# A Mean Field Game Approach to Equilibrium Pricing
with Market Clearing Condition 111 All the contents expressed in this research
are solely those of the author and do not represent any views or opinions of
any institutions. The author is not responsible or liable in any manner for
any losses and/or damages caused by the use of any contents in this research.

Masaaki Fujii222Quantitative Finance Course, Graduate School of Economics, The
University of Tokyo. , Akihiko Takahashi333Quantitative Finance Course,
Graduate School of Economics, The University of Tokyo.

( First version: 6 March, 2020
This version: 3 October, 2020 )

###### Abstract

In this work, we study an equilibrium-based continuous asset pricing problem
which seeks to form a price process endogenously by requiring it to balance
the flow of sales-and-purchase orders in the exchange market, where a large
number of agents $1\leq i\leq N$ are interacting through the market price.
Adopting a mean field game (MFG) approach, we find a special form of forward-
backward stochastic differential equations of McKean-Vlasov type with common
noise whose solution provides a good approximate of the market price. We show
the convergence of the net order flow to zero in the large $N$-limit and get
the order of convergence in $N$ under some conditions. We also extend the
model to a setup with multiple populations where the agents within each
population share the same cost and coefficient functions but they can be
different population by population.

Keywords : FBSDE of McKean-Vlasov type, common noise, general equilibrium

##  1 Introduction

One of the most important problems in the financial economics is to understand
how the asset price processes are formed through the interaction among a large
number of rational competitive agents. In this paper, using a stylized model
of security exchange, we try to explicitly form an approximate market price
process which balances the flow of sales-and-purchase orders from a large
number of rational financial institutions. If we directly force the price
process to balance the net order flow, the strategies of the agents become
strongly coupled and the problem is hardly solvable. In fact, it is even
unclear how to make the cost functions of the agents well-defined, since the
market price results in a very complicated recursive functional of strategies
of all the agents that makes it difficult to guarantee the convexity of the
cost functions. In order to circumvent this problem, we make use of the recent
developments of mean field games.

Since its inception brought by the pioneering works of Lasry & Lions [20, 21,
22] and Huang, Malhame & Caines [18], mean field game has rapidly developed
into one of the most actively studied topics in the field of probability
theory, applied mathematics, engineering, finance and economics. The greatest
strength of the mean field game approach is to render notoriously difficult
problems of stochastic differential games among many agents tractable by
transforming it to a simpler form of stochastic control problem. There exist
two approaches to the mean field games, one is analytic approach using partial
differential equations (PDEs), and the others is probabilistic approach based
on forward-backward stochastic differential equations (FBSDEs). For details of
analytic approach and its applications, the interested readers may consult the
works of Bensousssan, Frehse & Yam [3], Gomes, Nurbekyan & Pimentel [14],
Achdou et.al. [1], Gomes, Pimentel & Voskanyan [15] and also Kolokoltsov &
Malafeyev [19]. On the other hand, the probabilistic approach was developed by
the series of works of Carmona & Delarue [4, 5, 6] and the recent two volumes
of monograph [7, 8] provide its full mathematical details and many references
for a wide array of applications of mean field games.

Interestingly, from the perspective of equilibrium asset pricing, the number
of applications of mean field games is quite limited. In most of the existing
literature, the authors have given a response function of the price process
exogenously and searched an approximate Nash equilibrium among agents. See,
for example, applications to optimal trading as well as liquidation of
portfolio, exploitation of exhaustible resources and related issues among many
agents [10, 11, 12, 23], or an application to electricity pricing with smart
grids [2, 9]. In the work [17], the authors treat explicitly the balance of
demand and supply in the oil market, but the demand is exogenously given as a
function of the oil price. One notable exception is the work of Gomes & Saude
[16], in which the authors explicitly force demand and supply to balance and
endogenously construct the market clearing electricity price. They use the
analytic approach and the resultant equilibrium price process becomes
deterministic due to the absence of common noise.

In the current paper, we extend the work [16] by adopting the probabilistic
approach. In order to understand the price processes, in particular those of
financial assets, including systemic signals which impacts all the agents is
crucially important. We find an interesting form of FBSDEs of McKean-Vlasov
type with common noise as a limit problem. Although it involves dependence in
conditional law, it only appears as a conditional expectation. This allows us
to adopt the well-known Peng-Wu’s continuation method [24] to prove the
existence of a unique strong solution. The resultant candidate of the market
price process is derived completely endogenously by the optimal trading
strategies of the agents facing systemic information (including securities’
coupon stream) as well as idiosyncratic noise. Another benefit of
probabilistic approach is that it allows us to quantify the relation between
the actual game with finite number of agents and its large population limit.
In a similar manner to the standard mean field games in proving
$\varepsilon$-Nash equilibrium, we show that the solution of the mean-field
limit problem actually provides asymptotic market clearing in the large-$N$
limit. Under additional integrability conditions, Glivenko-Cantelli
convergence theorem in the Wasserstein distance even provides a specific order
of convergence in terms of the number of agents $N$. We also discuss the
extension of the model to the situation with multiple populations where the
agents share the same cost and coefficient functions within each population
but they can be different population by population. This will provide an
important tool to study the price formation in the presence of different type
of agents such as Buy-side and Sell-side institutions, for example.

The organization of the paper is as follows: After explaining the notation in
Section 2, we give an intuitive derivation of the limit problem from the game
of finite number of agents in Section 3, which motivates the readers to study
the special type of FBSDEs of MKV-type. The solvability of the FBSDE is
studied in Section 4. Using the derived regularity of the solution, we prove
the asymptotic market clearing in Section 5. In Section 6, we discuss the
extension of the model to the setup with multiple populations. Finally, in
Section 7, we give concluding remarks. We discuss further extensions of the
model and future directions of research.

##  2 Notation

We introduce (N+1) complete probability spaces:

$\displaystyle(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0})\quad{\rm{and}}\quad(\overline{\Omega}^{i},\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{i})_{i=1}^{N}~{},$

endowed with filtrations $\overline{\mathbb{F}}^{i}:=(\overline{{\cal
F}}_{t}^{i})_{t\geq 0}$, $i\in\\{0,\cdots,N\\}$. Here,
$\overline{\mathbb{F}}^{0}$ is the completion of the filtration generated by
$d^{0}$-dimensional Brownian motion $\boldsymbol{W}^{0}$ (hence right-
continuous) and, for each $i\in\\{1,\cdots,N\\}$, $\overline{\mathbb{F}}^{i}$
is the complete and right-continuous augmentation of the filtration generated
by $d$-dimensional Brownian motions $\boldsymbol{W}^{i}$ as well as a
$\boldsymbol{W}^{i}$-independent $n$-dimensional square-integrable random
variables $(\xi^{i})$. $(\xi^{i})_{i=1}^{N}$ are supposed to have the same
law. We also introduce the product probability spaces

$\Omega^{i}=\overline{\Omega}^{0}\times\overline{\Omega}^{i},\quad{\cal
F}^{i},\quad\mathbb{F}^{i}=({\cal F}_{t}^{i})_{t\geq
0},\quad\mathbb{P}^{i}~{},i\in\\{1,\cdots,N\\}$

where $({\cal F}^{i},\mathbb{P}^{i})$ is the completion of $(\overline{{\cal
F}}^{0}\otimes\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{0}\otimes\overline{\mathbb{P}}^{i})$ and
$\mathbb{F}^{i}$ is the complete and right-continuous augmentation of
$(\overline{{\cal F}}_{t}^{0}\otimes\overline{{\cal F}}_{t}^{i})_{t\geq 0}$.
In the same way, we define the complete probability space $(\Omega,{\cal
F},\mathbb{P})$ endowed with $\mathbb{F}=({\cal F}_{t})_{t\geq 0}$ satisfying
the usual conditions as a product of $(\overline{\Omega}^{i},\overline{{\cal
F}}^{i},\overline{\mathbb{P}}^{i};\overline{\mathbb{F}}^{i})_{i=0}^{N}$.

Throughout the work, the symbol $L$ and $L_{\varpi}$ denote given positive
constants, the symbol $C$ a general positive constant which may change line by
line. When we want to emphasize that $C$ depends only on some specific
variables, say $a$ and $b$, we use the symbol $C(a,b)$. For a given constant
$T>0$, we use the following notation for frequently encountered spaces:
$\bullet~{}$$\mathbb{L}^{2}({\cal G};\mathbb{R}^{d})$ denotes the set of
$\mathbb{R}^{d}$-valued ${\cal G}$-measurable square integrable random
variables.
$\bullet~{}$$\mathbb{S}^{2}(\mathbb{G};\mathbb{R}^{d})$ is the set of
$\mathbb{R}^{d}$-valued $\mathbb{G}$-adapted continuous processes
$\boldsymbol{X}$ satisfying

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

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

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

$\bullet~{}$${\cal L}(X)$ denotes the law of a random variable $X$.
$\bullet~{}{\cal P}(\mathbb{R}^{d})$ is the set of probability measures on
$(\mathbb{R}^{d},{\cal B}(\mathbb{R}^{d}))$.
$\bullet~{}{\cal P}_{p}(\mathbb{R}^{d})$ 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

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

We always assign ${\cal P}_{p}(\mathbb{R}^{d})$ with $(p\geq 1)$ the
$p$-Wasserstein distance $W_{p}$, which makes ${\cal P}_{p}(\mathbb{R}^{d})$ a
complete separable metric space. As an important property, for any
$\mu,\nu\in{\cal P}_{p}(\mathbb{R}^{d})$, we have

$\displaystyle
W_{p}(\mu,\nu)={\inf}\Bigl{\\{}\mathbb{E}[|X-Y|^{p}]^{\frac{1}{p}};{\cal
L}(X)=\mu,{\cal L}(Y)=\nu\Bigr{\\}}~{},$ ( 2.1)

where “$\inf$” is taken over all random variables with laws equal to $\mu$ and
$\nu$, respectively. For more details, see Chapter 5 in [7]. We frequently
omit the arguments such as $(\mathbb{G},\mathbb{R}^{d})$ in the above
definitions when there is no confusion from the context.

##  3 Intuitive Derivation of the Mean Field Problem

In this section, in order to introduce the special form of forward-backward
stochastic differential equations of McKean-Vlasov type to be studied in this
paper, we give a heuristic derivation of the mean-field limit problem from the
corresponding equilibrium problem with finite number of agents. As a
motivating example, we consider the equilibrium-based pricing problem of $n$
types of securities, which are continuously traded in the security exchange in
the presence of a large number of participating agents indexed by
$i\in\\{1,\cdots,N\\}$. Every agent is supposed to have many small clients who
can only trade directly to the agent via over-the-counter markets (OTC) and
have no access to the security exchange.

We suppose that each agent $i\in\\{1,\cdots,N\\}$ tries to solve the problem

$\inf_{\boldsymbol{\alpha^{i}}\in\mathbb{A}^{i}}J^{i}(\boldsymbol{\alpha}^{i})$
( 3.1)

with some cost functional

$\displaystyle
J^{i}(\boldsymbol{\alpha}^{i}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f(t,X_{t}^{i},\alpha_{t}^{i},\varpi_{t},c_{t}^{0},c_{t}^{i})dt+g(X_{T}^{i},\varpi_{T},c_{T}^{0},c_{T}^{i})\Bigr{]}~{},$

subject to the dynamic constraint:

$\displaystyle
dX_{t}^{i}=\Bigl{(}\alpha_{t}^{i}+l(t,\varpi_{t},c_{t}^{0},c_{t}^{i})\Bigr{)}dt+\sigma_{0}(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{i}$

with $X_{0}^{i}=\xi^{i}\in\mathbb{L}^{2}(\overline{{\cal
F}}_{0}^{i};\mathbb{R}^{n})$. Here, $(X_{t}^{i})_{t\geq 0}$ is an
$\mathbb{R}^{n}$-valued process denoting the time-$t$ position of the $n$
securities for the agent $i$ with the initial position $\xi^{i}$.
$(c_{t}^{0})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ denotes the
coupon payments from the securities or the market news commonly available to
all the agents, while $(c_{t}^{i})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{i};\mathbb{R}^{n})$ denotes some
independent factors affecting only on the agent $i$. Moreover,
$(c_{t}^{i})_{t\geq 0}$ are also assumed to have the common law for all $1\leq
i\leq N$. We further suppose $c_{T}^{0}$ and $c_{T}^{i}$ are square integrable
to handle the terminal cost $g$. Each agent controls $(\alpha_{t}^{i})_{t\geq
0}$ denoting the trading speed though the security exchange. The remaining
terms $(l,\sigma_{0},\sigma)$ denote the order flow to the agent from his/her
clients through over-the-counter (OTC) markets. $(\varpi_{t})_{t\geq 0}$ is
the market price of the $n$ securities. The space of admissible strategies
$\mathbb{A}^{i}$ of the agent $i$ is the set of processes
$(\alpha^{i}_{t})_{t\geq 0}$ adapted to the complete right-continuous
augmentation of the filtration $\bigl{(}\sigma\\{\varpi_{s}:s\leq
t\\})\vee{\cal F}_{t}^{i}\bigr{)}_{t\geq 0}$ satisfying

$\mathbb{E}\int_{0}^{T}|\alpha_{t}^{i}|^{2}dt<\infty~{}.$

In contrast to the standard optimization problems with a given market price
process, we want to understand the fundamental mechanism of financial market
which determines the market price by the equilibrium condition. The
equilibrium price $(\varpi_{t})_{t\geq 0}$ adapted to the filtration
$\mathbb{F}$ is determined endogenously so that the optimal strategies of the
agents $(\widehat{\alpha}^{i}_{t})_{i=1}^{N}$ satisfy the market clearing
condition for every $t\in[0,T]$, $\mathbb{P}$-a.s.

$\displaystyle\sum_{i=1}^{N}\widehat{\alpha}_{t}^{i}=0~{},$ ( 3.2)

which denotes the balance point of demand and supply at the security exchange.

Although we have already made simplistic assumptions such that the cost
functions as well as the coefficient functions of all the agents are common,
the problem is still hardly solvable. Due to the clearing condition
$(\ref{orig-clearing})$, we cannot adopt an open-loop equilibrium approach. In
particular, $(\varpi_{t})_{t\geq 0}$ becomes a complicated functional of the
agents’ trading strategies and hence the problem for each agent is highly
recursive with respect to $(\alpha^{i}_{t})_{t\geq 0,1\leq i\leq N}$. It is
even unclear how to guarantee the cost function well-defined by making it
convex with respect to the controls.

In order to obtain some insight, let us consider a much simpler situation. It
is natural to suppose that the impact to the market price
$(\varpi_{t})_{t\in[0,T]}$ from the individual agent becomes negligibly small
when $N$ is sufficiently large. Moreover, $(\varpi_{t})_{t\in[0,T]}$ is likely
to be given by $\overline{\mathbb{F}}^{0}$-progressively measurable process
since the effects from the idiosyncratic parts from many agents are expected
to be canceled out. If this is the case, the problem for each agent reduces to
the standard stochastic optimal control problem in a given random environment
$(\varpi_{t},c_{t}^{0},c_{t}^{i})_{t\in[0,T]}$ with $\mathbb{F}^{i}$-adapted
trading strategy $(\alpha^{i}_{t})_{t\in[0,T]}$. Let us first investigate this
simple problem in details. We introduce the cost functions:
$f:[0,T]\times(\mathbb{R}^{n})^{5}\rightarrow\mathbb{R}$,
$g:(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$,
$\overline{f}:[0,T]\times(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$ and
$\overline{g}:(\mathbb{R}^{n})^{3}\rightarrow\mathbb{R}$, which are measurable
functions such that

$\displaystyle
f(t,x,\alpha,\varpi,c^{0},c):=\langle\varpi,\alpha\rangle+\frac{1}{2}\langle\alpha,\Lambda\alpha\rangle+\overline{f}(t,x,\varpi,c^{0},c),$
$\displaystyle
g(x,\varpi,c^{0},c):=-\delta\langle\varpi,x\rangle+\overline{g}(x,c^{0},c)~{}.$

######  Assumption 3.1.

(MFG-a)
(i) $\Lambda$ is a positive definite $n\times n$ symmetric matrix with
$\underline{\lambda}I_{n\times n}\leq\Lambda\leq\overline{\lambda}I_{n\times
n}$ in the sense of 2nd-order form where $\underline{\lambda}$ and
$\overline{\lambda}$ are some constants satisfying
$0<\underline{\lambda}\leq\overline{\lambda}$.
(ii) For any $(t,x,\varpi,c^{0},c)$,

$\displaystyle|\overline{f}(t,x,\varpi,c^{0},c)|+|\overline{g}(x,c^{0},c)|\leq
L(1+|x|^{2}+|\varpi|^{2}+|c^{0}|^{2}+|c|^{2})~{}.$

(iii) $\overline{f}$ and $\overline{g}$ are continuously differentiable in $x$
and satisfy, for any $(t,x,x^{\prime},\varpi,c^{0},c)$,

$|\partial_{x}\overline{f}(t,x^{\prime},\varpi,c^{0},c)-\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}(x^{\prime},c^{0},c)-\partial_{x}\overline{g}(x,c^{0},c)|\leq
L|x^{\prime}-x|~{},$

and
$|\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}(x,c^{0},c)|\leq
L(1+|x|+|\varpi|+|c^{0}|+|c|)$.
(iv)The functions $\overline{f}$ and $\overline{g}$ are convex in $x$ in the
sense that for any $(t,x,x^{\prime},\varpi,c^{0},c)$,

$\displaystyle\overline{f}(t,x^{\prime},\varpi,c^{0},c)-\overline{f}(t,x,\varpi,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)\rangle\geq\frac{\gamma^{f}}{2}|x^{\prime}-x|^{2}~{},$
$\displaystyle\overline{g}(x^{\prime},c^{0},c)-\overline{g}(x,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{g}(x,c^{0},c)\rangle\geq\frac{\gamma^{g}}{2}|x^{\prime}-x|^{2}~{},$

with some constants $\gamma^{f},\gamma^{g}\geq 0$.
(v) $l,\sigma_{0},\sigma$ are the measurable functions defined on
$[0,T]\times(\mathbb{R}^{n})^{3}$ and are $\mathbb{R}^{n},\mathbb{R}^{n\times
d^{0}}$ and $\mathbb{R}^{n\times d}$-valued, respectively. Moreover they
satisfy the linear growth condition:

$\displaystyle|(l,\sigma_{0},\sigma)(t,\varpi,c^{0},c)|\leq
L(1+|\varpi|+|c^{0}|+|c|)$

for any $(t,\varpi,c^{0},c)$.
(vi) $\delta\in[0,1)$ is a given constant.

The first term $\langle\varpi,\alpha\rangle$ of $f$ denotes the direct cost
incurred by the sales and purchase of the securities and the second term
$\frac{1}{2}\langle\alpha,\Lambda\alpha\rangle$ is some fee to be paid to the
exchange depending on the trading speed, or may be interpreted as some
internal cost. The first term of $g$ denotes the mark-to-market value at the
closing time with some discount factor $\delta<1$.444We shall see that the
condition $\delta<1$ is necessary to obtain well-defined terminal condition
for the limit problem. $\overline{f}$ and $\overline{g}$ denote the running as
well as the terminal cost which are affected by the market price, coupon
streams, or the news.

######  Remark 3.1.

If we think $c^{0}$ as a coupon stream of the securities, one may consider for
example,

$\displaystyle\overline{f}(t,x,\varpi,c^{0},c)=-\langle
c^{0},x\rangle+\overline{f}^{\prime}(t,x,\varpi,c)$

as a running cost with an appropriate measurable function
$\overline{f}^{\prime}$. For securities with a given maturity $T$ with
exogenously specified payoff $c^{0}$, such as bonds and futures, it is natural
to consider

$\displaystyle g(x,c^{0})=\overline{g}(x,c^{0})=-\langle c^{0},x\rangle$

as the terminal cost.

For this problem, the (reduced) Hamiltonian is given by

$\displaystyle H(t,x,y,\alpha,\varpi,c^{0},c)=\langle
y,\alpha+l(t,\varpi,c^{0},c)\rangle+f(t,x,\alpha,\varpi,c^{0},c)~{}.$

Since
$\partial_{\alpha}H(t,x,y,\alpha,\varpi,c^{0},c)=y+\varpi+\Lambda\alpha$, the
minimizer of the Hamiltonian is

$\displaystyle\widehat{\alpha}(y,\varpi):=-\overline{\Lambda}(y+\varpi)$ (
3.3)

where $\overline{\Lambda}:=\Lambda^{-1}$. The adjoint FBSDE associated with
the stochastic maximal principle for each agent $1\leq i\leq N$ is thus given
by,

$\displaystyle
dX_{t}^{i}=\Bigl{(}\widehat{\alpha}(Y_{t}^{i},\varpi_{t})+l(t,\varpi_{t},c_{t}^{0},c_{t}^{i})\Bigr{)}dt+\sigma_{0}(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,\varpi_{t},c_{t}^{0},c_{t}^{i})dW_{t}^{i}~{},$
$\displaystyle
dY_{t}^{i}=-\partial_{x}\overline{f}(t,X_{t}^{i},\varpi_{t},c_{t}^{0},c_{t}^{i})dt+Z_{t}^{i,0}dW_{t}^{0}+Z_{t}^{i}dW_{t}^{i}~{},$
( 3.4)

with $X_{0}^{i}=\xi^{i}$ and
$Y_{T}^{i}=\partial_{x}g(X_{T}^{i},\varpi_{T},c_{T}^{0},c_{T}^{i})$.

######  Theorem 3.1.

Under Assumption (MFG-a) and a given
$(\varpi_{t})_{t\in[0,T]}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$,
the problem $(\ref{agent-P})$ for each agent is uniquely characterized by the
FBSDE $(\ref{agent-FBSDE})$ which is strongly solvable with a unique solution
$(X^{i},Y^{i},Z^{i,0},Z^{i})\in\mathbb{S}^{2}(\mathbb{F}^{i};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{i};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{i};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{i};\mathbb{R}^{n\times d})$.

###### Proof.

Since the cost functions are jointly convex with $(x,\alpha)$ and strictly
convex in $\alpha$, the problem is the special situation investigated in
Section 1.4.4 in [8]. Note that, in our case, the diffusion terms
$\sigma_{0},\sigma$ are independent of $(X^{i},\alpha^{i})$. The proof is the
direct result of Theorem 1.60 in the same reference. ∎

Using the above solution, the optimal strategy of each agent is given by

$\displaystyle\widehat{\alpha}^{i}_{t}=-\overline{\Lambda}(Y_{t}^{i}+\varpi_{t}),\quad
t\in[0,T]~{}.$

Let us check the market clearing condition. In the current situation,
$(\ref{orig-clearing})$ is equivalent to

$\displaystyle\varpi_{t}=-\frac{1}{N}\sum_{i=1}^{N}Y_{t}^{i}$

which is of course inconsistent with the our simplifying assumption that
requires $(\varpi_{t})_{t\geq 0}$ to be an $\overline{\mathbb{F}}^{0}$-adapted
process. However, in the current setup, for any $t\in[0,T]$,
$(Y^{i}_{t})_{i=1}^{N}$ are exchangeable random variables due to the
construction of the probability space, common coefficient functions, and the
fact that $(\xi^{i})_{i=1}^{N}$ as well as $(c_{t}^{i},t\in[0,T])_{i=1}^{N}$
are assumed to be i.i.d. Thus De Finetti’s theory of exchangeable sequence of
random variables tells,

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}Y_{t}^{i}=\mathbb{E}\Bigl{[}Y_{t}^{1}|\bigcap_{k\geq
1}\sigma\\{Y_{t}^{j},j\geq k\\}\Bigr{]}\quad{\rm a.s.}$

See for example Theorem 2.1 in [8]. It also seems natural to expect that the
tail $\sigma$-field is reduced to $\overline{{\cal F}}_{t}^{0}$. Therefore we
can expect that, in the large-$N$ limit, the market price of the securities
may be given by $\varpi_{t}=-\mathbb{E}[Y_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]$.

The above observation motivates us to consider the following FBSDE:

$\displaystyle
dX_{t}=\Bigl{(}\widehat{\alpha}\bigl{(}Y_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]\bigr{)}+l\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}\Bigr{)}dt$
$\displaystyle\qquad\quad+\sigma_{0}\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{0}+\sigma\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{1}~{},$ $\displaystyle
dY_{t}=-\partial_{x}\overline{f}\bigl{(}t,X_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dt+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$

with $X_{0}=\xi$ with $\displaystyle
Y_{T}=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})$. To
simplify the notation, we have omitted the superscript $1$ from $Y^{1}$,
$X^{1}$, $\xi^{1}$ and $c^{1}$. Let us remark on the terminal condition.
$Y_{T}=\partial_{x}g(X_{T},-\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}],c_{T}^{0},c_{T})$ is not yet fully specified. Taking the
conditional expectation in the both sides gives

$\displaystyle\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]=\delta\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]+\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}~{},$

which implies $\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}]=\frac{1}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}$. Substituting this expression for
$\mathbb{E}[Y_{T}|\overline{{\cal F}}_{T}^{0}]$ in $\partial_{x}g$, we get the
above specification of the terminal condition.

This is the FBSDE we are going to study in the following. It is of McKean-
Vlasov type with common noise, and similar to the FBSDEs relevant for the
extended mean field games. In the following, we are going to prove the
existence of a unique solution to the above FBSDE under appropriate conditions
and then show that $-\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}]$ is
actually a good approximate of the market price by investigating how
accurately it achieves the market clearing condition $(\ref{orig-clearing})$
when $N$ increases.

##  4 Solvability of the mean-field FBSDE

We now investigate the solvability of the FBSDE derived in the last section

$\displaystyle
dX_{t}=\Bigl{(}\widehat{\alpha}\bigl{(}Y_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]\bigr{)}+l\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}\Bigr{)}dt$
$\displaystyle\qquad\quad+\sigma_{0}\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{0}+\sigma\bigl{(}t,-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dW_{t}^{1}~{},$ $\displaystyle
dY_{t}=-\partial_{x}\overline{f}\bigl{(}t,X_{t},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}\bigr{)}dt+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$
( 4.1)

with $X_{0}=\xi$ with $\displaystyle
Y_{T}=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})$.
$\widehat{\alpha}$ is defined as in $(\ref{def-ha})$. $(c^{0}_{t})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ and
$(c_{t})_{t\geq 0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{1};\mathbb{R}^{n})$
with square integrable $c_{T}^{0},c_{T}$ are given as inputs. Let us remind
the notation to write $\xi=\xi^{1}$ and $c=c^{1}$.

###  4.1 Unique existence for small $T$

######  Assumption 4.1.

(MFG-b)
For any $(t,x,c^{0},c)\in[0,T]\times(\mathbb{R}^{n})^{3}$ and any
$\varpi,\varpi^{\prime}\in\mathbb{R}^{n}$, the coefficient functions
$l,\sigma_{0},\sigma$ and $\overline{f}$ satisfy

$\displaystyle|(l,\sigma_{0},\sigma)(t,\varpi,c^{0},c)-(l,\sigma_{0},\sigma)(t,\varpi^{\prime},c^{0},c)|+|\partial_{x}\overline{f}(t,x,\varpi,c^{0},c)-\partial_{x}\overline{f}(t,x,\varpi^{\prime},c^{0},c)|\leq
L_{\varpi}|\varpi-\varpi^{\prime}|~{}.$

Due to the Lipschitz continuity and the absence of $(Z^{0},Z)$ in the
diffusion coefficients of the forward SDE, we have the following short-term
existence result.

######  Theorem 4.1.

Under Assumptions (MFG-a,b), there exists some constant $\tau>0$ which depends
only on $(L,L_{\varpi},\underline{\lambda},\delta)$ such that for any
$T\leq\tau$, there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$.

###### Proof.

Although there exist terms involving $\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]$, one can adopt the standard technique for the Lipschitz FBSDE.
See, for example, the proof of Theorem 1.45 [8]. ∎

###  4.2 Unique existence for general $T$

In order to obtain existence result for general $T$, we are going to apply the
technique developed by Peng & Wu [24]. In the case of the standard
optimization problem, the joint convexity in the state and control variables
combined with strict convexity in the control variable are enough to obtain
the unique existence. Interestingly however, we need a strict convexity also
in the state variable $X$ in our problem. As we shall see, this is because the
term $-\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}]$ which appears due to the
clearing condition weakens the convexity.

######  Assumption 4.2.

(MFG-c1)
(i) The functions $\sigma_{0}$ and $\sigma$ are independent of the argument
$\varpi$.
(ii) For any $t\in[0,T]$, any random variables
$x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and any
sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the function $l$ satisfies the
monotone condition with some positive constant $\gamma^{l}>0$:

$\displaystyle\mathbb{E}\Bigl{[}\langle l(t,\mathbb{E}[x|{\cal
G}],c^{0},c)-l(t,\mathbb{E}[x^{\prime}|{\cal
G}],c^{0},c),x-x^{\prime}\rangle\Bigr{]}\geq\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[x-x^{\prime}|{\cal
G}]^{2}\bigr{]}~{}.$

(iii) There exists a strictly positive constant $\gamma$ satisfying
$0<\gamma\leq\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\wedge\gamma^{g}$.
Moreover, for any $x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal
F};\mathbb{R}^{n})$ and any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the
function $\overline{g}$ satisfies

$\displaystyle\gamma^{g}\mathbb{E}[|x-x^{\prime}|^{2}]+\frac{\delta}{1-\delta}\mathbb{E}\Bigl{[}\langle\mathbb{E}\bigl{[}\partial_{x}\overline{g}(x,c^{0},c)-\partial_{x}\overline{g}(x^{\prime},c^{0},c)|{\cal
G}\bigr{]},x-x^{\prime}\rangle\Bigr{]}\geq\gamma\mathbb{E}[|x-x^{\prime}|^{2}]~{}.$

######  Remark 4.1.

If $l$ and $\partial_{x}\overline{g}$ have separable forms such as
$h(x)+h^{c}(c^{0},c)$ with some functions $h$ and $h^{c}$, then the conditions
(ii) and (iii) are satisfied when the function $h$ is monotone. Economically
speaking, the condition (ii) implies that the demand from the individual OTC
clients of each agent toward the security decreases when its market price
rises.

The next theorem is the first main existence result.

######  Theorem 4.2.

Under Assumptions (MFG-a,b,c1), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$.

###### Proof.

In order to simplify the notation, let us define the functionals $B,F$ and $G$
for any $y,x,c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ by

$\displaystyle
B(t,y,c^{0},c):=\Bigl{(}-\overline{\Lambda}(y-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}])+l(t,-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}],c^{0},c)\Bigr{)},$ $\displaystyle
F(t,x,y,c^{0},c):=-\partial_{x}\overline{f}\bigl{(}t,x,-\mathbb{E}[y|\overline{{\cal
F}}_{t}^{0}],c^{0},c\bigr{)},$ $\displaystyle
G(x,c^{0},c):=\frac{\delta}{1-\delta}\mathbb{E}\bigl{[}\partial_{x}\overline{g}(x,c^{0},c)|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}(x,c^{0},c)~{}.$ ( 4.2)

With the convention $\Delta y:=y-y^{\prime}$, $\Delta x:=x-x^{\prime}$, one
can easily confirms

$\displaystyle\mathbb{E}\bigl{[}\langle
B(t,y,c^{0},c)-B(t,y^{\prime},c^{0},c),\Delta
y\rangle\bigr{]}\leq-\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\Delta
y|\overline{{\cal F}}_{t}^{0}]^{2}\bigr{]}~{},$
$\displaystyle\mathbb{E}\bigl{[}\langle
F(t,x,y,c^{0},c)-F(t,x^{\prime},y^{\prime},c^{0},c),\Delta
x\rangle\bigr{]}\leq-\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\mathbb{E}[|\Delta
x|^{2}]+\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\Delta y|\overline{{\cal
F}}_{t}^{0}]^{2}\bigr{]}~{},$ $\displaystyle\mathbb{E}\bigl{[}\langle
G(x,c^{0},c)-G(x^{\prime},c^{0},c),\Delta
x\rangle\bigr{]}\geq\gamma\mathbb{E}[|\Delta x|^{2}],$ ( 4.3)

where the first estimate follows from (MFG-c1)(ii) and Jensen’s inequality,
the second from (MFG-a)(iv), (MFG-b) and Cauchy-Schwarz inequality. The third
one is the direct consequence of (MFG-c)(iii).

We first make the following hypothesis: there exists some constant
$\varrho\in[0,1)$ such that, for any $(I_{t}^{b})_{t\geq 0}$,
$(I_{t}^{f})_{t\geq 0}$ in $\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})$ and
any $\eta\in\mathbb{L}^{2}({\cal F}_{T}^{1};\mathbb{R}^{n})$, there exists a
unique solution
$(x^{\varrho},y^{\varrho},z^{0,\varrho},z^{\varrho})\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE:

$\displaystyle dx_{t}^{\varrho}=\bigl{(}\varrho
B(t,y_{t}^{\varrho},c_{t}^{0},c_{t})+I_{t}^{b}\bigr{)}dt+\sigma_{0}(t,c_{t}^{0},c_{t})dW_{t}^{0}+\sigma(t,c_{t}^{0},c_{t})dW_{t}^{1}~{},$
$\displaystyle dy_{t}^{\varrho}=-\bigl{(}(1-\varrho)\gamma
x_{t}^{\varrho}-\varrho
F(t,x_{t}^{\varrho},y_{t}^{\varrho},c_{t}^{0},c_{t})+I_{t}^{f}\bigr{)}dt+z_{t}^{0,\varrho}dW_{t}^{0}+z_{t}^{\varrho}dW_{t}^{1}~{},$
( 4.4)

with $x_{0}^{\varrho}=\xi$ and $y_{T}^{\varrho}=\varrho
G(x_{T}^{\varrho},c_{T}^{0},c_{T})+(1-\varrho)x_{T}^{\varrho}+\eta$. Note that
when $\varrho=0$ we have a decoupled set of SDE and BSDE and hence the
hypothesis trivially holds. Our goal is to extend the $\varrho$ up to $1$ by
following Peng-Wu’s continuation method [24]. Now, for an arbitrary set of
inputs
$(x,y,z^{0},z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})^{2}\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ and
constant $\zeta\in(0,1)$, consider

$\displaystyle dX_{t}=\bigl{[}\varrho B(t,Y_{t},c_{t}^{0},c_{t})+\zeta
B(t,y_{t},c_{t}^{0},c_{t})+I_{t}^{b}\bigr{]}dt+\sigma_{0}(t,c_{t}^{0},c_{t})dW_{t}^{0}+\sigma(t,c_{t}^{0},c_{t})dW_{t}^{1}~{},$
$\displaystyle dY_{t}=-\bigl{[}(1-\varrho)\gamma X_{t}-\varrho
F(t,X_{t},Y_{t},c_{t}^{0},c_{t})+\zeta(-\gamma
x_{t}-F(t,x_{t},y_{t},c_{t}^{0},c_{t}))+I_{t}^{f}\bigr{]}dt$
$\displaystyle\qquad\quad+Z_{t}^{0}dW_{t}^{0}+Z_{t}dW_{t}^{1}~{},$ ( 4.5)

with $X_{0}=\xi$ and $Y_{T}=\varrho
G(X_{T},c_{T}^{0},c_{T})+(1-\varrho)X_{T}+\zeta(G(x_{T},c_{T}^{0},c_{T})-x_{T})+\eta$.
The existence of the solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\times\mathbb{S}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$
is guaranteed by the previous hypothesis. We are going to prove the map
$(x,y,z^{0},z)\mapsto(X,Y,Z^{0},Z)$ defined above becomes strict contraction
when $\zeta>0$ is chosen small enough.

For two set of inputs $(x,y,z^{0},z)$ and
$(x^{\prime},y^{\prime},z^{0\prime},z^{\prime})$, let us denote the
corresponding solutions to $(\ref{shifted-1})$ by $(X,Y,Z^{0},Z)$ and
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})$, respectively. We put $\Delta
X_{t}:=X_{t}-X_{t}^{\prime}$, $\Delta Y_{t}:=Y_{t}-Y_{t}^{\prime}$ and
similarly for the others. Applying Itô’s formula to $\langle\Delta
X_{t},\Delta Y_{t}\rangle$ and using the estimates $(\ref{peng-wu-
condition})$, we obtain

$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
Y_{T}\rangle\bigr{]}\leq-\gamma\mathbb{E}\int_{0}^{T}|\Delta X_{t}|^{2}dt$
$\displaystyle\hskip 85.35826pt+\zeta C\mathbb{E}\int_{0}^{T}\Bigl{[}|\Delta
Y_{t}|(|\Delta y_{t}|+\mathbb{E}[\Delta y_{t}|\overline{{\cal
F}}_{t}^{0}])+|\Delta X_{t}|(|\Delta x_{t}|+\mathbb{E}[|\Delta
y_{t}|\overline{{\cal F}}_{t}^{0}])\Bigr{]}dt~{},$
$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
Y_{T}\rangle\bigr{]}\geq(\varrho\gamma+(1-\varrho))\mathbb{E}[|\Delta
X_{T}|^{2}]-\zeta C\mathbb{E}\bigl{[}|\Delta X_{T}|(|\Delta
x_{T}|+\mathbb{E}[|\Delta x_{T}||\overline{{\cal F}}_{T}^{0}])\bigr{]}~{},$

with some $\varrho$-independent constant $C$. Let us set
$\gamma_{c}:=\min(1,\gamma)>0$. Then one easily confirms
$0<\gamma_{c}\leq\varrho\gamma+(1-\varrho)$ for any $\varrho\in[0,1)$. Then
the above estimates yields

$\displaystyle\gamma_{c}\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}$ $\displaystyle\leq$
$\displaystyle\zeta C\mathbb{E}\bigl{[}|\Delta X_{T}|(|\Delta
x_{T}|+\mathbb{E}[|\Delta x_{T}||\overline{{\cal F}}_{T}^{0}])\bigr{]}$
$\displaystyle+\zeta C\mathbb{E}\int_{0}^{T}\Bigl{[}|\Delta Y_{t}|(|\Delta
y_{t}|+\mathbb{E}[\Delta y_{t}|\overline{{\cal F}}_{t}^{0}])+|\Delta
X_{t}|(|\Delta x_{t}|+\mathbb{E}[|\Delta y_{t}|\overline{{\cal
F}}_{t}^{0}])\Bigr{]}dt~{}.$

Using Young’s inequality and and a new constant $C$, we get

$\displaystyle\mathbb{E}[|\Delta X_{T}|^{2}]+\mathbb{E}\int_{0}^{T}|\Delta
X_{t}|^{2}dtd\leq\zeta C\mathbb{E}\int_{0}^{T}\bigl{(}|\Delta
Y_{t}|^{2}+(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})\bigr{)}dt+\zeta
C\mathbb{E}[|\Delta x_{T}|^{2}]~{}.$ ( 4.6)

Treating $X,X^{\prime}$ as inputs, the standard estimates for the Lipschitz
BSDEs (see, for example, Theorem 4.2.3 in [25]) gives

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$
$\displaystyle\qquad\quad\leq C\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}+\zeta
C\mathbb{E}\Bigl{[}|\Delta x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta
y_{t}|^{2})dt\Bigr{]}~{}.$

Combining with $(\ref{eq-pw-1})$ and choosing $\zeta>0$ small, we obtain

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta
Z_{t}|^{2})dt\Bigr{]}\leq\zeta C\mathbb{E}\Bigl{[}|\Delta
x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})dt\Bigr{]}~{}.$
( 4.7)

By the similar procedures, we also have

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}\Bigr{]}\leq\zeta C\mathbb{E}\Bigl{[}|\Delta
x_{T}|^{2}+\int_{0}^{T}(|\Delta x_{t}|^{2}+|\Delta y_{t}|^{2})dt\Bigr{]}~{}.$
( 4.8)

From $(\ref{eq-pw-2})$ and $(\ref{eq-pw-3})$, we obtain

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\leq\zeta
C\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta x_{t}|^{2}+\sup_{t\in[0,T]}|\Delta
y_{t}|^{2}+\int_{0}^{T}(|\Delta z_{t}^{0}|^{2}+|\Delta z_{t}|^{2})dt\Bigr{]}.$

Thus there exists $\zeta>0$, being independent of the size of $\varrho$, that
makes the map $(x,y,z^{0},z)\mapsto(X,Y,Z^{0},Z)$ strict contraction.
Therefore the initial hypothesis holds true for $(\varrho+\zeta)$, which
establishes the existence. The uniqueness follows from the next proposition. ∎

######  Proposition 4.1.

Given two set of inputs $(\xi,c^{0},c),(\xi^{\prime},c^{0\prime},c^{\prime})$,
coefficients $(\delta,\Lambda),(\delta^{\prime},\Lambda^{\prime})$ and the
coefficient functions
$(l,\sigma_{0},\sigma,\overline{f},\overline{g}),(l^{\prime},\sigma_{0}^{\prime},\sigma^{\prime},\overline{f}^{\prime},\overline{g}^{\prime})$
satisfying Assumptions (MFG-a,b,c1), let us denote the corresponding solutions
to $(\ref{fbsde-single-p})$ by $(X,Y,Z^{0},Z)$ and
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})$, respectively. We also define
the functionals $(B,F,G)$and $(B^{\prime},F^{\prime},G^{\prime})$ by
$(\ref{BG-notation})$ with corresponding coefficients, respectively. Then, we
have the following stability result:

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\qquad\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+|\overline{G}|^{2}+\int_{0}^{T}\Bigl{(}|\overline{F}(t)|^{2}+|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\Bigr{)}dt\Bigr{]}~{},$

where $C$ is a constant depending only on $T$ as well as the Lipschitz
constants of the system, and

$\displaystyle\overline{B}(t):=B(t,Y_{t}^{\prime},c^{0}_{t},c_{t})-B^{\prime}(t,Y_{t}^{\prime},c_{t}^{0\prime},c_{t}^{\prime}),$
$\displaystyle\overline{F}(t):=F(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0},c_{t})-F^{\prime}(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0\prime},c_{t}^{\prime}),$
$\displaystyle(\overline{\sigma}_{0},\overline{\sigma})(t)=(\sigma_{0}(t,c_{t}^{0},c_{t})-\sigma_{0}^{\prime}(t,c_{t}^{0\prime},c_{t}^{\prime}),\sigma(t,c_{t}^{0},c_{t})-\sigma^{\prime}(t,c_{t}^{0\prime},c_{t}^{\prime})),$
$\displaystyle\overline{G}:=G(X_{T}^{\prime},c_{T}^{0},c_{T})-G^{\prime}(X_{T}^{\prime},c_{T}^{0\prime},c_{T}^{\prime})~{},$

and $\Delta\xi:=\xi-\xi^{\prime}$, $\Delta X_{t}:=X_{t}-X_{t}^{\prime}$ and
similarly for the other variables.

###### Proof.

Let us put $\Delta
B(t):=B(t,Y_{t},c_{t}^{0},c_{t})-B(t,Y_{t}^{\prime},c_{t}^{0},c_{t})$, $\Delta
F(t):=F(t,X_{t},Y_{t},c_{t}^{0},c_{t})-F(t,X_{t}^{\prime},Y_{t}^{\prime},c_{t}^{0},c_{t})$
and $\Delta G:=G(X_{T},c_{T}^{0},c_{T})-G(X_{T}^{\prime},c_{T}^{0},c_{T})$. We
get by Itô’s formula that

$\displaystyle\mathbb{E}\bigl{[}\langle\Delta X_{T},\Delta
G+\overline{G}\rangle\bigr{]}=\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle+\int_{0}^{T}\Bigl{(}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta Y_{t}\rangle$
$\displaystyle\qquad+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle+\bigl{(}\langle\Delta F(t),\Delta X_{t}\rangle+\langle\Delta
B(t),\Delta Y_{t}\rangle\bigr{)}\Bigr{)}dt\Bigr{]}~{}.$

Using $(\ref{peng-wu-condition})$, we obtain

$\displaystyle\gamma\mathbb{E}\Bigl{[}|\Delta X_{T}|^{2}+\int_{0}^{T}|\Delta
X_{t}|^{2}dt\Bigr{]}\leq\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle-\langle\Delta X_{T},\overline{G}\rangle$
$\displaystyle\qquad+\int_{0}^{T}\Bigl{(}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta
Y_{t}\rangle+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle\Bigr{)}dt\Bigr{]}~{}.$ ( 4.9)

On the other hand, the standard estimates for Lipschitz SDEs and BSDEs give

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
Y_{t}|^{2}+\int_{0}^{T}(|\Delta Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$
$\displaystyle\qquad\leq
C\mathbb{E}\Bigl{[}|\overline{G}|^{2}+\int_{0}^{T}|\overline{F}(t)|^{2}dt\Bigr{]}+C\mathbb{E}\Bigl{[}|\Delta
X_{T}|^{2}+\int_{0}^{T}|\Delta X_{t}|^{2}dt\Bigr{]}~{},$ ( 4.10)
$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta X_{t}|^{2}\Bigr{]}\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+\int_{0}^{T}\bigl{[}|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\bigr{]}dt\Bigr{]}+C\mathbb{E}\int_{0}^{T}|\Delta
Y_{t}|^{2}dt~{}.$

Combining the above inequalities $(\ref{stability-1})$ and
$(\ref{stability-y})$ gives

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|\Delta
X_{t}|^{2}+\sup_{t\in[0,T]}|\Delta Y_{t}|^{2}+\int_{0}^{T}(|\Delta
Z_{t}^{0}|^{2}+|\Delta Z_{t}|^{2})dt\Bigr{]}$ $\displaystyle\leq
C\mathbb{E}\Bigl{[}|\Delta\xi|^{2}+|\overline{G}|^{2}+\int_{0}^{T}\bigl{[}|\overline{F}(t)|^{2}+|\overline{B}(t)|^{2}+|\overline{\sigma}_{0}(t)|^{2}+|\overline{\sigma}(t)|^{2}\bigr{]}dt\Bigr{]}$
$\displaystyle+C\mathbb{E}\Bigl{[}\langle\Delta\xi,\Delta
Y_{0}\rangle-\langle\Delta
X_{T},\overline{G}\rangle+\int_{0}^{T}\bigl{[}\langle\overline{F}(t),\Delta
X_{t}\rangle+\langle\overline{B}(t),\Delta
Y_{t}\rangle+\langle\overline{\sigma}_{0}(t),\Delta
Z_{t}^{0}\rangle+\langle\overline{\sigma}(t),\Delta
Z_{t}\rangle\bigr{]}dt\Bigr{]}~{}.$

Now simple application of Young’s inequality establishes the claim. ∎

######  Corollary 4.1.

Under Assumptions (MFG-a,b,c1), the solution $(X,Y,Z^{0},Z)$ to the FBSDE
$(\ref{fbsde-single-p})$ satisfies the following estimate:

$\displaystyle\mathbb{E}\Bigl{[}\sup_{t\in[0,T]}|X_{t}|^{2}+\sup_{t\in[0,T]}|Y_{t}|^{2}+\int_{0}^{T}(|Z_{t}^{0}|^{2}+|Z_{t}|^{2})dt\Bigr{]}\leq
C\mathbb{E}\Bigl{[}|\xi|^{2}+|\partial_{x}\overline{g}(0,c_{T}^{0},c_{T})|^{2}$
$\displaystyle\hskip
113.81102pt+\int_{0}^{T}\Bigl{(}|\partial_{x}\overline{f}(t,0,0,c_{t}^{0},c_{t})|^{2}+|l(t,0,c_{t}^{0},c_{t})|^{2}+|(\sigma_{0},\sigma)(t,c_{t}^{0},c_{t})|^{2}\Bigr{)}dt\Bigr{]}~{},$

where $C$ is a constant depending only on $T,\delta$ and Lipschitz constants
of the system.

###### Proof.

By quick inspection of the proof for Proposition 4.1, one can confirm that as
long as there exists a solution
$(X^{\prime},Y^{\prime},Z^{0\prime},Z^{\prime})\in\mathbb{S}^{2}\times\mathbb{S}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$,
their coefficients need not satisfy Assumption (MFG-a,b,c1). In particular, by
putting $\xi^{\prime}$ and
$(l^{\prime},\sigma_{0}^{\prime},\sigma^{\prime},\overline{f}^{\prime},\overline{g}^{\prime})$
all zero, we have a trivial solution
$(X^{\prime},Y^{\prime},Z^{0^{\prime}},Z^{\prime})=(0,0,0,0)$. The desired
estimate now follows from Proposition 4.1. ∎

#### Securities of maturity $T$ with exogenously specified payoff

If we consider the exchange markets of bonds and futures, or other financial
derivatives with maturity $T$, those securities cease to exist at $T$ after
paying exogenously specified amount of cash $c_{T}^{0}$. In this case, it is
natural to consider with $\delta=0$ and

$\displaystyle g(x,c^{0})=\overline{g}(x,c^{0}):=-\langle c^{0},x\rangle~{},$
( 4.11)

since there is no reason to put penalty on the outstanding volume at $T$. In
this case, the terminal function $g$ in $(\ref{terminal-special})$ does not
have the strict convexity. Fortunately, even in this case, we can prove the
unique existence as well as the stability result of the same form.

######  Assumption 4.3.

(MFG-c2)
(i) The functions $\sigma_{0}$ and $\sigma$ are independent of the argument
$\varpi$.
(ii) For any $t\in[0,T]$, any random variables
$x,x^{\prime},c^{0},c\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and any
sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the function $l$ satisfies the
monotone condition with some positive constant $\gamma^{l}>0$:

$\displaystyle\mathbb{E}\Bigl{[}\langle l(t,\mathbb{E}[x|{\cal
G}],c^{0},c)-l(t,\mathbb{E}[x^{\prime}|{\cal
G}],c^{0},c),x-x^{\prime}\rangle\Bigr{]}\geq\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[x-x^{\prime}|{\cal
G}]^{2}\bigr{]}~{}.$

(iii) $\gamma:=\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}$ is strictly
positive and the terminal function $g$ is given by $(\ref{terminal-special})$
with $\delta=0$.

######  Theorem 4.3.

Under Assumptions (MFG-a,b,c2), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{1};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{1};\mathbb{R}^{n\times d})$ to the
FBSDE $(\ref{fbsde-single-p})$. Moreover, the same form of stability and
$\mathbb{L}^{2}$ estimates given in Proposition 4.1 and Corollary 4.1 hold.

###### Proof.

Note that, in this case, the terminal condition for the BSDE is independent of
$X_{T}$. Thus, as in Theorem 2.3 [24], we put
$y_{T}^{\varrho}=Y_{T}=-c_{T}^{0}$ in $(\ref{shifted-0})$ and
$(\ref{shifted-1})$, respectively. Using the fact that $\langle\Delta
X_{T},\Delta Y_{T}\rangle=0$, one can follow the same arguments to get the
desired result. The proof of the stability result can also be done in almost
exactly the same way. ∎

##  5 Asymptotic Market Clearing

We are now ready to investigate if our FBSDE $(\ref{fbsde-single-p})$ actually
provides a good approximate of the market price and if so, how accurate it is.
By Theorem 3.1, if we use $(-\mathbb{E}\bigl{[}Y_{t}|\overline{{\cal
F}}_{t}^{0}\bigr{]})_{t\in[0,T]}$ as the input $(\varpi_{t})_{t\in[0,T]}$,
where $(Y_{t})_{t\in[0,T]}$ is the unique solution to the FBSDE $(\ref{fbsde-
single-p})$ with the convention $\xi=\xi^{1}$ and $c=c^{1}$, the optimal
strategy of the individual agent is given by

$\displaystyle\widehat{\alpha}^{i}_{\rm{mf}}(t):=\widehat{\alpha}(Y_{t}^{i},-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])=-\overline{\Lambda}(Y_{t}^{i}-\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])$ ( 5.1)

where $(Y^{i}_{t})_{t\in[0,T]}$ is the solution to $(\ref{agent-FBSDE})$ with
$(\varpi_{t}=-\mathbb{E}\bigl{[}Y_{t}|\overline{{\cal
F}}_{t}^{0}\bigr{]})_{t\in[0,T]}$.

######  Theorem 5.1.

If the conditions for Theorem 4.1, Theorem 4.2 or Theorem 4.3 are satisfied
then we have

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt=0~{}.$

Moreover if there exists some constant $\Gamma$ such that
$\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|Y_{t}|^{q}\bigr{]}^{\frac{1}{q}}\leq\Gamma<\infty$
for some $q>4$, then there exists some constant $C$ independent of $N$ such
that

$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\Gamma^{2}\epsilon_{N},$ ( 5.2)

where
$\epsilon_{N}:=N^{-2/\max(n,4)}\bigl{(}1+\log(N)\mathbb{1}_{\\{n=4\\}}\bigr{)}$.

###### Proof.

Let us consider the following set of FBSDEs with $1\leq i\leq N$ on the
filtered probability space $(\Omega,{\cal F},\mathbb{P};\mathbb{F})$
constructed in Section 2.

$\displaystyle d\underline{X}_{t}^{i}$ $\displaystyle=$
$\displaystyle\Bigl{(}-\overline{\Lambda}(\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}])+l(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})\Bigr{)}dt$
$\displaystyle\qquad+\sigma_{0}(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dW_{t}^{0}+\sigma(t,-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dW_{t}^{i},$ $\displaystyle
d\underline{Y}_{t}^{i}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}(t,\underline{X}_{t}^{i},-\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{i})dt+\underline{Z}_{t}^{i,0}dW_{t}^{0}+\underline{Z}_{t}^{i}dW_{t}^{i},$

with $\underline{X}_{0}^{i}=\xi^{i}$ and
$\underline{Y}_{T}^{i}=\delta/(1-\delta)\mathbb{E}[\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})|\overline{{\cal
F}}_{T}^{0}]+\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})$.
Thanks to the existence of unique strong solution, Yamada-Watanabe Theorem for
FBSDEs (see, Theorem 1.33 [8]), there exists some measurable function $\Phi$
such that for every $1\leq i\leq N$,

$\displaystyle(\underline{X}^{i}_{t},\underline{Y}^{i}_{t})_{t\in[0,T]}=\Phi\Bigl{(}(c_{t}^{0})_{t\in[0,T]},(W_{t}^{0})_{t\in[0,T]},\xi^{i},(c_{t}^{i})_{t\in[0,T]},(W_{t}^{i})_{t\in[0,T]}\Bigr{)}~{}.$

Hence, conditionally on $\overline{{\cal F}}^{0}$, the set of proceses
$(\underline{X}^{i}_{t},\underline{Y}^{i}_{t})_{t\in[0,T]}$ with $1\leq i\leq
N$ are independently and identically distributed. In particular, we have
$\mathbb{P}$-a.s.

$\displaystyle\mathbb{E}[\underline{Y}_{t}^{i}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}],\quad\forall
t\in[0,T],$
$\displaystyle\mathbb{E}[\partial_{x}\overline{g}(\underline{X}_{T}^{i},c_{T}^{0},c_{T}^{i})|\overline{{\cal
F}}_{T}^{0}]=\mathbb{E}[\partial_{x}\overline{g}(X_{T},c_{T}^{0},c_{T})|\overline{{\cal
F}}_{T}^{0}]~{}.$ ( 5.3)

Note that, under the convention $\xi^{1}=\xi$ and $c^{1}=c$, we actually have
$(\underline{X}^{1},\underline{Y}^{1})=(X,Y)$. From $(\ref{law-identity})$, we
conclude that
$(X^{i}_{t},Y^{i}_{t},Z^{i,0}_{t},Z^{i}_{t})_{t\in[0,T]}=(\underline{X}^{i}_{t},\underline{Y}^{i}_{t},\underline{Z}^{i,0}_{t},\underline{Z}_{t}^{i})_{t\in[0,T]}$
in
$\mathbb{S}^{2}(\mathbb{F}^{i})\times\mathbb{S}^{2}(\mathbb{F}^{i})\times\mathbb{H}^{2}(\mathbb{F}^{i})\times\mathbb{H}^{2}(\mathbb{F}^{i})$.
Therefore,

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)=-\overline{\Lambda}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$ ( 5.4)

We can easily check that

$\displaystyle\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{|}\overline{{\cal
F}}_{t}^{0}\Bigr{]}\leq\frac{2}{N}\sum_{i=1}^{N}\mathbb{E}\bigl{[}|\underline{Y}_{t}^{i}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}+2\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}=4\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}|\overline{{\cal
F}}_{t}^{0}\bigr{]}~{}.$

Since $(\underline{Y}_{t}^{i})_{1\leq i\leq N}$ are $\overline{{\cal
F}}^{0}_{t}$-conditionally independently and identically distributed and also
$\underline{Y}^{1}\in\mathbb{S}^{2}$, the same arguments leading to $(2.14)$
in [8] imply that the pointwise convergence holds:

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}=0~{}.$ ( 5.5)

We are now going to show that the set of functions, $(f_{N})_{N\in\mathbb{N}}$
defined by

$\displaystyle[0,T]\ni t\mapsto
f_{N}(t):=\mathbb{E}\bigl{[}W_{2}\bigl{(}\overline{\mu}_{t},\mu_{t}\bigr{)}^{2}\bigr{]}\in\mathbb{R}$

with
$\overline{\mu}_{t}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}}$
and $\mu_{t}:={\cal L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})$ are
precompact in the set ${\cal C}([0,T];\mathbb{R})$ endowed with the topology
of uniform convergence. In fact, uniformly in $N$,

$\displaystyle\sup_{t\in[0,T]}|f_{N}(t)|\leq
4\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}|^{2}\bigr{]}\leq
C<\infty$ ( 5.6)

where $C$ is given by the estimate in Corollary 4.1. Moreover, for any $0\leq
t,s\leq T$, Cauchy-Schwarz, $(\ref{fN-inequality})$ and the triangular
inequalities give

$\displaystyle|f_{N}(t)-f_{N}(s)|\leq\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})+W_{2}(\overline{\mu}_{s},\mu_{2})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})-W_{2}(\overline{\mu}_{s},\mu_{2})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}$
$\displaystyle\quad\leq
C\mathbb{E}\Bigl{[}\Bigl{(}W_{2}(\overline{\mu}_{t},\mu_{t})-W_{2}(\overline{\mu}_{s},\mu_{s})\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\leq
C\mathbb{E}\Bigl{[}W_{2}(\overline{\mu}_{t},\overline{\mu}_{s})^{2}+W_{2}(\mu_{t},\mu_{s})^{2}\Bigr{]}^{\frac{1}{2}}~{}.$
$\displaystyle\quad\leq
C\mathbb{E}\Bigl{[}\frac{1}{N}\sum_{i=1}^{N}|\underline{Y}_{t}^{i}-\underline{Y}_{s}^{i}|^{2}+|\underline{Y}_{t}^{1}-\underline{Y}_{s}^{1}|^{2}\Bigr{]}^{\frac{1}{2}}$
$\displaystyle\quad\leq
C\mathbb{E}\bigl{[}|\underline{Y}_{t}^{1}-\underline{Y}_{s}^{1}|^{2}\bigr{]}^{\frac{1}{2}}~{},$

uniformly in $N$, where we have used the fact that $(\underline{Y}^{i})_{i\geq
1}$ are conditionally i.i.d at the last inequality. Since
$(\underline{Y}_{t}^{1})_{t\in[0,T]}$ is a continuous process, the above
estimate tells that $(f_{N})_{N\in\mathbb{N}}$ is equicontinuous, which is
also uniformly equicontinuous since we are working on the finite interval.
Now, Arzela-Ascoli theorem implies the desired precompactness.

Combining with the pointwise convergence $(\ref{pointwise-conv})$, we thus
conclude

$\displaystyle\lim_{N\rightarrow\infty}\sup_{t\in[0,T]}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}=0~{}.$ ( 5.7)

From the definition of Wasserstein distance $(\ref{def-W})$, we have

$\displaystyle\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\underline{Y}_{t}^{i}-\mathbb{E}[\underline{Y}_{t}^{1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{|}\leq
W_{1}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})\Bigr{)}~{},$

and hence, from $(\ref{alpha-m})$,

$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{i=1}^{N}\widehat{\alpha}^{i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\sup_{t\in[0,T]}\mathbb{E}\Bigl{[}W_{2}\Bigl{(}\frac{1}{N}\sum_{i=1}^{N}\delta_{\underline{Y}_{t}^{i}},{\cal
L}(\underline{Y}_{t}^{1}|\overline{{\cal F}}_{t}^{0})\Bigr{)}^{2}\Bigr{]}~{}.$
( 5.8)

The first conclusion now follows from $(\ref{law-conv})$. The latter claims
directly follows from the expression $(\ref{law-conv-2})$ and the (Fourth
Step) in the proof of Theorem 2.12 in [8]. ∎

Theorem 5.1 justifies our intuitive understanding and a special type of FBSDEs
$(\ref{fbsde-single-p})$ derived in Section 3 as a reasonable model to
approximate the market clearing price. When there exists higher integrability,
Glivenko-Cantelli convergence theorem in the Wasserstein distance even
provides a specific order $\epsilon_{N}$ of convergence in terms of the number
of agents $N$ $(\ref{Glivenko-Cantelli})$. See Theorem 5.8 and Remark 5.9 in
[7] for more details.

######  Remark 5.1.

Consider the situation treated in Theorem 4.3, for example, a market model of
a Futures contract. If the contract pays unit amount of the underlying asset
per contract whose value is exogenously given by $c_{T}^{0}$, our mean-field
limit model $(\ref{fbsde-single-p})$ gives $Y_{T}=-c_{T}^{0}$. This means that
the modeled Futures price satisfies
$\varpi_{T}=-\mathbb{E}[Y_{T}|\overline{{\cal F}}_{T}^{0}]=c_{T}^{0}$, which
guarantees the convergence of the modeled price to the value of the underlying
asset at the maturity $T$. This is a crucially important feature that any
market model of this type of securities must satisfy.

##  6 Extension to Multiple Populations

The main limitation of the last model is that there exists only one type of
agents who share the common cost functions as well as the coefficient
functions for their state dynamics. Interestingly, it is rather
straightforward to extend the model to the situation with multiple
populations, where the agents in each population share the same cost and
coefficient functions but they can be different population by population. From
the perspective of the practical applications, this is a big advantage since
we can analyze, for example, the interactions between the Sell-side and Buy-
side institutions for financial applications, or consumers and producers for
economic applications. For general issues of mean field games as well as mean
field type control problems in the presence of multiple populations without
common noise, see Fujii [13]. Although there exists a common noise in the
current model, the conditional law only enters as a form of expectation.
Therefore, as long as the system of FBSDEs is Lipschitz continuous, there
exists a unique strong solution at least for small $T$. For general $T$,
although it is rather difficult to find appropriate set of assumptions, it is
still possible for some simple cases. In this section, our main task is to
find an appropriate limit model that extends $(\ref{fbsde-single-p})$ for
multiple populations and the sufficient conditions that make appropriate
monotone conditions hold, which guarantees the existence of unique solution.

In the following, we shall treat $m$ populations indexed by
$p\in\\{1,\cdots,m\\}$. For each $p$, $N_{p}\geq 1$ agents are assumed to
belong to the population. We denote by $(p,i)$ the ith agent in the population
$p$. First, let us enlarge the probability space constructed in Section 2. In
addition to $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$, we introduce
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{P}}^{p,i})$ with
$1\leq i\leq N_{p}$ and $1\leq p\leq m$, each of which is generated by
$(\xi^{p,i},\boldsymbol{W}^{p,i})$ with $d$-dimensional Brownian motion
$\boldsymbol{W}^{p,i}$ and a $\boldsymbol{W}^{p,i}$-independent
$\mathbb{R}^{n}$-valued square integrable random variable $\xi^{p,i}$. For
each $p$, $(\xi^{p,i})_{i=1}^{N_{p}}$ are assumed to have the common law. We
define $(\Omega^{p,i},{\cal F}^{p,i},\mathbb{P}^{p,i};\mathbb{F}^{p,i})$ as
the product of $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{P}}^{p,i})$. Finally
$(\Omega,{\cal F},\mathbb{P};\mathbb{F})$ is defined as a product of all the
spaces $(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{F}}^{p,i})$, $1\leq
i\leq N_{p},1\leq p\leq m$, and $(\Omega^{i},{\cal
F}^{i},\mathbb{P}^{i};\mathbb{F}^{i})$ as a product of
$(\overline{\Omega}^{0},\overline{{\cal
F}}^{0},\overline{\mathbb{P}}^{0};\overline{\mathbb{F}}^{0})$ and
$(\overline{\Omega}^{p,i},\overline{{\cal
F}}^{p,i},\overline{\mathbb{P}}^{p,i};\overline{\mathbb{F}}^{p,i})$ with
$1\leq p\leq m$. Every probability space is assumed to be complete and every
filtration is assumed to be complete and right-continuously augmented to
satisfy the usual conditions.

As we have done in Section 3, we first assume that the market price of $n$
securities is given exogenously by
$\varpi_{t}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$. Under
this setup, we consider the control problem for each $(p,i)$ agent defined by

$\inf_{\boldsymbol{\alpha}^{p,i}\in\mathbb{A}^{p,i}}J^{p,i}(\boldsymbol{\alpha}^{p,i})~{},$
( 6.1)

with

$\displaystyle
J^{p,i}(\boldsymbol{\alpha}^{p,i}):=\mathbb{E}\Bigl{[}\int_{0}^{T}f_{p}(t,X_{t}^{p,i},\alpha_{t}^{p,i},\varpi_{t},c_{t}^{0},c_{t}^{p,i})dt+g_{p}(X_{T}^{p,i},\varpi_{T},c_{T}^{0},c_{T}^{p,i})\Bigr{]}~{},$

subject to the dynamic constraint:

$\displaystyle
dX_{t}^{p,i}=\Bigl{(}\alpha_{t}^{p,i}+l_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})\Bigr{)}dt+\sigma_{p,0}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{0}+\sigma_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{p,i}$

with $X_{0}^{p,i}=\xi^{p,i}$. As before we assume $(c_{t}^{0})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{0};\mathbb{R}^{n})$ and
$(c_{t}^{p,i})_{t\geq
0}\in\mathbb{H}^{2}(\overline{\mathbb{F}}^{p,i};\mathbb{R}^{n})$. In addition,
within each population $p$, the random sources $(c^{p,i}_{t})_{t\geq 0}$ are
assumed to have a common law $1\leq i\leq N_{p}$. Admissible strategies
$\mathbb{A}^{p,i}$ is the space
$\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})$. The measurable functions
$f_{p}:[0,T]\times(\mathbb{R}^{n})^{5}\rightarrow\mathbb{R}$,
$g_{p}:(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$,
$\overline{f}_{p}:[0,T]\times(\mathbb{R}^{n})^{4}\rightarrow\mathbb{R}$ and
$\overline{g}_{p}:(\mathbb{R}^{n})^{3}\rightarrow\mathbb{R}$ are given by

$\displaystyle
f_{p}(t,x,\alpha,\varpi,c^{0},c):=\langle\varpi,\alpha\rangle+\frac{1}{2}\langle\alpha,\Lambda_{p}\alpha\rangle+\overline{f}_{p}(t,x,\varpi,c^{0},c)~{},$
$\displaystyle
g_{p}(x,\varpi,c^{0},c):=-\delta\langle\varpi,x\rangle+\overline{g}_{p}(x,c^{0},c)~{}.$

######  Assumption 6.1.

(MFG-A) We assume the following conditions uniformly in
$p\in\\{1,\cdots,m\\}$.
(i) $\Lambda_{p}$ is a positive definite $n\times n$ symmetric matrix with
$\underline{\lambda}I_{n\times
n}\leq\Lambda_{p}\leq\overline{\lambda}I_{n\times n}$ in the sense of 2nd-
order form where $\underline{\lambda}$ and $\overline{\lambda}$ are some
constants satisfying $0<\underline{\lambda}\leq\overline{\lambda}$.
(ii) For any $(t,x,\varpi,c^{0},c)$,

$\displaystyle|\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\overline{g}_{p}(x,c^{0},c)|\leq
L(1+|x|^{2}+|\varpi|^{2}+|c^{0}|^{2}+|c|^{2})~{}.$

(iii) $\overline{f}_{p}$ and $\overline{g}_{p}$ are continuously
differentiable in $x$ and satisfy, for any $(t,x,x^{\prime},\varpi,c^{0},c)$,

$|\partial_{x}\overline{f}_{p}(t,x^{\prime},\varpi,c^{0},c)-\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}_{p}(x^{\prime},c^{0},c)-\partial_{x}\overline{g}_{p}(x,c^{0},c)|\leq
L|x^{\prime}-x|~{},$

and
$|\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)|+|\partial_{x}\overline{g}_{p}(x,c^{0},c)|\leq
L(1+|x|+|\varpi|+|c^{0}|+|c|)$.
(iv)The functions $\overline{f}_{p}$ and $\overline{g}_{p}$ are convex in $x$
in the sense that for any $(t,x,x^{\prime},\varpi,c^{0},c)$,

$\displaystyle\overline{f}_{p}(t,x^{\prime},\varpi,c^{0},c)-\overline{f}_{p}(t,x,\varpi,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)\rangle\geq\frac{\gamma^{f}}{2}|x^{\prime}-x|^{2}~{},$
$\displaystyle\overline{g}_{p}(x^{\prime},c^{0},c)-\overline{g}_{p}(x,c^{0},c)-\langle
x^{\prime}-x,\partial_{x}\overline{g}_{p}(x,c^{0},c)\rangle\geq\frac{\gamma^{g}}{2}|x^{\prime}-x|^{2}~{},$

with some constants $\gamma^{f},\gamma^{g}\geq 0$.
(v) $l_{p},\sigma_{p,0},\sigma_{p}$ are the measurable functions defined on
$[0,T]\times(\mathbb{R}^{n})^{3}$ and are $\mathbb{R}^{n},\mathbb{R}^{n\times
d^{0}}$ and $\mathbb{R}^{n\times d}$-valued, respectively. Moreover they
satisfy the linear growth condition:

$\displaystyle|(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi,c^{0},c)|\leq
L(1+|\varpi|+|c^{0}|+|c|)$

for any $(t,\varpi,c^{0},c)$.
(vi) $\delta\in[0,1)$ is a given constant.

Under Assumption (MFG-A), Theorem 3.1 guarantees that the control problem
$(\ref{control-pi})$ for each agent $(p,i)$ is uniquely characterized by

$\displaystyle
dX_{t}^{p,i}=\Bigl{(}\widehat{\alpha}_{p}(Y_{t}^{p,i},\varpi_{t})+l_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})\Bigr{)}dt+\sigma_{p,0}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{0}+\sigma_{p}(t,\varpi_{t},c_{t}^{0},c_{t}^{p,i})dW_{t}^{p,i},$
$\displaystyle
dY_{t}^{p,i}=-\partial_{x}\overline{f}_{p}(t,X_{t}^{p,i},\varpi_{t},c_{t}^{0},c_{t}^{p,i})dt+Z_{t}^{p,i,0}dW_{t}^{0}+Z_{t}^{p,i}dW_{t}^{p,i},$
( 6.2)

with $X_{0}^{p,i}=\xi^{p,i}$ and
$Y_{T}^{p,i}=-\delta\varpi_{T}+\partial_{x}\overline{g}_{p}(X_{T}^{p,i},c_{T}^{0},c_{T}^{p,i})$.
We have defined
$\widehat{\alpha}_{p}(y,\varpi):=-\overline{\Lambda}_{p}(y+\varpi)$ and
$\overline{\Lambda}_{p}:=(\Lambda_{p})^{-1}$ as before. There exists a unique
strong solution
$(X^{p,i}_{t},Y^{p,i}_{t},Z^{p,i,0}_{t},Z^{p,i}_{t})_{t\in[0,T]}\in\mathbb{S}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})\times\mathbb{S}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n})\times\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n\times
d^{0}})\times\mathbb{H}^{2}(\mathbb{F}^{p,i};\mathbb{R}^{n\times d})$, and the
optimal trading strategy for the agent $(p,i)$ is given by

$\displaystyle\widehat{\alpha}_{t}^{p,i}=\widehat{\alpha}_{p}(Y_{t}^{p,i},\varpi_{t})~{},\forall
t\in[0,T].$

Let us check the market clearing condition under this setup. In order to
balance the demand and supply of securities at the exchange, we need to have
$\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}(Y^{p,i}_{t},\varpi_{t})=0$.
This requires the market price to satisfy

$\displaystyle\varpi_{t}=-\Bigl{(}\sum_{p=1}^{m}n_{p}\overline{\Lambda}_{p}\Bigr{)}^{-1}\sum_{p=1}^{m}n_{p}\overline{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}\Bigr{)}~{},$

where $N=\sum_{p=1}^{m}N_{p}$ and $n_{p}:=N_{p}/N$. At the moment, this is
inconsistent to the initial assumption that requires $(\varpi_{t})_{t\geq 0}$
to be $\overline{\mathbb{F}}^{0}$-adapted. However, since for each $1\leq
p\leq m$, $(Y^{p,i}_{t})_{i=1}^{N_{p}}$ are $\overline{{\cal
F}}^{0}$-conditionally independently and identically distributed, we may
follow the same arguments used in Section 3. If we take $N\rightarrow\infty$
while keeping the relative size of populations $n_{p}$ constant, we can expect
to obtain

$\displaystyle\varpi_{t}=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]$ ( 6.3)

in the large population limit where

$\hat{\Lambda}_{p}:=n_{p}\overline{\Lambda}_{p},\qquad\hat{\Xi}:=\Bigl{(}\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigr{)}^{-1}~{}.$

######  Remark 6.1.

When $\Lambda_{p}=\Lambda$ for every population $p$, one can easily check that
$(\ref{market-price-guess})$ becomes

$\varpi_{t}=-\sum_{p=1}^{m}n_{p}\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]~{}.$

Since $Y$ of the adjoint equation represents the marginal cost i.e., the first
order derivative of the value function with respect to the state variable $x$,
the above expression of $\varpi$ implies that the market price may be given by
the population-weighted average of the marginal benefit (-cost) across the
entire populations.

###  6.1 Limit problem with multiple populations

By the observation we have just made, we are motivated to study the following
limit problem with $1\leq p\leq m$:

$\displaystyle dX_{t}^{p}$ $\displaystyle=$
$\displaystyle\Bigl{(}\widehat{\alpha}_{p}\bigl{(}Y_{t}^{p},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])\bigr{)}+l_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}\Bigr{)}dt$
$\displaystyle+\sigma_{p,0}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dW_{t}^{0}+\sigma_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dW_{t}^{p,1},$ $\displaystyle
dY_{t}^{p}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}_{p}\bigl{(}t,X_{t}^{p},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p}\bigr{)}dt+Z_{t}^{p,0}dW_{t}^{0}+Z_{t}^{p}dW_{t}^{p,1}~{},$
( 6.4)

with $X_{0}^{p}=\xi^{p}$ and

$\displaystyle
Y_{T}^{p}=\frac{\delta}{1-\delta}\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}\bigl{[}\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{}.$

We put as before $\xi^{p}:=\xi^{p,1}$ and $c^{p}:=c^{p,1}$ to lighten the
notation. Here,

$\displaystyle\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]):=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],\quad\widehat{\alpha}_{p}(y,\varpi):=-\overline{\Lambda}_{p}(y+\varpi)$

and hence $(\ref{fbsde-multiple-p})$ is actually an $m$-coupled system of
FBSDEs of McKean-Vlasov type. One can derive the terminal condition from

$Y_{T}^{p}=-\delta\varpi(\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}])+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{},$
( 6.5)

by summing over $1\leq p\leq m$ after taking conditional expectation given
$\overline{{\cal F}}_{T}^{0}$. In the following, we use the notation

$(X_{t},Y_{t},Z_{t}^{0},Z_{t})_{t\in[0,T]}=\Bigl{(}(X_{t}^{p})_{p=1}^{m},(Y_{t}^{p})_{p=1}^{m},(Z_{t}^{p,0})_{p=1}^{m},(Z_{t}^{p})_{p=1}^{m}\Bigr{)}_{t\in[0,T]}~{}.$
( 6.6)

###  6.2 Solvability for small $T$

For small $T$, Lipschitz continuity suffices to guarantee the existence of a
unique solution.

######  Assumption 6.2.

(MFG-B)
Uniformly in $p\in\\{1,\cdots,m\\}$, for any
$(t,x,c^{0},c)\in[0,T]\times(\mathbb{R}^{n})^{3}$ and any
$\varpi,\varpi^{\prime}\in\mathbb{R}^{n}$, the coefficient functions
$l_{p},\sigma_{p,0},\sigma_{p}$ and $\overline{f}_{p}$ satisfy

$\displaystyle|(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi,c^{0},c)-(l_{p},\sigma_{p,0},\sigma_{p})(t,\varpi^{\prime},c^{0},c)|$
$\displaystyle\qquad\qquad+|\partial_{x}\overline{f}_{p}(t,x,\varpi,c^{0},c)-\partial_{x}\overline{f}_{p}(t,x,\varpi^{\prime},c^{0},c)|\leq
L_{\varpi}|\varpi-\varpi^{\prime}|~{}.$

The following theorem follows exactly in the same way as Theorem 4.1.

######  Theorem 6.1.

Under Assumptions (MFG-A,B), there exists some constant $\tau>0$ which depends
only on $(L,L_{\varpi},\delta,n_{p},\Lambda_{p})$ such that for any
$T\leq\tau$, there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$.

######  Remark 6.2.

Note that the above system of FBSDEs becomes a linear-quadratic form by
choosing $(l_{p},\sigma_{p,0},\sigma_{p},\overline{f}_{p},\overline{g}_{p})$
appropriately. In this case, the problem reduces to solving ordinary
differential equations of Riccati type. Therefore, the existence of a solution
for a given $T$ can be tested, at lest numerically, by checking the absence of
a “blow up” in its solution.

###  6.3 Solvability for general $T$

We now move on to the existence result of a unique solution for general $T$.
It is very difficult to find general existence criteria for fully-coupled
multi-dimensional FBSDEs. A the moment, in order to apply well-known Peng-Wu’s
method, let us put the following simplifying assumptions.

######  Assumption 6.3.

(MFG-C1)
(i) For every $1\leq p\leq m$, the functions $\sigma_{p,0}$ and $\sigma_{p}$
are independent of the argument $\varpi$.
(ii) $\Lambda_{p}$=$\Lambda$ and $n_{p}=1/m$ for every $p$.
(iii) For any $t\in[0,T]$, any random variables
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the functions
$(l_{p})_{p=1}^{m}$ satisfy with some positive constant $\gamma^{l}>0$,

$\displaystyle\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\bigl{\langle}l_{p}\bigl{(}t,\mathbb{E}[\overline{x}|{\cal
G}],c^{0},c^{0}\bigr{)}-l_{p}\bigl{(}t,\mathbb{E}[\overline{x}^{\prime}|{\cal
G}],c^{0},c^{p}\bigr{)},x^{p}-x^{p\prime}\bigr{\rangle}\Bigr{]}\geq
m\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\overline{x}-\overline{x}^{\prime}|{\cal
G}]^{2}\bigr{]},$

where $\overline{x}:=\frac{1}{m}\sum_{p=1}^{m}x^{p}$ and similarly for
$\overline{x}^{\prime}$.
(iv) There exists a strictly positive constant $\gamma$ satisfying
$0<\gamma\leq\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\wedge\gamma^{g}$.
Moreover, the functions $(\overline{g}_{p})_{p=1}^{m}$ satisfy for any
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$,

$\displaystyle\frac{\delta}{1-\delta}m^{-1}\mathbb{E}\Bigl{[}\bigl{\langle}\sum_{p=1}^{m}\mathbb{E}[\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})-\partial_{x}\overline{g}_{p}(x^{p\prime},c^{0},c^{p})|{\cal
G}],\sum_{p=1}^{m}(x^{p}-x^{p\prime})\bigr{\rangle}\Bigr{]}$
$\displaystyle\qquad\quad+\gamma^{g}\sum_{p=1}^{m}\mathbb{E}[|x^{p}-x^{p\prime}|^{2}]\geq\gamma\sum_{p=1}^{m}\mathbb{E}[|x^{p}-x^{p\prime}|^{2}]~{}.$

######  Remark 6.3.

The conditions (iii) and (iv) in the above assumption are rather restrictive.
The condition (iii) is satisfied, for example, if $l_{p}$ has a separable form
$l_{p}=h(x)+h_{p}(c^{0}_{t},c_{t}^{p})$ with some function $h$, which is
common to every population and strictly monotone. (iv) is also satisfied by
requiring similar structure. Or, since $\partial_{x}\overline{g}_{p}$ is
Lipschitz continuous in $x$, the absolute value of the first term is bounded
by
$\frac{\delta}{1-\delta}\max((L_{p})_{p=1}^{m})\sum_{p=1}^{m}\mathbb{E}|x^{p}-x^{p\prime}|^{2}$,
where the $L_{p}$ is the Lipschitz constant for
$\partial_{x}\overline{g}_{p}$. Thus the condition (iv) is satisfied if
$\delta\max((L_{p})_{p=1}^{m})$ is sufficiently small.

The next result is the counterpart of Theorem 4.2.

######  Theorem 6.2.

Under Assumptions (MFG-A,B,C1), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$. Moreover, the same
form of stability and $\mathbb{L}^{2}$ estimates given in Proposition 4.1 and
Corollary 4.1 hold.

###### Proof.

Under Assumption (MFG-C1), $(\ref{fbsde-multiple-p})$ can be written as

$\displaystyle
dX_{t}^{p}=\Bigl{\\{}-\overline{\Lambda}\Bigl{(}Y_{t}^{p}-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}+l_{p}\Bigl{(}t,-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{p}\Bigr{)}\Bigr{\\}}dt$
$\displaystyle\qquad\qquad+\sigma_{p,0}(t,c_{t}^{0},c_{t}^{p})dW_{t}^{0}+\sigma_{p}(t,c_{t}^{0},c_{t}^{p})dW_{t}^{p,1},$
$\displaystyle
dY_{t}^{p}=-\partial_{x}\overline{f}_{p}\Bigl{(}t,X_{t}^{p},-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}],c_{t}^{0},c_{t}^{p}\Bigr{)}dt+Z_{t}^{p,0}dW_{t}^{0}+Z_{t}^{p}dW_{t}^{p,1},$

with $X_{0}^{p}=\xi^{p}$ and

$Y_{T}^{p}=\frac{\delta}{1-\delta}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}\bigl{[}\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})|\overline{{\cal
F}}_{T}^{0}\bigr{]}+\partial_{x}\overline{g}_{p}(X_{T}^{p},c_{T}^{0},c_{T}^{p})~{}.$

For each $p$, let us define the functionals $B_{p},F_{p}$ and $G_{p}$ for any
$y^{p},x^{p},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ with
$y:=(y^{p})_{p=1}^{m}$, $x:=(x^{p})_{p=1}^{m}$ and $c:=(c^{p})_{p=1}^{m}$ by

$\displaystyle
B_{p}(t,y,c^{0},c^{p}):=-\overline{\Lambda}\Bigl{(}y^{p}-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}+l_{p}\Bigl{(}t,-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}],c^{0},c^{p}\Bigr{)}$ $\displaystyle
F_{p}(t,x^{p},y,c^{0},c^{p}):=-\partial_{x}\overline{f}\Bigl{(}t,x^{p},-\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[y^{p}|\overline{{\cal
F}}_{t}^{0}],c^{0},c^{p}\Bigr{)},$ $\displaystyle
G_{p}(x,c^{0},c):=\frac{\delta}{1-\delta}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})|\overline{{\cal
F}}_{T}^{0}]+\partial_{x}\overline{g}_{p}(x^{p},c^{0},c^{p})~{},$

and set $B(t,y,c^{0},c):=(B_{p}(t,y,c^{0},c^{p}))_{p=1}^{m}$,
$F(t,x,y,c^{0},c):=(F_{p}(t,x^{p},y,c^{0},c^{p}))_{p=1}^{m}$ and
$G(x,c^{0},c):=(G_{p}(x,c^{0},c))_{p=1}^{m}$. With $\Delta y:=y-y^{\prime}$
and $\Delta x:=x-x^{\prime}$, we have from (MFG-C1)(iii),

$\displaystyle\mathbb{E}\Bigl{[}\langle
B(t,y,c^{0},c)-B(t,y^{\prime},c^{0},c),\Delta
y\rangle\Bigr{]}:=\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\langle
B_{p}(t,y,c^{0},c)-B_{p}(t,y^{\prime},c^{0},c),\Delta y^{p}\rangle\Bigr{]}$
$\displaystyle\leq-\sum_{p=1}^{m}\mathbb{E}[\langle\Delta
y^{p},\overline{\Lambda}\Delta
y^{p}\rangle]+\frac{1}{m}\mathbb{E}\Bigl{[}\bigl{\langle}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}],\overline{\Lambda}\sum_{p=1}^{m}\Delta
y^{p}\bigr{\rangle}\Bigr{]}-m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}$
$\displaystyle\leq-m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}.$ ( 6.7)

There exists a orthogonal matrix $P$ such that $P^{\top}\overline{\Lambda}P$
becomes diagonal. Then working on the new basis $\hat{y}^{p}=P^{\top}\Delta
y^{p}$, $1\leq p\leq m$, the last inequality of $(\ref{DelB-ineq})$ can be
checked component by component $1\leq i\leq n$ by the fact
$(\sum_{p=1}^{m}\hat{y}^{p}_{i})^{2}\leq
m\sum_{p=1}^{m}|\hat{y}^{p}_{i}|^{2}$. Second, from (MFG-A)(iv), (MFG-B) and
Cauchy-Schwarz inequality,

$\displaystyle\mathbb{E}\Bigl{[}\langle
F(t,x,y,c^{0},c)-F(t,x^{\prime},y^{\prime},c^{0},c),\Delta x\rangle\Bigr{]}$
$\displaystyle\hskip
85.35826pt\leq-\Bigl{(}\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}\Bigr{)}\mathbb{E}[|\Delta
x|^{2}]+m\gamma^{l}\mathbb{E}\Bigl{[}\Bigl{(}\frac{1}{m}\sum_{p=1}^{m}\mathbb{E}[\Delta
y_{t}^{p}|\overline{{\cal F}}_{t}^{0}]\Bigr{)}^{2}\Bigr{]}.$ ( 6.8)

Finally, from (MFG-A, C1)(iv), we immediately get

$\displaystyle\mathbb{E}\Bigl{[}\langle
G(x,c^{0},c)-G(x^{\prime},c^{0},c),\Delta
x\rangle\Bigr{]}\geq\gamma\mathbb{E}[|\Delta x|^{2}]~{}.$

Now we have established the monotone conditions corresponding to $(\ref{peng-
wu-condition})$ for the current model. We can now repeat the same procedures
in the proof of Theorem 4.2 and Proposition 4.1. ∎

Let us give the results for the securities of maturity $T$ with exogenously
specified payoff.

######  Assumption 6.4.

(MFG-C2)
(i) For every $1\leq p\leq m$, the functions $\sigma_{p,0}$ and $\sigma_{p}$
are independent of the argument $\varpi$.
(ii) $\Lambda_{p}$=$\Lambda$ and $n_{p}=1/m$ for every $p$.
(iii) For any $t\in[0,T]$, any random variables
$x^{p},x^{p\prime},c^{0},c^{p}\in\mathbb{L}^{2}({\cal F};\mathbb{R}^{n})$ and
any sub-$\sigma$-field ${\cal G}\subset{\cal F}$, the functions
$(l_{p})_{p=1}^{m}$ satisfy with some positive constant $\gamma^{l}>0$,

$\displaystyle\sum_{p=1}^{m}\mathbb{E}\Bigl{[}\bigl{\langle}l_{p}\bigl{(}t,\mathbb{E}[\overline{x}|{\cal
G}],c^{0},c^{0}\bigr{)}-l_{p}\bigl{(}t,\mathbb{E}[\overline{x}^{\prime}|{\cal
G}],c^{0},c^{p}\bigr{)},x^{p}-x^{p\prime}\bigr{\rangle}\Bigr{]}\geq
m\gamma^{l}\mathbb{E}\bigl{[}\mathbb{E}[\overline{x}-\overline{x}^{\prime}|{\cal
G}]^{2}\bigr{]},$

where $\overline{x}:=\frac{1}{m}\sum_{p=1}^{m}x^{p}$ and similarly for
$\overline{x}^{\prime}$.
(iv) $\gamma:=\gamma^{f}-\frac{L_{\varpi}^{2}}{4\gamma^{l}}$ is strictly
positive. Moreover, $\delta=0$ and the terminal function $g_{p}$ is given by

$\displaystyle g_{p}(x,c^{0})=\overline{g}_{p}(x,c^{0}):=-\langle
c^{0},x\rangle$ ( 6.9)

for every $1\leq p\leq m$.

######  Theorem 6.3.

Under Assumptions (MFG-A,B,C2), there exists a unique strong solution
$(X,Y,Z^{0},Z)\in\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{S}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d^{0}})^{m}\bigr{)}\times\mathbb{H}^{2}\bigl{(}\mathbb{F}^{1};(\mathbb{R}^{n\times
d})^{m}\bigr{)}$ to the FBSDE $(\ref{fbsde-multiple-p})$. Moreover, the same
form of the stability and $\mathbb{L}^{2}$ estimates given in Proposition 4.1
and Corollary 4.1 holds.

###### Proof.

Using the inequalities $(\ref{DelB-ineq})$ and $(\ref{DelF-ineq})$ with
$\sum_{p=1}^{m}\langle\Delta X_{T}^{p},\Delta Y_{T}^{p}\rangle=0$, we can
follow the same arguments in the proof of Theorem 4.3. ∎

###  6.4 Asymptotic market clearing for multi-population model

At the last part of this section, we investigate the asymptotic market
clearing in the presence of multiple populations. As in Section 5, we define
$(\varpi_{t})_{t\in[0,T]}$ using the solution to the system of the mean-field
FBSDEs:

$\displaystyle\varpi_{t}=\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]):=-\hat{\Xi}\sum_{p=1}^{m}\hat{\Lambda}_{p}\mathbb{E}\bigl{[}Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}\bigr{]}$

where $(Y_{t}^{p})_{p=1}^{m}$ is the solution of $(\ref{fbsde-multiple-p})$.
In order to test the accuracy of the above $(\varpi_{t})_{t\in[0,T]}$ as a
market clearing price, we solve the individual agent problem $(\ref{control-
pi})$ with this $\varpi$ as an input. The corresponding individual problem
$(\ref{control-pi})$ for the agent $(p,i)$ is given by the unique strong
solution $(X^{p,i},Y^{p,i},Z^{p,i,0},Z^{p,i})$ of $(\ref{fbsde-pi})$. The
optimal strategy for the agent $(p,i)$ is then given by

$\displaystyle\widehat{\alpha}_{\rm{mf}}^{p,i}(t):=-\overline{\Lambda}_{p}\Bigl{(}Y_{t}^{p,i}-\hat{\Xi}\sum_{q=1}^{m}\hat{\Lambda}_{q}\mathbb{E}\bigl{[}Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}\bigr{]}\Bigr{)}~{},\forall t\in[0,T]~{}.$

######  Theorem 6.4.

If the conditions for Theorem 6.1, Theorem 6.2 or Theorem 6.3 are satisfied
then we have

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)\Bigr{|}^{2}dt=0~{},$

where $N:=\sum_{p=1}^{m}N_{p}$ and the limit is taken while keeping
$(n_{p}:=N_{p}/N)_{1\leq p\leq m}$ constant. Moreover if there exists some
constant $\Gamma$ such that
$\sup_{t\in[0,T]}\mathbb{E}\bigl{[}|Y_{t}|^{q}\bigr{]}^{\frac{1}{q}}\leq\Gamma<\infty$
for some $q>4$, then there exists some constant $C$ independent of $N$ such
that

$\displaystyle\mathbb{E}\int_{0}^{T}\Bigl{|}\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)\Bigr{|}^{2}dt\leq
C\Gamma^{2}\epsilon_{N},$

where
$\epsilon_{N}:=N^{-2/\max(n,4)}\bigl{(}1+\log(N)\mathbb{1}_{\\{n=4\\}}\bigr{)}$.

###### Proof.

By the definition of $\widehat{\alpha}^{p,i}_{\rm{mf}}$, we have

$\displaystyle\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)$
$\displaystyle=$
$\displaystyle-\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\overline{\Lambda}_{p}\Bigl{(}Y_{t}^{p,i}-\hat{\Xi}\sum_{q=1}^{m}\hat{\Lambda}_{q}\mathbb{E}[Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}$ ( 6.10) $\displaystyle=$
$\displaystyle-\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}-\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$

On the other hand, we have for each $1\leq p\leq m$, $1\leq i\leq N_{p}$,

$\displaystyle dX_{t}^{p,i}$ $\displaystyle=$
$\displaystyle\Bigl{(}\widehat{\alpha}_{p}\bigl{(}Y_{t}^{p,i},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}])\bigr{)}+l_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}\Bigr{)}dt$
$\displaystyle+\sigma_{p,0}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dW_{t}^{0}+\sigma_{p}\bigl{(}t,\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dW_{t}^{p,i},$ $\displaystyle
dY_{t}^{p,i}$ $\displaystyle=$
$\displaystyle-\partial_{x}\overline{f}_{p}\bigl{(}t,X_{t}^{p,i},\varpi(\mathbb{E}[Y_{t}|\overline{{\cal
F}}_{t}^{0}]),c_{t}^{0},c_{t}^{p,i}\bigr{)}dt+Z_{t}^{p,i,0}dW_{t}^{0}+Z_{t}^{p,i}dW_{t}^{p,i}~{},$

with $X_{0}^{p,i}=\xi^{p,i}$,

$\displaystyle Y_{T}^{p,i}=-\delta\varpi(\mathbb{E}[Y_{T}|\overline{{\cal
F}}_{T}^{0}])+\partial_{x}\overline{g}_{p}(X_{T}^{p,i},c_{T}^{0},c_{T}^{p,i})~{}.$

By the unique strong solvability, Yamada-Watanabe theorem implies that there
exists some function $\Phi_{p}$ for each $1\leq p\leq m$ such that for every
$1\leq i\leq N_{p}$,

$\displaystyle(Y_{t}^{p,i})_{t\in[0,T]}=\Phi_{p}\Bigl{(}c^{0},(W_{t}^{0})_{t\in[0,T]},(\mathbb{E}[Y_{t}^{q}|\overline{{\cal
F}}_{t}^{0}]_{t\in[0,T]})_{1\leq q\leq
m},\xi^{p,i},(c_{t}^{p,i})_{t\in[0,T]},(W^{p,i}_{t})_{t\in[0,T]}\Bigr{)}.$

Hence $(Y_{t}^{p,i})_{t\in[0,T],1\leq i\leq N_{p}}$ are independently and
identically distributed conditionally on $\overline{{\cal F}}^{0}$. In
particular, we have $\mathbb{E}[Y_{t}^{p,i}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal F}}_{t}^{0}]$.

We now compare
$(X^{p,1}_{t},Y^{p,1}_{t},Z^{p,1,0}_{t},Z^{p,1}_{t})_{t\in[0,T]}$ with
$(X^{p}_{t},Y^{p}_{t},Z^{p,0}_{t},Z^{p}_{t})_{t\in[0,T]}$ by treating
$\varpi(\mathbb{E}[Y_{t}|\overline{{\cal F}}_{t}^{0}])$ as external inputs.
Note that the terminal condition of the latter satisfies the relation
$(\ref{terminal-pre})$. Then the standard stability result of the Lipschitz
FBSDEs implies $(Y^{p,1}_{t})_{t\in[0,T]}=(Y^{p}_{t})_{t\in[0,T]}$ in
$\mathbb{S}^{2}(\mathbb{F}^{p,1};\mathbb{R}^{n})$. As a result we have
obtained $\mathbb{E}[Y_{t}^{p}|\overline{{\cal
F}}_{t}^{0}]=\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal F}}_{t}^{0}]$. Using the
expression $(\ref{clearing-mp})$, we obtain

$\displaystyle\frac{1}{N}\sum_{p=1}^{m}\sum_{i=1}^{N_{p}}\widehat{\alpha}^{p,i}_{\rm{mf}}(t)$
$\displaystyle=$
$\displaystyle-\sum_{p=1}^{m}\hat{\Lambda}_{p}\Bigl{(}\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}Y_{t}^{p,i}-\mathbb{E}[Y_{t}^{p,1}|\overline{{\cal
F}}_{t}^{0}]\Bigr{)}~{}.$

We can now repeat the last part of the proof for Theorem 5.1. ∎

##  7 Concluding Remarks and Further Extensions

In this work, we have studied endogenous formation of market clearing price
using a stylized model of a security exchange. We have derived a special type
of FBSDE of McKean-Vlasov type with common noise whose solution provides a
good approximate of the equilibrium price. In addition to the existence of
strong unique solution to the FBSDE, we have proved that the modeled price
asymptotically clear the market in the large $N$-limit. We also gave the order
of convergence $\epsilon_{N}$ when the solution of the FBSDE possesses higher
order of integrability. In the following, let us list up of a further
extension of our technique and some interesting topics for future projects:

$\bullet~{}$Dependence on the conditional law of the state: For applications
to energy and commodity markets, or economic models with producers and
consumers, one may want to study the cost functions
$(\overline{f},\overline{g})$ depending on the empirical distribution of the
sate $X$ of the agents such as
${\overline{f}}\Bigl{(}t,X_{t}^{i},\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{t}^{j}},\varpi_{t},c^{0}_{t},c^{i}_{t}\Bigr{)}$.
Under the setup with conditional independence, the cost function for the limit
problem is naturally given by ${\overline{f}}\Bigl{(}t,X_{t},{\cal
L}(X_{t}|\overline{{\cal F}}_{t}^{0}),\varpi_{t},c^{0}_{t},c_{t}\Bigr{)}$.
Even in this case, the resultant FBSDE $(\ref{fbsde-single-p})$ is solvable,
at least for small $T$, if
$(\partial_{x}\overline{f},\partial_{x}\overline{g})$ are Lipschitz continuous
in the measure argument with respect to $W_{2}$-distance. Under the stronger
assumption guaranteeing the monotone conditions $(\ref{peng-wu-condition})$,
one can even achieve the existence of unique solution in general $T$. As long
as the source of common noise is solely from the filtration
$\overline{\mathbb{F}}^{0}$ generated by $\boldsymbol{W}^{0}$, we can avoid
subtleties regarding the admissibility (so-called $H$-hypothesis). See Remark
2.10 in [8] as a useful summary for this issue.

$\bullet~{}$Explicit solution: If we chose $\overline{f},\overline{g}$ as
quadratic functions and $l,\sigma_{0},\sigma$ as affine functions, we obtain a
linear-quadratic mean field game with common noise. In this case, an explicit
solution may be available where the coefficients functions are given as the
solutions to differential equations of Riccati type.

$\bullet~{}$Property of market price process: It seems interesting to study
the properties of the market clearing price theoretically and numerically. For
example, if $n=d^{0}$ the equivalent martingale measure (EMM) can be uniquely
determined. Based on the payoff distribution $c^{0}$ and the cost functions of
the agents $(\overline{f},\overline{g})$, one may study how the market price
process under the EMM behaves, for example, the relation between the skew of
its implied volatility and the risk-averseness of the agents.

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