta p)|^{2}+|\mathfrak{m}_{2}(\Delta
r)|^{2})+C\frac{N_{1}}{N_{2}}\sum_{i=1}^{N_{1}}(|\Delta\widehat{\beta}^{i}|^{2}+|\Delta
P^{i}|^{2}),\end{split}$ ( 3.18)

where we have used Assumption 3.3 (vi) in the second inequality. Note that,
the constant $C$ depends on $L$ and convexity parameters but not on
$(N_{1},N_{2})$. Now we choose
$\delta_{0}^{F}:=\min\bigl{(}\frac{\lambda_{\beta}}{2C},\frac{\gamma_{2}^{f}}{6C}\bigr{)}$.
Then, for any $\delta_{\mathfrak{n}}=N_{1}/N_{2}$ satisfying
$\delta_{\mathfrak{n}}\leq\delta_{0}^{F}$, we have

$\small\begin{split}{\cal
R}(t)\leq&\sum_{i=1}^{N_{1}}\bigl{[}\frac{\gamma_{1}^{f}}{2}|\Delta
X^{i}|^{2}+\frac{\lambda_{\varpi}}{2}|\Delta\varpi|^{2}+\lambda_{\beta}|\Delta\widehat{\beta}^{i}|^{2}\bigr{]}+\gamma_{2}^{f}\sum_{j=1}^{N_{2}}\bigl{[}\frac{1}{2}|\Delta
x^{j}|^{2}+\frac{2}{3}|\Delta r^{j}|^{2}\bigr{]}\\\
&+C\delta_{\mathfrak{n}}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y^{j}|^{2}+|\Delta
p^{j}|^{2})\bigr{]}.\end{split}$

By inserting into $(\ref{eq-cald})$, we obtain the desired estimate on ${\cal
D}(t)$. ∎

We now provide the first main result. Since Peng-Wu’s monotonicity does not
hold, we need in general to make all the Lipschitz constants small enough to
guarantee the well-posedness of FBSDEs for a given duration. However, the
result of Lemma 3.3, which shows the violation of the monotonicity is
proportional to $\delta_{\mathfrak{n}}$, suggests that the well-posedness of
the system $(\ref{fbsde-full-finite})$ may be available as long as the ratio
$\delta_{\mathfrak{n}}=N_{1}/N_{2}$ is small enough. We shall show that this
is actually the case by the next theorem.

######  Theorem 3.4.

Let Assumptions 3.2-3.3 hold. Then, there exists some positive constant
$\delta_{*}^{F}\leq\delta_{0}^{F}$ such that, for any $(N_{1},N_{2})$
satisfying $\delta_{\mathfrak{n}}\leq\delta_{*}^{F}$, there exists a unique
square integrable solution to the system of FBSDEs $(\ref{fbsde-full-finite})$
with
$u:=(X,x,y,P,p,r)\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{1}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{1}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}\times\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{N_{2}}$.

###### Proof.

First, we put
$\gamma:=\min\bigl{(}\frac{\gamma_{1}^{f}}{2},\frac{\gamma_{2}^{f}}{3},\gamma_{1}^{g},\gamma_{2}^{g}\bigr{)}$.
Let $I_{X^{i}},I_{x^{j}},I_{r^{j}},I_{P^{i}},I_{y^{j}},I_{p^{j}}$ be in
$\mathbb{H}^{2}(\mathbb{F};\mathbb{R}^{n})$ and
$\theta_{P^{i}},\theta_{y^{j}},\theta_{p^{j}}$ be in $\mathbb{L}^{2}({\cal
F}_{T},\mathbb{R}^{n})$ for every $1\leq i\leq N_{1},1\leq j\leq N_{2}$. We
hypothesize that, for any choice of
$(I_{X^{i}},I_{x^{j}},I_{y^{j}},I_{P^{i}},I_{p^{j}},I_{r^{j}})$ and
$(\theta_{P^{i}},\theta_{y^{j}},\theta_{p^{j}})$, there exists some
$\varrho\in[0,1)$ such that there exists a unique square integrable solution
with
$\overline{u}=(\overline{X},\overline{x},\overline{y},\overline{P},\overline{p},\overline{r})\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$
to the following system of FBSDEs, $1\leq i\leq N_{1},1\leq j\leq N_{2}$:

$\small\begin{split}&d\overline{X}_{t}^{i}=(\varrho
B_{X^{i}}(t,\overline{u}_{t})+I_{X^{i}}(t))dt+\sigma_{1}^{i,0}(t)dW_{t}^{0}+\sigma_{1}^{i}(t)dW_{t}^{i},\\\
&d\overline{x}_{t}^{j}=(\varrho
B_{x^{j}}(t,\overline{u}_{t})+I_{x^{j}}(t))dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
&d\overline{r}_{t}^{j}=(\varrho
B_{r^{j}}(t,\overline{u}_{t})+I_{r^{j}}(t))dt,\\\
&d\overline{P}_{t}^{i}=-\bigl{[}(1-\varrho)\gamma\overline{X}_{t}^{i}-\varrho
F_{P^{i}}(t,\overline{u}_{t})+I_{P^{i}}(t)\bigr{]}dt+\overline{Q}_{t}^{i,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\overline{Q}_{t}^{i,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}\overline{\mathfrak{Q}}_{t}^{i,k}dB_{t}^{k},\\\
&d\overline{y}_{t}^{j}=-\bigl{[}(1-\varrho)\gamma\overline{x}_{t}^{j}-\varrho
F_{y^{j}}(t,\overline{u}_{t})+I_{y^{j}}(t)\bigr{]}dt+\overline{z}_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\overline{\mathfrak{z}}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}\overline{z}_{t}^{j,k}dB_{t}^{k},\\\
&d\overline{p}_{t}^{j}=-\bigl{[}-(1-\varrho)\gamma\overline{r}_{t}^{j}-\varrho
F_{p^{j}}(t,\overline{u}_{t})+I_{p^{j}}(t)\bigr{]}dt+\overline{q}_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\overline{\mathfrak{q}}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}\overline{q}_{t}^{j,k}dB_{t}^{k},\end{split}$

with
$\overline{X}_{0}^{i}=\xi^{i},~{}\overline{x}_{0}^{j}=\eta^{j},~{}\overline{r}_{0}^{j}=0$
and $\overline{P}_{T}^{i}=\varrho
G_{P^{i}}(\overline{u}_{T})+(1-\varrho)\overline{X}_{T}^{i}+\theta_{P^{i}},~{}\overline{y}_{T}^{j}=\varrho
G_{y^{j}}(\overline{u}_{T})+(1-\varrho)\overline{x}_{T}^{j}+\theta_{y^{j}},~{}\overline{p}_{T}^{j}=\varrho
G_{p^{j}}(\overline{u}_{T})-(1-\varrho)\overline{r}_{T}^{j}+\theta_{p^{j}}.$
The hypothesis clearly holds when $\varrho=0$ since then the system becomes
the decoupled Lipschitz FBSDEs.

Now, with a given input process
$\overline{u}\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$, we
consider the perturbed FBSDEs on $u=(X,x,y,P,p,r)$ with $\zeta\in(0,1)$,
$1\leq i\leq N_{1},1\leq j\leq N_{2}$:

$\small\begin{split}&dX_{t}^{i}=\bigl{[}\varrho B_{X^{i}}(t,u_{t})+\zeta
B_{X^{i}}(t,\overline{u}_{t})+I_{X^{i}}(t)\bigr{]}dt+\sigma_{1}^{i,0}(t)dW_{t}^{0}+\sigma_{1}^{i}(t)dW_{t}^{i},\\\
&dx_{t}^{j}=\bigl{[}\varrho B_{x^{j}}(t,u_{t})+\zeta
B_{x^{j}}(t,\overline{u}_{t})+I_{x^{j}}(t)\bigr{]}dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
&dr_{t}^{j}=\bigl{[}\varrho B_{r^{j}}(t,u_{t})+\zeta
B_{r^{j}}(t,\overline{u}_{t})+I_{r^{j}}(t)\bigr{]}dt,\\\
&dP_{t}^{i}=-\bigl{[}(1-\varrho)\gamma X_{t}^{i}-\varrho
F_{P^{i}}(t,u_{t})+\zeta(-\gamma\overline{X}_{t}^{i}-F_{P^{i}}(t,\overline{u}_{t}))+I_{P^{i}}(t)\bigr{]}dt\\\
&\qquad\quad+Q_{t}^{i,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}Q_{t}^{i,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}\mathfrak{Q}_{t}^{i,k}dB_{t}^{k},\\\
&dy_{t}^{j}=-\bigl{[}(1-\varrho)\gamma x_{t}^{j}-\varrho
F_{y^{j}}(t,u_{t})+\zeta(-\gamma\overline{x}_{t}^{j}-F_{y^{j}}(t,\overline{u}_{t}))+I_{y^{j}}(t)\bigr{]}dt\\\
&\qquad\quad+z_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{z}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}z_{t}^{j,k}dB_{t}^{k},\\\
&dp_{t}^{j}=-\bigl{[}-(1-\varrho)\gamma r_{t}^{j}-\varrho
F_{p^{j}}(t,u_{t})+\zeta(\gamma\overline{r}_{t}^{j}-F_{p^{j}}(t,\overline{u}_{t}))+I_{p^{j}}(t)\bigr{]}dt\\\
&\qquad\quad+q_{t}^{j,0}dW_{t}^{0}+\sum_{k=1}^{N_{1}}\mathfrak{q}_{t}^{j,k}dW_{t}^{k}+\sum_{k=1}^{N_{2}}q_{t}^{j,k}dB_{t}^{k},\end{split}$
( 3.19)

with the boundary conditions
$X_{0}^{i}=\xi^{i},~{}x^{j}_{0}=\eta^{j},~{}r_{0}^{j}=0$ and
$P_{T}^{i}=\varrho
G_{P^{i}}(u_{T})+(1-\varrho)X_{T}^{i}+\zeta(G_{P^{i}}(\overline{u}_{T})-\overline{X}_{T}^{i})+\theta_{P^{i}}$,
$y_{T}^{j}=\varrho
G_{y^{j}}(u_{T})+(1-\varrho)x_{T}^{j}+\zeta(G_{y^{j}}(\overline{u}_{T})-\overline{x}_{T}^{j})+\theta_{y^{j}}$
and $p_{T}^{j}=\varrho
G_{p^{j}}(u_{T})-(1-\varrho)r_{T}^{j}+\zeta(G_{p^{j}}(\overline{u}_{T})+\overline{r}_{T}^{j})+\theta_{p^{j}}$.
Note that, by the initial hypothesis, there exists a unique solution
$u=(X,x,y,P,p,r)\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$
to the FBSDEs $(\ref{fbsde-shifted})$. This defines a map
$\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}\ni\overline{u}\mapsto
u\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$. If we show
that thie map is a contraction for some $\zeta>0$ independent of $\varrho$,
then we can extend the hypothesis from $\varrho$ to $(\varrho+\zeta)$. Then we
are done, because $\varrho=1$ and $I,\theta=0$ give the interested FBSDEs
$(\ref{fbsde-full-finite})$. In the following, we shall show that if
$\delta_{\mathfrak{n}}\leq\delta_{*}^{F}$ with some $\delta_{*}^{F}>0$
determined independently from $\varrho$, there exists a sufficiently small
$\zeta>0$ independent of $\varrho$ such that the above map
$\overline{u}\mapsto u$ becomes a contraction as desired.

First step: Let us denote by
$u,\acute{u}\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$ the
solutions to $(\ref{fbsde-shifted})$ with two input processes
$\overline{u},\acute{\overline{u}}\in\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$,
respectively. As in Lemmas 3.2-3.3, we use the notations: $\Delta
u:=u-\acute{u}$, $\Delta\overline{u}:=\overline{u}-\acute{\overline{u}}$,
$\Delta B_{X^{i}}(t):=B_{X^{i}}(t,u_{t})-B_{X^{i}}(t,\acute{u}_{t})$,
$\Delta\overline{B}_{X^{i}}(t):=B_{X^{i}}(t,\overline{u}_{t})-B_{X^{i}}(t,\acute{\overline{u}}_{t})$,
$\Delta F_{P^{i}}(t):=F_{P^{i}}(t,u_{t})-F_{P^{i}}(t,\acute{u}_{t})$,
$\Delta\overline{F}_{P^{i}}(t):=F_{P^{i}}(t,\overline{u}_{t})-F_{P^{i}}(t,\acute{\overline{u}}_{t})$,
$\Delta G_{P^{i}}:=G_{P^{i}}(u_{T})-G_{P^{i}}(\acute{u}_{T})$,
$\Delta\overline{G}_{P^{i}}:=G_{P^{i}}(\overline{u}_{T})-G_{P^{i}}(\acute{\overline{u}}_{T})$,
etc. An application of Itô formula with the result of Lemma 3.3 gives

$\small\begin{split}&\sum_{i=1}^{N_{1}}\mathbb{E}\bigl{[}\langle\Delta
X_{T}^{i},\Delta
P_{T}^{i}\rangle\bigr{]}+\sum_{j=1}^{N_{2}}\mathbb{E}\bigl{[}\langle\Delta
x_{T}^{j},\Delta y_{T}^{j}\rangle+\langle(-1)\Delta r_{T}^{j},\Delta
p_{T}^{j}\rangle\bigr{]}\\\
&\leq-\gamma\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{t}^{j}|^{2}+|\Delta
r_{t}^{j}|^{2})\bigr{]}dt\\\
&+\delta_{\mathfrak{n}}C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt+\zeta\mathbb{E}\int_{0}^{T}\sum_{i=1}^{N_{1}}\bigl{[}\langle\Delta\overline{B}_{X^{i}}(t),\Delta
P_{t}^{i}\rangle+\langle\gamma\Delta\overline{X}_{t}^{i}+\Delta\overline{F}_{P^{i}}(t),\Delta
X_{t}^{i}\rangle\bigr{]}dt\\\
&+\zeta\mathbb{E}\int_{0}^{T}\sum_{j=1}^{N_{2}}\bigl{[}\langle\Delta\overline{B}_{x^{j}}(t),\Delta
y_{t}^{j}\rangle+\langle\gamma\Delta\overline{x}_{t}^{j}+\Delta\overline{F}_{y^{j}}(t),\Delta
x_{t}^{j}\rangle+\langle(-1)\Delta\overline{B}_{r^{j}}(t),\Delta
p_{t}^{j}\rangle+\langle\gamma\Delta\overline{r}_{t}^{j}-\Delta\overline{F}_{p^{j}}(t),\Delta
r_{t}^{j}\rangle\bigr{]}dt.\end{split}$

We also have, from Lemma 3.2,

$\small\begin{split}&\sum_{i=1}^{N_{1}}\mathbb{E}\bigl{[}\langle\Delta
X_{T}^{i},\Delta
P_{T}^{i}\rangle\bigr{]}+\sum_{j=1}^{N_{2}}\mathbb{E}\bigl{[}\langle\Delta
x_{T}^{j},\Delta y_{T}^{j}\rangle+\langle(-1)\Delta r_{T}^{j},\Delta
p_{T}^{j}\rangle\bigr{]}\\\
&\geq(\varrho\gamma+(1-\varrho))\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{T}^{j}|^{2}+|\Delta
r_{T}^{j}|^{2})\bigr{]}\\\
&\quad+\zeta\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}\langle\Delta\overline{G}_{P^{i}}-\Delta\overline{X}_{T}^{i},\Delta
X_{T}^{i}\rangle+\sum_{j=1}^{N_{2}}(\langle\Delta\overline{G}_{y^{j}}-\Delta\overline{x}_{T}^{j},\Delta
x_{T}^{j}\rangle+\langle\Delta\overline{G}_{p^{j}}+\Delta\overline{r}_{T}^{j},(-1)\Delta
r_{T}^{j}\rangle\bigr{]}.\end{split}$

Put $\gamma_{c}:=\min(1,\gamma)$, which satisfies
$0<\gamma_{c}\leq\varrho\gamma+(1-\varrho)$, then the above two inequalities
yield

$\small\begin{split}&\gamma_{c}\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{T}^{j}|^{2}+|\Delta
r_{T}^{j}|^{2})\Bigr{]}+\gamma_{c}\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{t}^{j}|^{2}+|\Delta
r_{t}^{j}|^{2})\bigr{]}dt\\\
&\leq\delta_{\mathfrak{n}}C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt+\zeta\mathbb{E}\int_{0}^{T}\sum_{i=1}^{N_{1}}\bigl{[}\langle\Delta\overline{B}_{X^{i}}(t),\Delta
P_{t}^{i}\rangle+\langle\gamma\Delta\overline{X}_{t}^{i}+\Delta\overline{F}_{P^{i}}(t),\Delta
X_{t}^{i}\rangle\bigr{]}dt\\\
&\quad+\zeta\mathbb{E}\int_{0}^{T}\sum_{j=1}^{N_{2}}\bigl{[}\langle\Delta\overline{B}_{x^{j}}(t),\Delta
y_{t}^{j}\rangle+\langle\gamma\Delta\overline{x}_{t}^{j}+\Delta\overline{F}_{y^{j}}(t),\Delta
x_{t}^{j}\rangle+\langle(-1)\Delta\overline{B}_{r^{j}}(t),\Delta
p_{t}^{j}\rangle+\langle\gamma\Delta\overline{r}_{t}^{j}-\Delta\overline{F}_{p^{j}}(t),\Delta
r_{t}^{j}\rangle\bigr{]}dt.\\\
&\quad-\zeta\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}\langle\Delta\overline{G}_{P^{i}}-\Delta\overline{X}_{T}^{i},\Delta
X_{T}^{i}\rangle+\sum_{j=1}^{N_{2}}(\langle\Delta\overline{G}_{y^{j}}-\Delta\overline{x}_{T}^{j},\Delta
x_{T}^{j}\rangle+\langle\Delta\overline{G}_{p^{j}}+\Delta\overline{r}_{T}^{j},(-1)\Delta
r_{T}^{j}\rangle\bigr{]}.\end{split}$

We now expand the right hand side by using Lipschitz continuity of functions,
Lemma 3.1 and Cauchy-Schwarz inequality. By choosing $\zeta$ small enough to
absorb the terms $(|\Delta X|^{2},|\Delta x|^{2},|\Delta r|^{2})$ into the
left hand side, we obtain

$\small\begin{split}&\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{T}^{j}|^{2}+|\Delta
r_{T}^{j}|^{2})\bigr{]}+\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{t}^{j}|^{2}+|\Delta
r_{t}^{j}|^{2})\bigr{]}dt\\\
&\leq(\zeta+\delta_{\mathfrak{n}})C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt\\\ &\quad+\zeta
C\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta\overline{X}_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{T}^{j}|^{2}+|\Delta\overline{r}_{T}^{j}|^{2})\bigr{]}\\\
&\quad+\zeta
C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}(|\Delta\overline{X}_{t}^{i}|^{2}+|\Delta\overline{P}_{t}^{i}|^{2})+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{t}^{j}|^{2}+|\Delta\overline{r}_{t}^{j}|^{2}+|\Delta\overline{y}_{t}^{j}|^{2}+|\Delta\overline{p}_{t}^{j}|^{2})\bigr{]}dt.\end{split}$
( 3.20)

Here, the constant $C$ depends only on $L$ and the convexity parameters such
as $\gamma,\lambda$. In particular, $C$ and the $\zeta$ used in the above
transformation are independent of $\varrho$ and the population sizes.

Second step: We need to get estimates on the backward components $(|\Delta
P|^{2},|\Delta y|^{2},|\Delta p|^{2})$ in the right hand side of $(\ref{first-
step-finite})$. In addition to $(\overline{u},\acute{\overline{u}})$, let us
now treat $(X,x,r)$ as well as $(\acute{X},\acute{x},\acute{r})$ as exogenous
input processes. We consider $(P,y,p)$ as the solution to the BSDEs in
$(\ref{fbsde-shifted})$ with the input processes $(\overline{u},X,x,r)$, and
$(\acute{P},\acute{y},\acute{p})$ with
$(\acute{\overline{u}},\acute{X},\acute{x},\acute{r})$, respectively. We can
then apply the standard stability result for Lipschitz BSDEs (see, for
example, [57, Theorem 4.4.4] or [50, Theorem 5.21]) to obtain

$\small\begin{split}&\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt\\\ &\leq
C\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{T}^{j}|^{2}+|\Delta
r_{T}^{j}|^{2})\bigr{]}+C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{t}^{j}|^{2}+|\Delta
r_{t}^{j}|^{2})\bigr{]}dt\\\ &\quad+\zeta
C\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta\overline{X}_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{T}^{j}|^{2}+|\Delta\overline{r}_{T}^{j}|^{2})\bigr{]}\\\
&\quad+\zeta
C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}(|\Delta\overline{X}_{t}^{i}|^{2}+|\Delta\overline{P}_{t}^{i}|^{2})+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{t}^{j}|^{2}+|\Delta\overline{r}_{t}^{j}|^{2}+|\Delta\overline{y}_{t}^{j}|^{2}+|\Delta\overline{p}_{t}^{j}|^{2})\bigr{]}dt,\end{split}$

where $C$ depends also on $T$ in addition to the Lipschitz constants. This
result combined with $(\ref{first-step-finite})$ then implies

$\small\begin{split}&\mathbb{E}\Bigl{[}\sum_{i=1}^{N_{1}}|\Delta
X_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta x_{T}^{j}|^{2}+|\Delta
r_{T}^{j}|^{2})+\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}(|\Delta
X_{t}^{i}|^{2}+|\Delta P_{t}^{i}|^{2})+\sum_{j=1}^{N_{2}}(|\Delta
x_{t}^{j}|^{2}+|\Delta r_{t}^{j}|^{2}+|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt\Bigr{]}\\\
&\leq(\zeta+\delta_{\mathfrak{n}})C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta
P_{t}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta y_{t}^{j}|^{2}+|\Delta
p_{t}^{j}|^{2})\bigr{]}dt\\\ &\quad+\zeta
C\mathbb{E}\bigl{[}\sum_{i=1}^{N_{1}}|\Delta\overline{X}_{T}^{i}|^{2}+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{T}^{j}|^{2}+|\Delta\overline{r}_{T}^{j}|^{2})\bigr{]}\\\
&\quad+\zeta
C\mathbb{E}\int_{0}^{T}\bigl{[}\sum_{i=1}^{N_{1}}(|\Delta\overline{X}_{t}^{i}|^{2}+|\Delta\overline{P}_{t}^{i}|^{2})+\sum_{j=1}^{N_{2}}(|\Delta\overline{x}_{t}^{j}|^{2}+|\Delta\overline{r}_{t}^{j}|^{2}+|\Delta\overline{y}_{t}^{j}|^{2}+|\Delta\overline{p}_{t}^{j}|^{2})\bigr{]}dt.\end{split}$
( 3.21)

Choose $\delta_{*}^{F}\leq\delta_{0}^{F}$ so that $C\delta_{*}^{F}<1$. Then,
for any $\delta_{\mathfrak{n}}=N_{1}/N_{2}\leq\delta_{*}^{F}$, there exists
$\zeta>0$ sufficiently small so that

$\small\begin{split}(\zeta+\delta_{\mathfrak{n}})C<1,~{}\text{and}~{}\frac{\zeta
C}{1-(\zeta+\delta_{\mathfrak{n}})C}<1.\end{split}$

For such a $\zeta$, the inequality $(\ref{second-step-finite})$ implies that
the map $\overline{u}\mapsto u$ is a contraction with respect to the norm
$\mathbb{E}[|u_{T}|^{2}+\int_{0}^{T}|u_{t}|^{2}dt]^{\frac{1}{2}}$. By a simple
application of Burkholder-Davis-Gundy (BDG) inequality, one can also show that
it is a contraction map in the space
$\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})^{2N_{1}+4N_{2}}$. ∎

By Theorems 3.3 and 3.4,
$(\widehat{\beta}_{t}^{i}:=\widehat{\beta}^{i}(t,u_{t}),i=1,\ldots,N_{1})_{t\in[0,T]}$
and $(\varpi_{t}(y_{t},\widehat{\beta}_{t}))_{t\in[0,T]}$ defined by the
solution $u$ of $(\ref{fbsde-full-finite})$ give the unique optimal solution
to $(\ref{central-problem})$ and the associated equilibrium price process,
respectively. As one can see from the form of $\varpi_{t}(\cdot,\cdot)$, the
cooperative agents gain more market power, i.e. larger price impacts, as their
relative population size $\delta_{\mathfrak{n}}$ grows. If the power is too
strong, it may allow the cooperative agents to make arbitrary amount of profit
by some sort of price manipulation. In fact, in the literature on the optimal
trade executions, it has been known that large price impacts may cause a so-
called “hot-potato” problem of strongly oscillating sale and purchase. (See,
for example, Schied & Zhang (2017) [53].) It seems that this is one of the
economic reasons why we need a small $\delta_{\mathfrak{n}}$ for the well-
posedness.

Since $(\ref{fbsde-full-finite})$ has a strong unique $\mathbb{F}$-adapted
solution, Yamada-Watanabe Theorem for FBSDEs [15, Theorem 1.33] implies that
there exists a some measurable function $\Phi$ satisfying

$\small\begin{split}\bigl{(}(X^{i},P^{i})_{\\{i=1,\ldots,N_{1}\\}},(x^{j},y^{j},p^{j},r^{j})_{\\{j=1,\ldots,N_{2}\\}}\bigr{)}=\Phi\bigl{(}W^{0},(\xi^{i},W^{i})_{\\{i=1,\ldots,N_{1}\\}},(\eta^{j},B^{j})_{\\{j=1,\ldots,N_{2}\\}}\bigr{)}.\end{split}$

By the symmetry in the interactions of the agents (see the last statement of
Lemma 3.1) with the form of empirical means, as well as the symmetry in their
heterogeneities as mentioned in Remarks 3.1 and 3.6, the function $\Phi$ does
not depend on the order of indexes $i$ and $j$. Hence, due to the i.i.d.
property of $(\xi^{i},W^{i})_{\\{i=1,\ldots,N_{1}\\}}$ and
$(\eta^{j},B^{j})_{\\{j=1,\ldots,N_{2}\\}}$, we can see that the joint
distributions of
$(X^{i},P^{i},\xi^{i},c^{i},\widehat{\beta}^{i},W^{i})_{\\{i=1,\ldots,N_{1}\\}}$
as well as
$(x^{j},y^{j},p^{j},r^{j},\eta^{j},\mathfrak{c}^{j},B^{j})_{\\{j=1,\ldots,N_{2}\\}}$
are invariant under the permutation of their indexes respectively. In other
words, they are the exchangeable random variables. Since $W^{0}$ is
independent from the other inputs, the exchangeability also holds ${\cal
F}^{0}$-conditionally. Moreover, the permutation of
$(x^{j},y^{j},p^{j},r^{j},\eta^{j},\mathfrak{c}^{j},B^{j})_{\\{j=1,\ldots,N_{2}\\}}$
does not affect the distribution of
$(X^{i},P^{i},\xi^{i},c^{i},\widehat{\beta}^{i},W^{i})_{\\{i=1,\ldots,N_{1}\\}}$
and vice versa.

##  4 Mean-field equilibrium

###  4.1 Notations and preliminary remarks

Before setting up the mean-field model, let us introduce some notations to be
used in this section. Since we will single out an arbitrarily chosen pair of
representative agents $(i,j)$ from the two populations, and the other agents
are to be decoupled, it is useful to work on smaller probability spaces.

  * •

$(\overline{\Omega},\overline{{\cal F}},\overline{\mathbb{P}})$ is a
probability space defined by the product
$\overline{\Omega}:=\overline{\Omega}^{1,i}\times\overline{\Omega}^{2,j}$ with
$(\overline{{\cal F}},\overline{\mathbb{P}})$ the completion of
$(\overline{{\cal F}}^{1,i}\otimes\overline{{\cal
F}}^{2,j},~{}\overline{\mathbb{P}}^{1,i}\otimes\overline{\mathbb{P}}^{2,j})$.
$\overline{\mathbb{F}}:=(\overline{{\cal F}}_{t})_{t\geq 0}$ denotes the
complete and right-continuous augmentation of $(\overline{{\cal
F}}_{t}^{1,i}\otimes\overline{{\cal F}}_{t}^{2,j})_{t\geq 0}$.

  * •

$(\Omega,{\cal F},\mathbb{P})$ is a probability space defined by the product
$\Omega:=\Omega^{0}\times\overline{\Omega}$ with $({\cal F},\mathbb{P})$ the
completion of $({\cal F}^{0}\otimes\overline{{\cal
F}},~{}\mathbb{P}^{0}\otimes\overline{\mathbb{P}})$. $\mathbb{F}:=({\cal
F}_{t})_{t\geq 0}$ denotes the complete and right-continuous augmentation of
$({\cal F}_{t}^{0}\otimes\overline{{\cal F}}_{t})_{t\geq 0}$. Here,
$(\Omega^{0},{\cal F}^{0},\mathbb{P}^{0})$ with $\mathbb{F}^{0}$ is defined in
the same way as in Section 2.

$\mathbb{E}[\cdot]$ denotes the expectation with respect to $\mathbb{P}$, and
$\omega=(\omega^{0},\overline{\omega})$ with $\omega^{0}\in\Omega^{0}$ and
$\overline{\omega}\in\overline{\Omega}$ a generic element of $\Omega$. Recall
that $\mathbb{F}^{0}$ denotes the filtration generated by the common noise
$W^{0}$. We also need the following:

  * •

$(\Omega^{1,i},{\cal F}^{1,i},\mathbb{P}^{1,i})$ is a probability space
defined by the product $\Omega^{1,i}:=\Omega^{0}\times\overline{\Omega}^{1,i}$
with $({\cal F}^{1,i},\mathbb{P}^{1,i})$ the completion of $({\cal
F}^{0}\otimes\overline{{\cal
F}}^{1,i},\mathbb{P}^{0}\otimes\overline{\mathbb{P}}^{1,i})$.
$\mathbb{F}^{1,i}:=({\cal F}^{1,i}_{t})_{t\geq 0}$ denotes the complete and
right-continuous augmentation of $({\cal F}^{0}_{t}\otimes\overline{{\cal
F}}_{t}^{1,i})_{t\geq 0}$. $(\Omega^{2,j},{\cal F}^{2,j},\mathbb{P}^{2,j})$
with $\mathbb{F}^{2,j}:=({\cal F}^{2,j}_{t})_{t\geq 0}$ is defined similarly
as above with $(\overline{\Omega}^{1,i},\overline{{\cal
F}}^{1,i},\overline{\mathbb{P}}^{1,i})$ and $\overline{\mathbb{F}}^{1,i}$
replaced by $(\overline{\Omega}^{2,j},\overline{{\cal
F}}^{2,j},\overline{\mathbb{P}}^{2,j})$ and $\overline{\mathbb{F}}^{2,j}$.

Let us give some remarks on the properties of ${\cal F}^{0}$-conditional
distributions and expectations. By [15, Lemma 2.4], it is shown that for any
$\mathbb{R}^{n}$-valued random variable $X$ on $(\Omega,{\cal F},\mathbb{P})$,
$X(\omega^{0},\cdot)$ is a random variable on
$(\overline{\Omega},\overline{{\cal F}},\overline{\mathbb{P}})$ for
$\mathbb{P}^{0}$-a.e. $\omega^{0}\in\Omega^{0}$. In particular, by assigning
an arbitrary ${\cal P}(\mathbb{R}^{n})$ value for the exceptional set, the map
defined by $\omega^{0}\mapsto{\cal L}^{0}(X)(\omega^{0}):={\cal
L}(X(\omega^{0},\cdot))$ becomes a ${\cal P}(\mathbb{R}^{n})$-valued random
variable on $(\Omega^{0},{\cal F}^{0},\mathbb{P}^{0})$, and ${\cal L}^{0}(X)$
provides the conditional law of $X$ given ${\cal F}^{0}$. Moreover, if
$X=(X_{t})_{t\geq 0}$ is a stochastic process in
$\mathbb{S}^{2}(\mathbb{F};\mathbb{R}^{n})$ then [15, Lemma 2.5] shows that we
can find a version of $({\cal L}^{0}(X_{t}))_{t\geq 0}$ such that the process
$({\cal L}^{0}(X_{t}))_{t\geq 0}$ is $\mathbb{F}^{0}$-adapted and has
continuous paths in ${\cal P}_{2}(\mathbb{R}^{n})$. We denote by
$\mathbb{E}^{0}[\cdot]$ the ${\cal F}^{0}$-conditional expectation, or
equivalently the integration over $\overline{\Omega}$ by
$\overline{\mathbb{P}}$. For any integrable $\mathbb{F}$-adapted process
$X=(X_{t})_{t\geq 0}$, we have
$\mathbb{E}^{0}[X_{t}](\omega^{0})=\mathbb{E}[X_{t}(\omega^{0},\cdot)]=\mathbb{E}[X_{t}|{\cal
F}_{t}^{0}](\omega^{0})=\mathbb{E}[X_{t}|{\cal F}^{0}](\omega^{0})$ since
$\sigma(W_{s}^{0}-W_{t}^{0}),~{}s>t$ is independent of $\sigma(X_{t})$. In
[15, Section 4.3.1], it is shown that we can find a version of
$(\mathbb{E}^{0}[X_{t}])_{t\geq 0}$ which is $\mathbb{F}^{0}$-progressively
measurable by modifying $dt\otimes d\mathbb{P}^{0}$-null set when the process
$X=(X_{t})_{t\geq 0}$ is $\mathbb{F}$-progressively measurable. In the
remainder, we always use such a version.

As we have discussed in Subsection 3.2.1, we need to lift a function on
probability measures to that on $\mathbb{L}^{2}$-random variables to define
the $L$-derivative. To do this, we need another probability space.

  * •

$(\widetilde{\overline{\Omega}},\widetilde{\overline{{\cal
F}}},\widetilde{\overline{\mathbb{P}}})$ is a copy of the probability space
$(\overline{\Omega},\overline{{\cal F}},\overline{\mathbb{P}})$. For any
random variable $X$ on $\Omega=\Omega^{0}\times\overline{\Omega}$, the symbol
$\widetilde{X}$ is used to denote a random variable defined as a copy of $X$
on the space
$\widetilde{\Omega}:=\Omega^{0}\times\widetilde{\overline{\Omega}}$. We also
use the abbreviations such as,
$\widetilde{f}_{1}^{i}(t,\cdot):=f_{1}(t,\cdot,c_{t}^{0},\widetilde{c}_{t}^{i})$
and
$\widetilde{g}_{1}^{i}(\cdot):=g_{1}(\cdot,c_{T}^{0},\widetilde{c}_{T}^{i})$
to ease the notations.

In particular, conditionally on ${\cal F}^{0}$ (i.e. for each
$\omega^{0}\in\Omega^{0},\mathbb{P}^{0}$-a.e.), $X$ and $\widetilde{X}$ are
independent and have the same distribution ${\cal L}^{0}(X)={\cal
L}^{0}(\widetilde{X})$. The corresponding expectations
$\widetilde{\mathbb{E}}[\cdot]$ and $\widetilde{\mathbb{E}}^{0}[\cdot]$ are
defined in the same way as $\mathbb{E}[\cdot]$ and $\mathbb{E}^{0}[\cdot]$,
respectively. When we need to put both $X$ and $\widetilde{X}$ on the common
space, we always assume that they are extended naturally to the space
$\Omega^{0}\times\overline{\Omega}\times\widetilde{\overline{\Omega}}$ and
still use the same symbols. We have
$\mathbb{E}\widetilde{\mathbb{E}}[\cdot]=\mathbb{E}\widetilde{\mathbb{E}}^{0}[\cdot]=\widetilde{\mathbb{E}}\mathbb{E}^{0}[\cdot]$.
In particular, if $F:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
is a measurable function such that $F(X,\widetilde{X})$ is integrable, then
Fubini’s theorem implies
$\mathbb{E}\widetilde{\mathbb{E}}^{0}[F(X,\widetilde{X})]=\mathbb{E}\widetilde{\mathbb{E}}^{0}[F(\widetilde{X},X)]$.
Throughout this section, we work under Assumptions 3.2-3.3.

###  4.2 Mean-field problem

Firstly, let us provide a formal derivation of the corresponding mean field
problem. To do so, it is beneficial for us to recall the famous result of De
Finetti’s theory of exchangeable sequence of random variables:

######  Theorem 4.1 ([15][Theorem 2.1]).

For any exchangeable sequence of random variables $(X_{n})_{n\geq 1}$ with
$\mathbb{E}[|X_{1}|]<\infty$, it holds, $\mathbb{P}$-a.s.,

$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}X_{i}=\mathbb{E}[X_{1}|{\cal
F}_{\infty}],$

where ${\cal F}_{\infty}$ is the tail $\sigma$-field ${\cal
F}_{\infty}=\bigcap_{n\geq 1}\sigma\\{X_{k},k\geq n\\}$.

Suppose that $(y^{j})_{j=1,\ldots,N_{2}}$ and
$(X^{i},\beta^{i})_{i=1,\ldots,N_{1}}$ are exchangeable sequences of random
variables respectively. If we take $N_{1},N_{2}\rightarrow\infty$ with
$\delta_{\mathfrak{n}}:=N_{1}/N_{2}$ kept fixed, then we expect to have, from
Theorem 4.1 with ${\cal F}_{\infty}$ replaced by ${\cal F}^{0}$,

$\varpi_{t}(y_{t},\beta_{t}):=-\mathfrak{m}_{2}(y_{t})+\delta_{\mathfrak{n}}\Lambda_{t}\mathfrak{m}_{1}(\beta_{t})\longrightarrow-\mathbb{E}^{0}[y_{t}^{1}]+\delta_{\mathfrak{n}}\Lambda_{t}\mathbb{E}^{0}[\beta_{t}^{1}]~{}{\text{$\mathbb{P}$-a.s.
as $N_{1},N_{2}\rightarrow\infty$}}.$

It is also natural to suppose

$\nu_{t}^{N_{1}}:=\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}\delta_{X^{i}}\longrightarrow{\cal
L}^{0}(X_{t}^{1})~{}{\text{$\mathbb{P}$-a.s. as $N_{1}\rightarrow\infty$}}.$

Considering the exchangeability of the solution to the system of equations
$(\ref{fbsde-full-finite})$ as discussed at the end of Section 3.2.3, the
above observation motivates us to study the following problem: Solve

$\small\inf_{\beta^{i}\in\mathbb{A}_{1}^{i}}{\cal J}_{1}^{i}(\beta^{i}),$ (
4.1)

with the cost functional defined by

$\small\begin{split}{\cal
J}_{1}^{i}(\beta^{i}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f_{1}^{i}(t,X_{t}^{i},{\cal
L}^{0}(X_{t}^{i}),\pi_{t}(y_{t}^{j},\beta_{t}^{i}),\beta_{t}^{i})dt+g_{1}^{i}(X_{T}^{i},{\cal
L}^{0}(X_{T}^{i}))\Bigr{]},\end{split}$

under the dynamic constraints:

$\small\begin{split}&dX^{i}_{t}=(\beta_{t}^{i}+\rho_{1}(t)\pi_{t}(y_{t}^{j},\beta_{t}^{i})+l_{1}^{i}(t))dt+\sigma_{1}^{i,0}(t)dW_{t}^{0}+\sigma_{1}^{i}(t)dW_{t}^{i},\\\
&dx_{t}^{j}=(-\overline{\Lambda}_{t}(y_{t}^{j}+\pi_{t}(y_{t}^{j},\beta_{t}^{i}))+\rho_{2}(t)\pi_{t}(y_{t}^{j},\beta_{t}^{i})+l_{2}^{j}(t))dt+\sigma_{2}^{j,0}(t)dW_{t}^{0}+\sigma_{2}^{j}(t)dB_{t}^{j},\\\
&dy_{t}^{j}=-(\partial_{x}\overline{f}_{2}^{j}(t,x_{t}^{j})-b_{t}\pi_{t}(y_{t}^{j},\beta_{t}^{i}))dt+z_{t}^{j,0}dW_{t}^{0}+z_{t}^{j,j}dB_{t}^{j},\end{split}$
( 4.2)

with the boundary conditions; $X^{i}_{0}=\xi^{i},~{}x^{j}_{0}=\eta^{j}$ and
$y^{j}_{T}=\partial_{x}g_{2}^{j}(x_{T}^{j})$. Here,
$\mathbb{A}_{1}^{i}:=\mathbb{H}^{2}(\mathbb{F}^{1,i};A_{1})$ is the space of
admissible controls, and $\pi_{t}(\cdot,\cdot)$ is defined by

$\small\pi_{t}(y,\beta):=-\mathbb{E}^{0}[y]+\delta_{\mathfrak{n}}\Lambda_{t}\mathbb{E}^{0}[\beta].$
( 4.3)

In this problem, we have chosen a pair $\bf{(i,j)}$ as the representative
agents from the two populations and they are kept fixed throughout this
section. The rigorous discussions on the relation between this problem with
the finite-agent equilibrium discussed in the previous sections will be the
topic in the next section.

######  Remark 4.1.

Let Assumptions 3.2-3.3 hold. Then, for a given
$\beta^{i}\in\mathbb{A}_{1}^{i}$, there exists a unique solution
$x^{j},y^{j}\in\mathbb{S}^{2}(\mathbb{F}^{2,j};\mathbb{R}^{n})$ of $(\ref{mfg-
c-dynamics})$. This can be shown by the Peng-Wu’s continuation method in a
completely parallel manner to Theorem 3.2. In fact, this is a special
situation handled in [28, Theorem 4.2]. Therefore, for each
$\beta^{i}\in\mathbb{A}_{1}^{i}$, the state processes
$(X^{i},x^{j},y^{j})\in\mathbb{S}^{2}(\mathbb{F}^{1,i};\mathbb{R}^{n})\times(\mathbb{S}^{2}(\mathbb{F}^{2,j};\mathbb{R}^{n}))^{2}$
are defined uniquely.

######  Remark 4.2.

Note that, the symmetric interaction contained in $\varpi_{t}(\cdot,\cdot)$ as
the empirical means does not exist in $\pi_{t}(\cdot,\cdot)$ anymore. This
induces a decoupling among the agents, which can also be observed in the
information structure. Since $X^{i}$ is adapted to $\mathbb{F}^{1,i}$, the
representative agent $i$ cares only about the common information
$\mathbb{F}^{0}$ and her own idiosyncratic information
$\overline{\mathbb{F}}^{1,i}$ (see the definition of $\mathbb{A}_{1}^{i}$
given above). The same observation also holds for the agent $j$ of the second
population. In fact, the optimal control of the agent $j$ is now given by
$\widehat{\alpha}_{t}^{j}=-\overline{\Lambda}_{t}(y_{t}^{j}+\pi_{t}(y_{t}^{j},\beta_{t}^{i}))$,
which is adapted to $\mathbb{F}^{0}\otimes\overline{\mathbb{F}}^{2,j}$.

The involvement of $\mathbb{E}^{0}[\beta^{i}_{t}]$-term in the cost function
makes our problem the conditional extended mean-field control [2]. For this
type of problem, it is convenient to work on the lifted Hamiltonian ${\cal H}$
defined on the $\mathbb{L}^{2}$-space. We define the map ${\cal
H}:[0,T]\times\mathbb{L}^{2}({\cal
F}_{t}^{1,i};\mathbb{R}^{n})\times(\mathbb{L}^{2}({\cal
F}_{t}^{2,j};\mathbb{R}^{n}))^{2}\times\mathbb{L}^{2}({\cal
F}_{t}^{1,i};\mathbb{R}^{n})\times(\mathbb{L}^{2}({\cal
F}_{t}^{2,j};\mathbb{R}^{n}))^{2}\times\mathbb{L}^{2}({\cal
F}_{t}^{1,i};A_{1})\ni(t,X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t},\beta^{i}_{t})\mapsto{\cal
H}(t,X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t},\beta^{i}_{t})\in\mathbb{R}$
by

$\small\begin{split}&{\cal
H}(t,X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t},\beta^{i}_{t})\\\
&:=\mathbb{E}\Bigl{[}\langle
P^{i}_{t},\beta^{i}_{t}+\rho_{1}(t)\pi_{t}(y^{j}_{t},\beta^{i}_{t})+l_{1}^{i}(t)\rangle+f_{1}^{i}(t,X^{i}_{t},{\cal
L}^{0}(X^{i}_{t}),\pi_{t}(y^{j}_{t},\beta^{i}_{t}),\beta^{i}_{t})\\\
&\quad+\overline{\delta}_{\mathfrak{n}}\bigl{\\{}\langle
p^{j}_{t},-\overline{\Lambda}_{t}(y^{j}_{t}+\pi_{t}(y^{j}_{t},\beta^{i}_{t}))+\rho_{2}(t)\pi_{t}(y^{j}_{t},\beta^{i}_{t})+l_{2}^{j}(t)\rangle+\langle
r_{t}^{j},-\partial_{x}\overline{f}_{2}^{j}(t,x^{j}_{t})+b_{t}\pi_{t}(y_{t}^{j},\beta^{i}_{t})\rangle\bigr{\\}}\Bigr{]},\end{split}$
( 4.4)

where $\overline{\delta}_{\mathfrak{n}}:=1/\delta_{\mathfrak{n}}=N_{2}/N_{1}$.
The dependence on the conditional distribution ${\cal L}^{0}(X^{i}_{t})$ as
well as the conditional expectations
$(\mathbb{E}^{0}[y^{j}_{t}],\mathbb{E}^{0}[\beta^{i}_{t}])$ in
$\pi_{t}(\cdot,\cdot)$ is treated as a function on the random variables
$X_{t}^{i}$ and $(y_{t}^{j},\beta^{i}_{t})$, respectively. Since there is no
information about the size of populations in the dynamics $(\ref{mfg-c-
dynamics})$, it must be given by hand to the Hamiltonian ${\cal H}$ using the
coefficient $\overline{\delta}_{\mathfrak{n}}$. The way in which ${\cal H}$
depends on $\overline{\delta}_{\mathfrak{n}}$ may be guessed from $H$ in
$(\ref{H-1-finite})$. As before, with slight abuse of notation, we write the
lifted Hamiltonian as ${\cal
H}(t,X_{t}^{i},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t},\pi_{t},\beta^{i}_{t})$
when we treat $\pi_{t}\in\mathbb{L}^{2}({\cal F}_{t}^{0};\mathbb{R}^{n})$ as a
separate argument.

For each
$u_{t}:=(X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t})$, we
want to find the minimizer:

$\small\begin{split}\widehat{\beta}^{i}(t,u_{t}):={\rm argmin}\bigl{(}{\cal
H}(t,X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t},\beta_{t}^{i});\beta^{i}_{t}\in\mathbb{L}^{2}({\cal
F}_{t}^{1,i};A_{1})\bigr{)}.\end{split}$

The optimization problem in a Hilbert space has been well studied and we can
follow the standard technique using the weak sequential lower-semicontinuity.
See, for example, [48] and a brief monograph [51]. Below, we often use the
following results on Frechet derivatives of ${\cal H}$: with
$\pi_{t}:=\pi_{t}(y^{j}_{t},\beta^{i}_{t})$ and $\mu_{t}:={\cal
L}^{0}(X^{i}_{t})$,

$\small\begin{split}D_{\beta^{i}}{\cal
H}(t,u_{t},\beta^{i}_{t})&=\partial_{\beta^{i}}{\cal
H}(t,u_{t},\pi_{t},\beta^{i}_{t})+\delta_{\mathfrak{n}}\Lambda_{t}\mathbb{E}^{0}[\partial_{\pi}{\cal
H}(t,u_{t},\pi_{t},\beta^{i}_{t})]\\\
&=P_{t}^{i}+\partial_{\beta}f_{1}^{i}(t,X^{i}_{t},\mu_{t},\pi_{t},\beta^{i}_{t})+\delta_{\mathfrak{n}}\Lambda_{t}\mathbb{E}^{0}[\partial_{\varpi}f_{1}^{i}(t,X_{t}^{i},\mu_{t},\pi_{t},\beta_{t}^{i})]\\\
&\quad+\delta_{\mathfrak{n}}\rho_{1}(t)\Lambda_{t}\mathbb{E}^{0}[P_{t}^{i}]+(-I+\rho_{2}(t)\Lambda_{t})\mathbb{E}^{0}[p_{t}^{j}]+b_{t}\Lambda_{t}\mathbb{E}^{0}[r^{j}_{t}],\\\
D_{X^{i}}{\cal
H}(t,u_{t},\beta_{t}^{i})&=\partial_{x}f_{1}^{i}(t,X^{i}_{t},\mu_{t},\pi_{t},\beta^{i}_{t})+\widetilde{\mathbb{E}}^{0}[\partial_{\mu}\widetilde{f}_{1}^{i}(t,\widetilde{X}^{i}_{t},\mu_{t},\pi_{t},\widetilde{\beta}^{i}_{t})(X^{i}_{t})],\\\
D_{x^{j}}{\cal H}(t,u_{t},\beta^{i}_{t})&=\partial_{x^{j}}{\cal
H}(t,u_{t},\beta_{t}^{i})=-c_{f}^{j}(t)r^{j}_{t},\\\ D_{y^{j}}{\cal
H}(t,u_{t},\beta^{i}_{t})&=-\overline{\delta}_{\mathfrak{n}}\overline{\Lambda}_{t}p^{j}_{t}-\mathbb{E}^{0}[\partial_{\pi}{\cal
H}(t,u_{t},\pi_{t},\beta^{i}_{t})]\\\
&=-\mathbb{E}^{0}[\partial_{\varpi}f_{1}^{i}(t,X^{i}_{t},\mu_{t},\pi_{t},\beta^{i}_{t})]-\rho_{1}(t)\mathbb{E}^{0}[P^{i}_{t}]+\overline{\delta}_{\mathfrak{n}}\bigl{(}\overline{\Lambda}_{t}(\mathbb{E}^{0}[p^{j}_{t}]-p^{j}_{t})-\rho_{2}(t)\mathbb{E}^{0}[p^{j}_{t}]-b_{t}\mathbb{E}^{0}[r^{j}_{t}]\bigr{)}.\end{split}$
( 4.5)

######  Lemma 4.1.

Under Assumptions 3.2-3.3, for each $t\in[0,T]$ and a given set of adjoint
variables $(P_{t}^{i},p^{j}_{t},r^{j}_{t})\in\mathbb{L}^{2}({\cal
F}_{t}^{1,i};\mathbb{R}^{n})\times(\mathbb{L}^{2}({\cal
F}_{t}^{2,j};\mathbb{R}^{n}))^{2}$, the lifted Hamiltonian ${\cal H}$ defined
by $(\ref{lifted-Hamiltonian})$ is jointly convex in
$(X^{i}_{t},x^{j}_{t},y^{j}_{t},\beta^{i}_{t})\in\mathbb{L}^{2}({\cal
F}_{t}^{1,i};\mathbb{R}^{n})\times(\mathbb{L}^{2}({\cal
F}_{t}^{2,j};\mathbb{R}^{n}))^{2}\times\mathbb{L}^{2}({\cal
F}_{t}^{1,i};A_{1})$ and strictly so in $(X^{i}_{t},\beta^{i}_{t})$. In
particular, for any $t\in[0,T]$ and input
$u_{t}:=(X_{t}^{i},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t})$, there
exists a unique minimizer
$\widehat{\beta}_{t}^{i}:=\widehat{\beta}^{i}(t,u_{t})\in\mathbb{L}^{2}({\cal
F}_{t}^{1,i};A_{1})$ that satisfies the following linear growth and the
Lipschitz continuity properties:

$\small\begin{split}&\mathbb{E}\bigl{[}|\widehat{\beta}^{i}(t,u_{t})|^{2}\bigr{]}\leq
C\mathbb{E}\Bigl{[}1+|X_{t}^{i}|^{2}+|P_{t}^{i}|^{2}+|\mathbb{E}^{0}[y_{t}^{j}]|^{2}+|\mathbb{E}^{0}[p_{t}^{j}]|^{2}+|\mathbb{E}^{0}[r_{t}^{j}]|^{2}+|c_{t}^{0}|^{2}+|c_{t}^{i}|^{2}\Bigr{]},\\\
&\mathbb{E}\bigl{[}|\widehat{\beta}^{i}(t,u_{t})-\widehat{\beta}^{i}(t,\acute{u}_{t})|^{2}\bigr{]}\leq
C\mathbb{E}\Bigl{[}|X^{i}_{t}-\acute{X}^{i}_{t}|^{2}+|P_{t}^{i}-\acute{P}^{i}_{t}|^{2}+|\mathbb{E}^{0}[y^{j}_{t}-\acute{y}^{j}_{t}]|^{2}+|\mathbb{E}^{0}[p_{t}^{j}-\acute{p}^{j}_{t}]|^{2}+|\mathbb{E}^{0}[r_{t}^{j}-\acute{r}^{j}_{t}]|^{2}\Bigr{]}.\end{split}$

where
$\acute{u}_{t}:=(\acute{X}^{i}_{t},\acute{x}^{j}_{t},\acute{y}^{j}_{t},\acute{P}^{i}_{t},\acute{p}^{j}_{t},\acute{r}^{j}_{t})$
is another input and $C$ is some positive constant independent of
$\delta_{\mathfrak{n}}$.

###### Proof.

Firstly, we check the relevant convexity of ${\cal H}$. For a given set of
adjoint variables $(P_{t}^{i},p_{t}^{j},r_{t}^{j})$, let us consider two
arbitrary inputs
$(X^{i}_{t},x^{j}_{t},y^{j}_{t}),(\acute{X}^{i}_{t},\acute{x}^{j}_{t},\acute{y}^{j}_{t})\in\mathbb{L}^{2}({\cal
F}^{1,i}_{t};\mathbb{R}^{n})\times(\mathbb{L}^{2}({\cal
F}_{t}^{2,j};\mathbb{R}^{n}))^{2}$ and
$\beta^{i}_{t},\acute{\beta}^{i}_{t}\in\mathbb{L}^{2}({\cal
F}^{1,i}_{t};A_{1})$. For notational simplicity, we put
$u_{t}:=(X^{i}_{t},x^{j}_{t},y^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t})$ and
$\acute{u}_{t}:=(\acute{X}^{i}_{t},\acute{x}^{j}_{t},\acute{y}^{j}_{t},P^{i}_{t},p^{j}_{t},r^{j}_{t})$,
$\mu_{t}:={\cal L}^{0}(X^{i}_{t}),\acute{\mu}_{t}:={\cal
L}^{0}(\acute{X}^{i}_{t}),\pi_{t}:=\pi_{t}(y^{j}_{t},\beta^{i}_{t}),\acute{\pi}_{t}:=\pi_{t}(\acute{y}^{j}_{t},\acute{\beta}^{i}_{t})$,
$\Delta
X^{i}_{t}:=\acute{X}^{i}_{t}-X^{i}_{t},\Delta\beta_{t}^{i}:=\acute{\beta}^{i}_{t}-\beta^{i}_{t}$
and $\Delta\pi_{t}:=\acute{\pi}_{t}-\pi_{t}=-\mathbb{E}^{0}[\Delta
y^{j}_{t}]+\delta_{\mathfrak{n}}\Lambda_{t}\mathbb{E}^{0}[\Delta\beta^{i}_{t}]$,
etc. Then, the Frechet derivatives $(\ref{H-Frechet-D})$, Assumption 3.3 (iv)
and Fubini’s theorem yield

$\small\begin{split}&{\cal H}(t,\acute{u}_{t},\acute{\beta}^{i}_{t})-{\cal
H}(t,u_{t},\beta^{i}_{t})-\mathbb{E}\bigl{[}\langle D_{X^{i}}{\cal