# Equilibrium in Functional Stochastic Games with Mean-Field Interaction

Eduardo Abi Jaber The first author is grateful for the financial support from
the Chaires FiME-FDD, Financial Risks, Deep Finance & Statistics and Machine
Learning and systematic methods in finance at Ecole Polytechnique. Ecole
Polytechnique, CMAP  Eyal Neuman  Department of Mathematics, Imperial College
London  Moritz Voß  Department of Mathematics, University of California Los
Angeles

###### Abstract

We consider a general class of finite-player stochastic games with mean-field
interaction, in which the linear-quadratic cost functional includes linear
operators acting on controls in $L^{2}$. We propose a novel approach for
deriving the Nash equilibrium of the game explicitly in terms of operator
resolvents, by reducing the associated first order conditions to a system of
stochastic Fredholm equations of the second kind and deriving their closed
form solution. Furthermore, by proving stability results for the system of
stochastic Fredholm equations we derive the convergence of the equilibrium of
the $N$-player game to the corresponding mean-field equilibrium. As a by-
product we also derive an $\varepsilon$-Nash equilibrium for the mean-field
game, which is valuable in this setting as we show that the conditions for
existence of an equilibrium in the mean-field limit are less restrictive than
in the finite-player game. Finally we apply our general framework to solve
various examples, such as stochastic Volterra linear-quadratic games, models
of systemic risk and advertising with delay, and optimal liquidation games
with transient price impact.

Mathematics Subject Classification (2010):

49N80, 49N90, 93E20, 91G80

JEL Classification:

C02, C61, C73, G11, G32

Keywords:

mean-field games, Nash equilibrium, Volterra stochastic control, optimal
portfolio liquidation, systemic risk, price impact

###### Contents

  1. 1 Introduction
  2. 2 The Finite-Player Game
    1. 2.1 Function spaces, integral operators
    2. 2.2 Definition of the $N$-player game
    3. 2.3 Main results for the $N$-player game
  3. 3 Important Examples
    1. 3.1 Stochastic Volterra Linear-Quadratic games
    2. 3.2 Propagator models: optimal liquidation with transient impact and signals
    3. 3.3 Delay in the control: an inter-bank lending and borrowing model
    4. 3.4 Delay in the state: an advertising model with mean-field effect
  4. 4 The Mean-Field Game
    1. 4.1 The Infinite-Player Game Formulation
    2. 4.2 Convergence and Approximation Results
  5. 5 Stochastic Fredholm equations
  6. 6 Proofs of Theorems 2.7 and 2.8
  7. 7 Proofs of the results from Section 4

##  1 Introduction

Large population stochastic games and their mean-field limits have attracted a
considerable attention in the past decades since the pioneering work of Lasry
and Lions [29] and Huang, Malhamé, and Caines [24], which was further
developed by Bardi and Priuli [5], Gomes et al. [18], Carmona and Delarue [8],
Lacker [25], Feleqi [15], Fischer [16], Cardaliaguet et al. [7], among others.
The tractability and convergence properties of this versatile class of
interacting particle systems has applications in various research areas such
as mathematical finance, economics, population modeling, and marketing (see,
e.g., Chapter 1 of [8]).

One of the main challenges in the area of finite population stochastic games
is to derive explicitly the Nash equilibrium of the system. Motivating
examples from mathematical finance include price impact games with competition
for liquidity between agents [36, 12, 14, 35, 32], systemic risk games
introduced by Carmona et al. [9, 10], Fouque and Zhang [17], as well as
optimal investment problems studied in Lacker and Zariphopoulou [28], Lacker
and Soret [27]. In various important extensions of the aforementioned models
it turns out that the state variables of the players and their objective
functionals naturally depend on the entire trajectory of the controls. These
generalizations give rise to infinite dimensional dynamic stochastic games. In
such a setting, deriving a Nash equilibrium of the system is in general
considered to be intractable, and solutions to such games only appear in very
particular examples; see Section 3 for a comprehensive survey of such
problems.

In this work we develop a novel approach for solving a general class of
infinite-dimensional stochastic games with mean-field interaction between the
players. One of the key ideas of our method hinges upon writing the infinite-
dimensional $N$-player game in a static form. Specifically, we consider
$N$-player stochastic games in which each agent $i$ has an objective
functional of the form

$\displaystyle J^{i}(u^{i}):=$
$\displaystyle\,{\mathbb{E}}\left[-\langle\bar{u},\bm{A}_{1}\bar{u}\rangle_{L^{2}}-\langle
u^{i},\bm{A}_{2}u^{i}\rangle_{L^{2}}-\langle
u^{i},(\bm{A}_{3}+\bm{A}^{*}_{3})\bar{u}\rangle_{L^{2}}\right.$ (1.1)
$\displaystyle\hskip 14.0pt\left.+\langle b^{i},u^{i}\rangle_{L^{2}}+\langle
b^{0},\bar{u}\rangle_{L^{2}}+c^{i}\right],$

where $u^{i}$ represents the agent’s control and
$\bar{u}=N^{-1}\sum_{i=1}^{N}u^{i}$ captures a mean-field interaction between
all agents. Here, the symbols $\bm{A}_{i}$, $i=1,2,3$ denote non-anticipative
linear operators on $L^{2}([0,T],{\mathbb{R}})$;
$(b^{i}_{s})_{s\in[0,T]},(b^{0}_{s})_{s\in[0,T]}$ are progressively measurable
stochastic processes; and $c^{i}$ is a random variable. The inner product is
defined in the usual sense as $\langle
f,g\rangle_{L^{2}}:=\int_{0}^{T}f(s)g(s)ds$ for $f,g\in
L^{2}([0,T],{\mathbb{R}})$.

The main goal of this work is to derive explicitly the Nash equilibrium of
finite-player games with reward functionals (1.1). In order to achieve this,
we develop a new technique to solve simultaneously for all agents
$i\in\\{1,\ldots,N\\}$ their individual path-dependent stochastic control
problem

$J^{i}(u^{i})\rightarrow\max_{u^{i}}$ (1.2)

over progressively measurable and square-integrable $u^{i}$’s. Moreover, we
also apply our method to solve the corresponding simpler limiting mean-field
game when the number of agents $N$ tends to infinity and provide a convergence
result of the finite-player equilibrium strategies towards their mean-field
limits.

Our approach uses variational calculus in order to establish sufficient first
order conditions for a Nash equilibrium which take the form of stochastic
Fredholm equations with both forward and backward components. The derivation
of the equilibrium then relies on explicitly solving this system of equations
for which we develop a new method of solution. The approach is versatile
enough to allow for non-Markovian and non-semimartingale settings with no
additional effort, in the sense that the controls are allowed to depend on the
entire trajectories of the inputs $(b_{i})_{i=0}^{N}$ which are only assumed
to be progressively measurable; that is, neither necessarily semimartingales
nor independent. Our framework unifies and extends major examples of
stochastic dynamical games that have appeared in the literature beyond the
Markovian and semimartingale case and were often considered as intractable
(see Section 3). In the following, we give a concrete summary of the
contribution of the methods developed in this paper.

Our contributions. In Theorems 2.7 and 2.8, we use calculus of variations to
compute the first order conditions of the optimization problems in (1.2),
which leads to the following stochastic Fredholm equation for $\bar{u}$ with
both forward and backward components:

$\displaystyle\bar{u}_{t}=f_{t}-\int_{0}^{t}K(t,r)\bar{u}_{r}dr-\int_{t}^{T}L(r,t)\mathbb{E}_{t}\bar{u}_{r}dr,\quad
t\in[0,T],$ (1.3)

where $f$ is progressively measurable and $K,L$ are deterministic kernels
determined by the inputs of the model
$(\bm{A}_{1},\bm{A}_{2},\bm{A}_{3},b^{i},b^{0},c^{i})$. One of our main
innovations is that we derive an explicit solution to (1.3) using a novel
approach (see Proposition 5.1). Then, we utilize the solution to (1.3) to
disentangle and solve the optimisation problem of each player and hence to
derive explicitly the Nash equilibrium. In addition to solving (1.3), we
derive a stability result for (1.3) (see Proposition 5.3). This stability
result is the crucial ingredient for deriving the convergence of the finite-
player game equilibrium toward its mean-field game equilibrium limit in
Theorem 4.14.

The framework developed in this paper is unique in that the solvability,
stability, and consistency of finite-player games with mean-field interaction
boil down to the study of the stochastic Fredholm equation (1.3). To the best
of our knowledge, our approach gives the first canonical method for deriving
explicitly the unique Nash equilibrium to this general class of stochastic
games. The closest result to Theorems 2.14 and 2.8 appeared in Huang et al.
[23], where the authors derived a first order condition for a special case of
(1.1) in terms of a system of stochastic integral equations but they did not
obtain a solution to the system. We also refer to Bensoussan et al. [6] where
first order conditions for linear-quadratic stochastic games with delays in
the state and the control were derived. In Section 3.4 we show that we can
compute explicit solutions to this class of games as a corollary of our main
results.

In Section 3, we highlight the versatility of the objective functional in
(1.1) by showing that it nests several important, challenging and diverse
examples of dynamical stochastic games beyond the Markovian and semimartingale
case. In particular, we introduce in Section 3.1 a general class of Linear-
Quadratic Stochastic Volterra games whose objective functional is shown to be
equivalent to (1.1). Then, we show how such framework accommodates, extends
and solves three major examples that appeared in the literature: (i) inter-
bank lending and borrowing models with delay in the control which were studied
in Carmona et al. [10], Fouque and Zhang [17] (see Section 3.3); (ii)
advertising models à la Nerlove and Arrow [33], Gozzi and Marinelli [19] with
mean-field effect and delay in the state (Section 3.4); (iii) multi-player
price impact games with a general propagator in the spirit of Abi Jaber and
Neuman [2], Neuman and Voß [35] (Section 3.2).

Convergence results of the finite-player Nash equilibrium to the corresponding
mean-field equilibrium in Markovian settings have attracted considerable
attention recently. Laurière and Tangpi [30] proved such convergence results
for open-loop equilibria of games with idiosyncratic noise for each of the
players. Neuman and Voß [35] studied the corresponding problem for portfolio
liquidation games with common noise. Lacker and Le Flem [26] and Djete [11]
proved the convergence of closed-loop solutions (under the a priori assumption
that they exist) to the mean-field solution in the case where each player is
influenced by idiosyncratic and common noise. In these papers (except for
[35]) the presence of idiosyncratic noise is crucial to establish the
convergence. In order to illustrate the strength of our methods we provide in
Theorems 4.6 and 4.11 the equilibrium of the infinite-player mean-field game
in our non-Markovian setting and show that the conditions to obtain the latter
are less restrictive than in the corresponding finite $N$-player game in
Theorems 2.7 and 2.8. Moreover, in Theorem 4.14 we derive the aforementioned
convergence which does not require the presence of idiosyncratic noise.
Indeed, we show that convergence is obtained as a corollary of a stability
result for the associated stochastic Fredholm equation (see Proposition 5.3).
For the sake of completeness, we also provide in Theorem 4.16 the associated
$\varepsilon$-Nash equilibrium result to our class of infinite-dimensional
games. In other words, we show that the mean-field equilibrium strategy gives
close to optimal rewards for each player in the $N$-player game when $N$ is
sufficiently large.

Finally, let us illustrate in the following toy example that our choice of
objective functionals of type (1.1), which we refer to as static games, also
include dynamical games, as the objectives explicitly contain state variables.
We refer to Section 3 for the generalisation of this equivalence to functional
games as in (1.1).

Motivating example. Let us consider a simple dynamical Markovian $N$-player
game with controlled state variables given by

$\displaystyle dX^{i}_{t}=\left(aX^{i}_{t}+u^{i}_{t}\right)dt+dW^{i}_{t},\quad
X^{i}_{0}=0,\quad i=1,\ldots,N,$ (1.4)

where $u^{i}$ is the control of the $i$-th player, $a\in{\mathbb{R}}$ and
$(W^{1},\ldots,W^{N})$ is an $N$-dimensional Brownian motion. Each player
$i\in\\{1,\ldots,N\\}$ wishes to maximize following objective functional with
mean-field interaction in the controls

$J^{i}(u^{i})=\mathbb{E}\left[\int_{0}^{T}\left(-(u^{i}_{t})^{2}+\frac{1}{N}\sum_{j=1}^{N}u^{j}_{t}\right)dt+(X^{i}_{T})^{2}\right].$
(1.5)

We will show how this dynamical game can be transformed into a static game
without state variables as in (1.1). Indeed, note that we can rewrite
$(X^{i}_{T})^{2}$ by first using the variation of constants formula to solve
for $X^{i}$ in (1.4), i.e.,

$\displaystyle
X^{i}_{t}=\int_{0}^{t}e^{a(t-s)}u_{s}^{i}ds+\int_{0}^{t}e^{a(t-s)}dW_{s}^{i},$

and then apply Fubini’s theorem to obtain

$\displaystyle(X^{i}_{T})^{2}$
$\displaystyle=2\int_{0}^{T}u^{i}_{s}\left(\int_{0}^{s}e^{a(2T-s-r)}u^{i}_{r}dr\right)ds$
(1.6)
$\displaystyle\quad\quad+2\int_{0}^{T}\left(\int_{0}^{T}e^{a(2T-s-r)}dW^{i}_{r}\right)u^{i}_{s}ds+\left(\int_{0}^{T}e^{a(T-s)}dW_{s}^{i}\right)^{2}.$
(1.7)

This shows that the objective functional (1.5) can be written in the form

$\displaystyle J^{i}(u^{i})=\mathbb{{\mathbb{E}}}\left[-\langle
u^{i},\bm{A}_{2}u^{i}\rangle_{L^{2}}+\langle
b^{i},u^{i}\rangle_{L^{2}}+\langle b^{0},\bar{u}\rangle_{L^{2}}+c^{i}\right],$
(1.8)

where

$(\bm{A}_{2}u)(s)=u_{s}-{2}\int_{0}^{s}e^{a(2T-s-r)}u_{r}dr,\quad s\leq T,$

and

$b^{i}_{s}=2\int_{0}^{T}e^{a(2T-s-r)}dW^{i}_{r},\quad b^{0}_{s}=1,\quad
c^{i}=\left(\int_{0}^{T}e^{a(T-s)}dW_{s}^{i}\right)^{2},$

and $\bar{u}=\frac{1}{N}\sum_{j=1}^{N}u^{j}$. Hence, we observe that the
objective functional in (1.8) corresponds to a static game without state
variables. Moreover, this representation very naturally lends itself to
directly employing first order variational calculus. This motivates us to
study more general static objective criterions in the spirit of (1.8).

Structure of the paper. In Section 2 we present the class of finite-player
stochastic games and state our main result on the explicit derivation of the
Nash equilibrium. In Section 3 we describe important multi-agent models that
are included in our framework. Section 4 is dedicated to the derivation of the
Nash equilibrium in the corresponding infinite-player mean-field game, as well
as related $\varepsilon$-Nash equilibrium and convergence results. In Section
5 we derive some essential results on the solution and stability of stochastic
Fredholm equations. Finally, Sections 6–7 are dedicated to the proofs of our
main results.

##  2 The Finite-Player Game

In this section we derive explicitly the Nash equilibrium of (1.1). Before
stating this result, we introduce some essential definitions of function
spaces and integral operators.

###  2.1 Function spaces, integral operators

We denote by $\langle\cdot,\cdot\rangle_{L^{2}}$ the inner product on
$L^{2}([0,T],{\mathbb{R}})$, that is

$\langle f,g\rangle_{L^{2}}=\int_{0}^{T}f(s)g(s)ds,\quad f,g\in
L^{2}\left([0,T],\mathbb{R}\right),$ (2.1)

where $\|\cdot\|_{L^{2}}$ the induced norm. We define
$L^{2}\left([0,T]^{2},\mathbb{R}\right)$ to be the space of measurable kernels
$G:[0,T]^{2}\to{\mathbb{R}}$ such that

$\displaystyle\int_{0}^{T}\int_{0}^{T}|G(t,s)|^{2}dtds<\infty.$

For any kernel $G\in L^{2}\left([0,T]^{2},\mathbb{R}\right)$, we denote by
$\bm{G}$ the integral operator induced by the kernel $G$ that is

$\displaystyle({\bm{G}}f)(s)=\int_{0}^{T}G(s,u)f(u)du,\quad f\in
L^{2}\left([0,T],\mathbb{R}\right).$

$\bm{G}$ is a linear bounded operator from
$L^{2}\left([0,T],\mathbb{R}\right)$ into itself. We denote by $G^{*}$ the
adjoint kernel of $G$ for $\langle\cdot,\cdot\rangle_{L^{2}}$, that is

$\displaystyle G^{*}(s,u)$ $\displaystyle=\;G(u,s),\quad(s,u)\in[0,T]^{2},$

and by $\bm{G}^{*}$ the corresponding adjoint integral operator.

###  2.2 Definition of the $N$-player game

We present the class of functional stochastic games which are studied in this
paper. Let $T>0$ denote a finite deterministic time horizon and let $N\geq 2$
be an integer. We fix a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F}:=(\mathcal{F}_{t})_{0\leq t\leq
T},\mathbb{P})$ satisfying the usual conditions of right continuity and
completeness and use the notation
$\mathbb{E}_{t}=\mathbb{E}[\cdot|\mathcal{F}_{t}]$ to represent the
conditional expectation with respect to $\mathcal{F}_{t}$. Let
$b^{i}=(b^{i}_{t})_{0\leq t\leq T}$, $i=0,\ldots,N$ be progressively
measurable processes satisfying

$\int_{0}^{T}\mathbb{E}\left[(b_{s}^{i})^{2}\right]ds<\infty,\quad\mathbb{E}[c^{i}]<\infty,\quad
i=0,\ldots,N.$ (2.2)

We say that a measurable Volterra kernel $G:[0,T]^{2}\to{\mathbb{R}}$, i.e.
such that $G(t,s)=0$ whenever $s\geq t$, is nonnegative definite if for every
$f\in L^{2}\left([0,T],\mathbb{R}\right)$ we have

$\int_{0}^{T}\int_{0}^{T}\big{(}G(t,s)+G(s,t)\big{)}f(s)f(t)dsdt\geq 0.$ (2.3)

Next we define the class of Volterra kernels which will be used in our
setting.

######  Definition 2.1 (Class of admissible kernels $\mathcal{G}$).

We say that a nonnegative definite Volterra kernel
$G:[0,T]^{2}\mapsto{\mathbb{R}}_{+}$ is in the class of kernels $\mathcal{G}$
if it satisfies the following conditions:

$\displaystyle\sup_{t\leq T}\int_{0}^{T}|G(t,s)|^{2}ds+\sup_{s\leq
T}\int_{0}^{T}|G(t,s)|^{2}dt<\infty.$ (2.4)

######  Definition 2.2 (Admissible Volterra operator).

We say that an integral operator $\bm{G}$ in an admissible Volterra operator
if it is induced by the kernel $G\in\mathcal{G}$.

######  Remark 2.3.

Observe that if $G\in\mathcal{G}$, property (2.3) implies for any $f\in
L^{2}\left([0,T],\mathbb{R}\right)$,

$\langle f,\bm{G}f\rangle_{L^{2}}=\frac{1}{2}\langle
f,(\bm{G}+\bm{G}^{*})f\rangle_{L^{2}}\geq 0.$ (2.5)

We consider $N$ agents $i\in\\{1,\ldots,N\\}$ who select their controls
$u^{i,N}$ from the admissible set

$\mathcal{U}:=\left\\{u\,:\,\mathbb{F}\textrm{-progressively measurable s.t.
}\int_{0}^{T}\mathbb{E}\left[u_{s}^{2}\right]ds<\infty\right\\}$ (2.6)

and let

$u^{-i,N}:=(u^{1,N},\ldots,u^{i-1,N},u^{i+1,N},\ldots,u^{N,N}),\quad\bar{u}^{N}:=\frac{1}{N}\sum_{j=1}^{N}u^{j,N}.$
(2.7)

In the following, we consider operators
$(\bm{{A}}_{1},\bm{{A}}_{2},\bm{{A}}_{3})$ which satisfy the following
assumptions.

######  Assumption 2.4.

We assume that $\bm{A}_{1}$ and $\bm{A}_{3}$ are admissible Volterra
operators, and $\bm{A}_{2}$ is given by

$\bm{A}_{2}:=\lambda{\rm id}+\bm{\hat{A}}_{2}$ (2.8)

where $\bm{\hat{A}}_{2}$ is an admissible Volterra operator, $\lambda>0$ is a
constant and ${\rm id}$ is the identity operator, i.e.

$({\rm id}f)(t)=f(t),\quad\textrm{for all }t\in[0,T],\,f\in
L^{2}\left([0,T],\mathbb{R}\right).$

Each agent $i\in\\{1,\ldots,N\\}$ has the following individual performance
functional

$\displaystyle J^{i,N}(u^{i,N};u^{-i,N}):=$
$\displaystyle\,{\mathbb{E}}\left[-\langle\bar{u}^{N},\bm{A}_{1}\bar{u}^{N}\rangle_{L^{2}}-\langle
u^{i,N},\bm{A}_{2}u^{i,N}\rangle_{L^{2}}-\langle
u^{i,N},(\bm{A}_{3}+\bm{A}^{*}_{3})\bar{u}^{N}\rangle_{L^{2}}\right.$ (2.9)
$\displaystyle\hskip 14.0pt\left.+\langle b^{i},u^{i,N}\rangle_{L^{2}}+\langle
b^{0},\bar{u}^{N}\rangle_{L^{2}}+c^{i}\right],$

where $\bar{u}^{N}\in\mathcal{U}$ describes the mean-field interaction between
the agents.

######  Remark 2.5.

Note that in the objective functional in (2.9) the processes
$(b^{i}_{t})_{0\leq t\leq T}$, $i=0,\ldots,N$, are fairly general. They can be
thought of as incorporating simultaneously different sources of noise: i.e., a
common noise affecting all $N$ players as well as player $i$’s independent
individual source of noise; and also other random or deterministic factors
idiosyncratic to player $i$. We refer to Section 3 below for specific
examples.

Our main goal in this section is to solve simultaneously for each agent
$i\in\\{1,\ldots,N\\}$ their individual optimal stochastic control problem

$J^{i,N}(u^{i,N};u^{-i,N})\rightarrow\max_{u^{i}\in\mathcal{U}}.$ (2.10)

This solution will establish a Nash equilibrium in the following usual sense.

######  Definition 2.6.

A set of strategies
$\hat{u}^{N}=(\hat{u}^{1,N},\ldots,\hat{u}^{N,N})\in\mathcal{U}^{N}$ is called
a Nash equilibrium if for all $i\in\\{1,\ldots,N\\}$ and for all admissible
strategies $v\in\mathcal{U}$ it holds that

$J^{i,N}(\hat{u}^{i,N};\hat{u}^{-i,N})\geq J^{i,N}(v;\hat{u}^{-i,N}).$

###  2.3 Main results for the $N$-player game

For convenience, we introduce the following operators,

$\displaystyle\bm{G}$
$\displaystyle:=\frac{\bm{A}_{1}}{N^{2}}+2\frac{\bm{A}_{3}}{N}+\bm{\hat{A}}_{2},$
(2.11) $\displaystyle\bm{H}$ $\displaystyle:=\frac{\bm{A}_{1}}{N}+\bm{A}_{3}.$

Now we are ready to state our main results for the finite-players game. The
first result derives an explicit closed form solution to the average control
$\bar{u}^{N}$ in the Nash equilibrium of the game. To formulate the result, we
define for any $G\in\mathcal{G}$ the kernel

$G_{t}(s,r)=G(s,r)1_{\\{r\geq t\\}}$

where $1_{\\{\cdot\\}}$ is the indicator set function. We denote by
${\bm{G}}_{t}$ the associated Volterra operator.

We also set

$\bar{b}_{t}:=\frac{1}{N}\sum_{i=1}^{N}b^{i}_{t},\qquad 0\leq t\leq T.$ (2.12)

######  Theorem 2.7.

Under Assumption 2.4 and (2.2) any Nash equilibrium
$\hat{u}^{N}\in\mathcal{U}^{N}$ to the game (2.7) is such that
$\bar{u}^{N}:=\frac{1}{N}\sum_{j=1}^{N}\hat{u}^{j,N}$ is given by

$\displaystyle\bar{u}^{N}_{t}=\left(({\rm
id}-\overline{\bm{B}})^{-1}\overline{a}\right)(t),\qquad 0\leq t\leq T,$
(2.13)

where

$\displaystyle\overline{a}_{t}$
$\displaystyle:=\frac{1}{2\lambda}\left(\bar{b}_{t}+\frac{1}{N}b^{0}_{t}-\left\langle
1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,t),\overline{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}\left[\bar{b}_{\cdot}+\frac{1}{N}b^{0}_{\cdot}\right]\right\rangle_{L^{2}}\right),$
(2.14) $\displaystyle\overline{B}(t,s)$
$\displaystyle:=1_{\\{s<t\\}}\frac{1}{2\lambda}\bigg{(}\left\langle
1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,t),\overline{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,s)\right\rangle_{L^{2}}-\overline{K}(t,s)\bigg{)},$
$\displaystyle\overline{K}(t,s)$ $\displaystyle:=\frac{N-1}{N}H(t,s)+G(t,s),$

and

$\overline{\bm{D}}_{t}:=2\lambda{\rm
id}+\frac{N-1}{N}(\bm{H}_{t}+{\bm{H}^{*}_{t}})+\bm{G}_{t}+\bm{G}^{*}_{t},$

for all $t\in[0,T]$.

Using the result of Theorem 2.7 we derive each player’s Nash equilibrium
strategy $\hat{u}^{i}$.

######  Theorem 2.8.

Assume that (2.2) and Assumption 2.4 are satisfied and let $\bar{u}$ be as in
(2.13). The unique Nash equilibrium of the game (2.7) is given by

$\displaystyle\hat{u}^{i,N}_{t}=\left(({\rm
id}-\bm{B})^{-1}a^{i}\right)(t),\qquad 0\leq t\leq T,$ (2.15)

where

$\displaystyle a^{i}_{t}$
$\displaystyle:=\frac{1}{2\lambda}\bigg{(}{\mathbb{E}}_{t}\left[b^{i}_{t}+\frac{1}{N}b^{0}_{t}-((\bm{H}+\bm{H}^{*})\bar{u}^{N})(t)\right]$
(2.16)
$\displaystyle\qquad\qquad-\left\langle{1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,t)},\widetilde{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}\left[b^{i}_{\cdot}+\frac{1}{N}b^{0}_{\cdot}-((\bm{H}+\bm{H}^{*})\bar{u}^{N})(\cdot)\right]\right\rangle_{L^{2}}\bigg{)},$
$\displaystyle B(t,s)$
$\displaystyle:=1_{\\{s<t\\}}\frac{1}{2\lambda}\bigg{(}\left\langle
1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,t),\widetilde{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,s)\right\rangle_{L^{2}}-\hat{K}(t,s)\bigg{)},$
$\displaystyle\hat{K}(t,s)$ $\displaystyle=G(t,s)-\frac{1}{N}H(t,s)$

and

$\widetilde{\bm{D}}_{t}:=2\lambda{\rm
id}-\frac{1}{N}(\bm{H}_{t}+\bm{H}_{t}^{*})+\bm{G}_{t}+\bm{G}^{*}_{t},$

for all $t\in[0,T]$.

The proofs of Theorems 2.7 and 2.8 are given in Section 6.

######  Remark 2.9.

Our assumptions in Theorems 2.7 and 2.8 ensure that the operators
$\\{\overline{\bm{D}}_{t}\\}_{t\in[0,T]}$,
$\\{\widetilde{\bm{D}}_{t}\\}_{t\in[0,T]}$, $({\rm id}-\overline{\bm{B}})$ and
$({\rm id}-{\bm{B}})$ are invertible so that $(\bar{u},u)$ in (2.13) and
(2.15) are well defined.

######  Remark 2.10.

The results of Theorems 2.7 and 2.8 significantly improve various results for
linear-quadratic (LQ) stochatic mean-field games that have appeared recently
in the literature. In [22, 23] a class of LQ stochastic game with state
variables that satisfy Volterra integral equations was studied. The solution
to the game is described in terms of a system of stochastic Volterra
equations. Our results in this section provide explicit solutions to this
class of problems. See Section 3.1 for additional details. In [20] a class of
stochastic control problems with delay in the controls was studied. The
solution to the control problem was characterised in terms of a system of
integral equations. Our framework provides a solution for a single agent and
multiplayer games with delay in the controls, in the states and in the mean-
field interaction. Our framework also allows common and non-semimartingale
noise which are not straightforward extensions in these models. See Sections
3.3 and 3.4 for an elaborate discussion on our contribution to stochastic
games with delays in the controls and the states.

######  Remark 2.11.

In Section 3 we apply our general framework in order to solve various well-
known models, such as stochastic Volterra linear-quadratic games, models of
systemic risk and advertising with delay and optimal liquidation games with
transient price impact. As described in Section 3, many of the models in these
classes do not have a closed form solution, and our framework establishes a
new approach to deriving them. See Remarks 3.6, 3.8 and 3.11 for additional
details.

######  Remark 2.12.

The expressions (2.15) and (2.16) that determine the optimal strategy
$\hat{u}^{i}$ lend themselves naturally to numerical discretization schemes
using the so-called Nyström method where the time interval $[0,T]$ is
discretized and the operators/kernels are approximated by corresponding
matrices, see for instance [2, Section 5] and the Jupyter notebook cited
therein for an example of implementation in a single player case.

##  3 Important Examples

In this section we demonstrate the outreach and versatility of our general
results in Theorems 2.7 and 2.8, by showing that it solves several important,
challenging and diverse examples of stochastic dynamical games including:

  * •

Delay in the control,

  * •

Delay in the state and in the mean-field interaction,

  * •

General propagator models,

  * •

Mean-field interaction of controls (extended mean-field games),

  * •

Common noise.

We first introduce in Section 3.1 a generic Linear-Quadratic Stochastic
Volterra game whose objective functional is shown to be of the form (2.9).
Then, we show how such framework accommodates and extends three major examples
that have appeared in the literature: optimal liquidation games with general
transient price impact kernels (also known as propagators) and signals
(Section 3.2), an inter-bank lending and borrowing model with delay in the
control (Section 3.3) and advertising models with mean-field effect and delay
in the state (Section 3.4). For the sake of readability we omit the
superscript $N$ from $u^{i,N},\bar{u}^{N}$ and $u^{-i,N}$ throughout this
section.

###  3.1 Stochastic Volterra Linear-Quadratic games

We show that generic stochastic Volterra Linear-quadratic games are included
in the general framework which was developed in Section 2.

Define the following controlled Volterra state variables

$\displaystyle X^{i}_{t}$
$\displaystyle=P^{i}_{t}+\int_{0}^{t}G_{2}(t,s)u^{i}_{s}ds+\int_{0}^{t}G_{3}(t,s)\bar{u}_{s}ds,\quad
i=1,\ldots,N,$ (3.1) $\displaystyle Y^{i}_{t}$
$\displaystyle=R^{i}_{t}+\int_{0}^{t}G_{1}(t,s)\bar{u}_{s}ds,\quad
i=1,\ldots,N,$ (3.2)

for some progressively measurable processes $P^{i},R^{i}$ and kernels
$G_{1},G_{2},G_{3}$. Set $Z^{i}=(X^{i},Y^{i})^{\top}$ so that

$\displaystyle
Z^{i}_{t}=d^{i}_{t}+\int_{0}^{t}D(t,s)\begin{pmatrix}u^{i}_{s}\\\
\bar{u}_{s}\end{pmatrix}ds,$ (3.3)

with

$\displaystyle d^{i}_{t}=\begin{pmatrix}P^{i}_{t}\\\
R^{i}_{t}\end{pmatrix}\quad\mbox{and}\quad
D(t,s)=\begin{pmatrix}{G_{2}}(t,s)&G_{3}(t,s)\\\ 0&G_{1}(t,s)\end{pmatrix},$

and consider the following objective functional to be maximized by the $i$-th
player:

$\displaystyle\mathcal{J}^{i}_{Vol}(u^{i},u^{-i})=\mathbb{E}\left[\int_{0}^{T}f^{Vol}_{i}(Z^{i}_{t},u^{i}_{t})dt+g^{Vol}_{i}(Z^{i}_{T})\right],$
(3.4)

where the running and terminal criterion have a linear-quadratic dependence in
$(Z^{i},u^{i})$ of the form

$\displaystyle f^{Vol}_{i}(z,u)$
$\displaystyle=-pu^{2}-z^{\top}Qz+uz^{\top}q,$ (3.5) $\displaystyle
g^{Vol}_{i}(z)$ $\displaystyle=-z^{\top}Sz+z^{\top}s^{i},$ (3.6)

such that $p\geq 0$, $Q,S\in\mathbb{R}^{2\times 2}$, $q\in{\mathbb{R}}^{2}$
and $s^{i}$ is an $\mathcal{F}_{T}$-measurable random variable in
${\mathbb{R}}^{2}$, $i=1,\ldots,N$.

Since the dynamics of each $Z^{i}$ in (3.3) are linear in $u^{i}$ and
$\bar{u}$ and $f^{Vol}_{i}$ and $g_{i}^{Vol}$ are linear-quadratic in
$(Z^{i},u^{i})$ it is clear that $\mathcal{J}^{i}_{Vol}$ is a linear-quadratic
functional in $(u^{i},\bar{u})$ as in (2.9). This is summarized in the
following Lemma.

######  Lemma 3.1.

The objective functional (3.4) for the stochastic Volterra game can be written
in the form of (2.9) with the following coefficients

$\displaystyle\begin{pmatrix}\hat{A}_{2}(t,s)&A_{3}(t,s)\\\
A_{3}(t,s)&A_{1}(t,s)\end{pmatrix}$
$\displaystyle=1_{\\{s<t\\}}D(T,t)^{\top}(S+S^{\top})D(T,s)$
$\displaystyle\quad+1_{\\{s<t\\}}\int_{0}^{T}D(r,t)^{\top}(Q+Q^{\top})D(r,s)dr-\frac{1_{\\{s<t\\}}}{2}\begin{pmatrix}q^{\top}D(t,s)\\\
0_{{\mathbb{R}}^{2}}^{\top}\end{pmatrix}$
$\displaystyle\quad-\frac{1_{\\{s<t\\}}}{2}\begin{pmatrix}D(t,s)^{\top}q&0_{{\mathbb{R}}^{2}}\end{pmatrix},$
$\displaystyle\begin{pmatrix}b^{i}_{t}\\\ b^{0}_{t}\end{pmatrix}$
$\displaystyle=D(T,t)^{\top}\left({\mathbb{E}}_{t}[s^{i}]-(S+S^{\top}){\mathbb{E}}_{t}[d^{i}_{T}]\right)-\int_{t}^{T}D(r,t)^{\top}(Q+Q^{\top}){\mathbb{E}}_{t}[d^{i}_{r}]dr$
$\displaystyle\quad\qquad\quad+\begin{pmatrix}(d^{i}_{t})^{\top}q\\\
0\end{pmatrix},$ $\displaystyle c^{i}$
$\displaystyle=\mathbb{-}\int_{0}^{T}(d_{t}^{i})^{\top}Qd_{t}^{i}dt-(d_{T}^{i})^{\top}Sd_{T}^{i}+(d^{i}_{T})^{\top}s^{i},\quad
i=1,\ldots,N,$ $\displaystyle\lambda$ $\displaystyle=p.$

###### Proof.

The proof is straightforward aplication of Fubini’s Theorem and the tower
property of the conditional expectation. ∎

######  Remark 3.2.

The solutions that we derived in (2.15) and (2.16) apply to the particular
case of stochastic Volterra LQ games of the form (3.1)-(3.2). Solutions to
Volterra LQ control problems can be characterised in some cases in terms of
solutions to operator Riccati equations and to $L^{2}$-valued BSDEs; see [2,
Section 6]. In other cases they can be related to infinite dimensional Riccati
equations, see [3, 22, 37] for single-player examples. Our derivation provides
explicit operator solutions to such Riccati equations for the case of dynamics
without control in the volatility. Furthermore, we point out that our
expressions share some similarities with formulas that have recently appeared
in the computations of Laplace transforms of some quadratic functionals of
non-controlled Volterra processes [1, 4].

In the following we will show that the dynamics (3.1) includes any stochastic
Volterra equation for the sate variables $X^{i}$, where the drift has linear
dependence in $X^{i}$ itself and in $\bar{X}:=\frac{1}{N}\sum_{j=1}^{N}X^{j}$.
In order to do that we first introduce the notion of the resolvent of a
kernel.

We recall the definition of the product of kernels. For any $G,H\in
L^{2}\left([0,T]^{2},\mathbb{R}\right)$ we define the $\star$-product as
follows

$\displaystyle(G\star
H)(s,u)=\int_{0}^{T}G(s,z)H(z,u)dz,\quad(s,u)\in[0,T]^{2},$

which is a well-defined kernel in $L^{2}\left([0,T]^{2},\mathbb{R}\right)$ due
to Cauchy-Schwarz inequality. Denoting by $\bm{G}$ and $\bm{H}$ the two
integral operators induced by the kernels $G$ and $H$, we get that
$\bm{G}\bm{H}$ is an integral operator induced by the kernel $G\star H$.

For a kernel $K\in L^{2}([0,T]^{2},{\mathbb{R}})$, we define its resolvent
$R_{T}\in L^{2}([0,T]^{2},{\mathbb{R}})$ by the unique solution to

$\displaystyle R_{T}=K+K\star R_{T},\quad\quad K\star R_{T}=R_{T}\star K.$
(3.7)

In terms of integral operators, this translates into

$\displaystyle\bm{R}_{T}=\bm{K}+\bm{K}\bm{R}_{T},\quad\bm{K}\bm{R}_{T}=\bm{R}_{T}\bm{K}.$

In particular, if $K$ admits a resolvent, $({{\rm id}}-\bm{K})$ is invertible
and

$\displaystyle({{\rm id}}-\bm{K})^{-1}={\rm id}+\bm{R}_{T}.$ (3.8)

######  Lemma 3.3.

For $i=1,\ldots,N$, let $M^{i}$ be a progressively measurable process with
sample paths in $L^{2}([0,T],{\mathbb{R}})$ and $K,H:[0,T]^{2}\to\mathbb{R}$
be two Volterra kernels in $\mathcal{G}$. The $N$-dimensional linear system of
Volterra equations

$\displaystyle
X^{i}_{t}=M^{i}_{t}+\int_{0}^{t}K(t,s)X^{i}_{s}ds+\int_{0}^{t}H(t,s)\bar{X}_{s}ds,\quad
i=1,\ldots,N,$ (3.9)

where $\bar{X}=\frac{1}{N}\sum_{j=1}^{N}X^{j}$, admits a unique solution given
by

$\displaystyle
X^{i}_{t}=\widetilde{M}^{i}_{t}+\int_{0}^{t}R^{K}(t,s)\widetilde{M}^{i}_{s}ds,\quad
i=1,\ldots,N,$ (3.10)

with

$\displaystyle\widetilde{M}^{i}_{t}$
$\displaystyle=M^{i}_{t}+\int_{0}^{t}H(t,s)\bar{M}_{s}ds+\int_{0}^{t}\left(H\star
R^{K+H}\right)(t,s)\bar{M}_{s}ds,\quad i=1,\ldots,N,$
$\displaystyle\bar{M}_{t}$ $\displaystyle=\frac{1}{N}\sum_{j=1}^{N}M^{j}_{t},$

and $R^{K},R^{K+H}$ the resolvents of $K$ and $K+H$, respectively.

###### Proof.

We first observe that any Volterra kernel in $\mathcal{G}$ admits a resolvent
kernel, see for instance Corollary 9.3.16 in [21]. Thanks to the resolvent
equation (3.8), the explicit solution to the linear Volterra equation
satisfied by the mean process $\bar{X}$,

$\displaystyle\bar{X}_{t}=\bar{M}_{t}+\int_{0}^{t}\left(K(t,s)+H(t,s)\right)\bar{X}_{s}ds,$

is given by

$\displaystyle\bar{X}_{t}=\bar{M}_{t}+\int_{0}^{t}R^{K+H}(t,s)\bar{M}_{s}ds.$
(3.11)

Plugging this expression back in (3.9) together with another application of
(3.8) yields (3.10). ∎

######  Remark 3.4.

A particular case of interest in our application of Lemma 3.3 to controlled
Volterra processes, is the case where

$\displaystyle
M^{i}_{t}=P^{i}_{t}+\int_{0}^{t}G(t,s)u^{i}_{s}ds+\int_{0}^{t}\bar{G}(t,s)\bar{u}_{s}ds,$

where $P^{i}$ is progressively measurable, $G$, $\bar{G}$ Volterra kernels in
$\mathcal{G}$ and $u$ is the control. In particular, we will apply Lemma 3.3
with $\bar{G}=K\equiv 0$ so that, using the fact that $H+H\star R^{H}=R^{H}$,
the solution in (3.9) simplifies to the form of (3.1), that is,

$\displaystyle X^{i}_{t}$ $\displaystyle=M^{i}_{t}+\int_{0}^{t}\left(H+H\star
R^{H}\right)(t,s)\bar{M}_{s}ds$ (3.12)
$\displaystyle=M^{i}_{t}+\int_{0}^{t}R^{H}(t,s)\bar{M}_{s}ds$ (3.13)
$\displaystyle=P^{i}_{t}+\int_{0}^{t}R^{H}(t,s)\frac{1}{N}\sum_{j=1}^{N}P_{t}^{j}ds+\int_{0}^{t}(R^{H}\star
G)(t,s)\bar{u}_{s}ds+\int_{0}^{t}G(t,s)u^{i}_{s}ds,$ (3.14)

for $i=1,\ldots,N.$

###  3.2 Propagator models: optimal liquidation with transient impact and
signals

The framework for optimal liquidation with transient impact and price
predicting signals was first introduced in [31]. The stochastic game, where
several financial agents aim to liquidate in an optimal way their positions in
a risky asset was presented in [35]. In the following we briefly describe the
model.

The model. We consider $N$ traders with an initial position of
$x^{i}_{0}\in\mathbb{R}$ shares in a risky asset for the $i$-th trader,
$i=1,\ldots,N$. The number of shares the trader holds at time $t$ is
prescribed as

$\displaystyle\widetilde{X}^{i}_{t}=x^{i}_{0}-\int_{0}^{t}u^{i}_{s}ds,$ (3.15)

where $(u^{i}_{s})_{s\in[0,T]}$ denotes the selling speed which is chosen by
the trader. We assume that the traders’ individual and aggregated trading
activity causes price impact on the risky asset’s execution price. The actual
price at which the orders are executed for the $i$-th trader is given by

$S^{i}_{t}:=P^{i}_{t}-\lambda u^{i}_{t}-\widetilde{Y}^{\bar{u}}_{t},\qquad
0\leq t\leq T,$ (3.16)

where $N^{i}$ denotes some unaffected progressively measurable price process
incorporating an individual (or common) trading signal. The process

$\displaystyle\widetilde{Y}_{t}^{\bar{u}}=\int_{0}^{t}G(t,s)\bar{u}_{s}ds,\qquad
0\leq t\leq T,$ (3.17)

captures an aggregated linear decaying price impact with a general propagator
kernel $G$ and $\lambda>0$ captures instantaneous slippage costs which are
incurred by the $i$-th trader’s execution strategy.

######  Remark 3.5.

In [34, 35] an exponential decaying kernel of the form $G(t)=e^{-\rho t}$ for
some $\rho>0$ is specified, so the control problem can be regarded as
Markovian. However, in practice, the decay of the price impact has been shown
to be slower than exponential, and more realistically modeled by a power-law
kernel, which makes the problem non-Markovian and quite challenging to solve.
For the single player case, the problem with general propagator
$G\in\mathcal{G}$ and a progressively measurable $P$ was solved recently in
[2]. We refer to [2, Example 2.5] for some examples of realistic kernels and
to the references therein for the empirical motivation.

The objective criterion.  The aim of each trader is to maximize the profit and
loss from implementing the trading strategy $u^{i}$ given by

$-\int_{0}^{T}S^{i}_{t}d\widetilde{X}^{i}_{t}+P^{i}_{T}\widetilde{X}^{i}_{T}=-\int_{0}^{T}S^{i}_{t}u^{i}_{t}dt+P^{i}_{T}\widetilde{X}^{i}_{T}$

in the presence of running and terminal penalizations on the trader’s
inventory of the form $\phi(x^{i}_{t})^{2}$ and $\varrho(x^{i}_{T})^{2}$ with
$\phi,\varrho>0$, which encourage the trader to liquidate her position. This
leads to the following performance functional

$J^{i}_{liq}(u^{i})=\mathbb{E}\Bigg{[}\int_{0}^{T}f^{liq}_{i}(t,\widetilde{X}^{i}_{t},\widetilde{Y}^{\bar{u}}_{t},u^{i}_{t})dt+g^{liq}_{i}(\widetilde{X}^{i}_{T})\Bigg{]},$
(3.18)

with

$\displaystyle f^{liq}_{i}(t,x,y,u)$ $\displaystyle=-\lambda u^{2}-\phi
x^{2}-uy+P^{i}_{t}u,\quad g^{liq}_{i}(x)=-\varrho x^{2}+P^{i}_{T}x.$ (3.19)

The correspondence with our Volterra game framework.

The optimal liquidation problem with aggregated transient price impact and
signals corresponds to a Volterra game in the sense of Section 3.1 with
controlled variables $X^{i}$ in (3.1) and $Y^{i}$ in (3.2) given by
$(X^{i},Y^{i})=(\widetilde{X}^{i},P^{i}-\widetilde{Y}^{\bar{u}})$ with
$G_{2}(t,s)=-1_{s\leq t},\quad G_{3}=0\quad G_{1}=-G\quad
P^{i}=x_{0}^{i},\quad R^{i}=P^{i}.$ The objective criterion in (3.18) can be
written as in (3.4) in terms of $f^{Vol}$ in (3.5) and $g^{Vol}$ in (3.6)
where $\displaystyle p$ $\displaystyle=\lambda,\quad
Q={\begin{pmatrix}\phi&0\\\ 0&0\end{pmatrix}},\quad q={(0,1)^{\top}},\quad
S={\begin{pmatrix}\varrho&0\\\ 0&0\end{pmatrix}},\quad
s^{i}=(P^{i}_{T},0)^{\top}.$

######  Remark 3.6 (Improvements with respect to the related literature).

In [35] the above game was explicitly solved for an exponential price impact
kernels $G(t,s)=1_{\\{s<t\\}}e^{-\rho(t-s)}$ for some constant $\rho>0$. Such
kernels turns the model into a Markovian game, which can be solved via
forward-backward stochastic differential equations. The problem becomes quite
intricate for more general kernels $G\in\mathcal{G}$ which are allowed in our
setup; e.g., power law kernels of the form
$G(t,s)=1_{\\{s<t\\}}(t-s)^{-\alpha}$ for some $\alpha\in(0,1/2)$. In the
single player case, this problem was solved only very recently in [2] by
making an ansatz on the value function combined with a martingale verification
argument. Our approach not only extends such results of [2] to a multi-player
price impact game but also provides a new and direct argument for deriving the
solution for the single player case.

###  3.3 Delay in the control: an inter-bank lending and borrowing model

In this subsection we show how our results extend models on systemic risk with
delay studied in [10, 17].

The model. The log-monetary reserves $x^{i}$ of $N$ banks, $i=1,\ldots,N$, are
modeled as follows

$\displaystyle
dx^{i}_{t}=\left(h_{i}(t)+\int_{[0,t]}\nu(ds)u^{i}_{t-s}\right)dt+\sigma_{i}dW^{i}_{t},\quad
x^{i}_{0}\in{\mathbb{R}},$ (3.20)

with $h_{i}$ a progressively measurable process, $(W^{1},\ldots,W^{N})$ an
$N$-dimensional Brownian motion and $\nu$ a (signed) measure on $[0,T]$ of
locally bounded variation. The control $u^{i}$ of the $i$-th bank corresponds
to the rate of lending or borrowing (depending on the sign of $u^{i}$) from a
central bank. The delay in the control reflects the repayments after a fixed
time. The main example is given by $u^{i}_{t}-u^{i}_{t-\tau}$ which
corresponds to the case $\nu=\delta_{0}-\delta_{\tau}$ for some fixed time
$\tau\geq 0$. In this case, if the $i$-th bank borrowed from the central bank
an amount $u_{t}dt$ at some time $t$, then $x^{i}_{t}$ increases by $u_{t}dt$.
In addition, since the amount needs to be paid back to the central bank at a
later time $t+\tau$, the log-monetary reserve $x^{i}_{t+\tau}$ decreases by
$u^{i}_{(t+\tau)-\tau}dt=u^{i}_{t}dt$, which explains the form of the drift
$u^{i}_{t}-u^{i}_{t-\tau}$ at time $t$.

######  Remark 3.7.

We note that in [10, 17], the dynamics of $x^{i}$ are stated slightly
differently using the convention of delayed differential equations:

$\displaystyle
dx^{i}_{t}=\left(\int_{[0,\tau]}\theta(ds)u^{i}_{t-s}\right)dt+\sigma_{i}dW^{i}_{t},\quad
x^{i}_{0}\in\mathbb{R},$

for some fixed $\tau\geq 0$, with an initialization of the control of the form

$\quad u^{i}_{s}=\phi^{i}(s),\quad\mbox{for }s\in[-\tau,0],$

for some specified function $\phi$ and measure $\theta$. Setting
$\nu(ds)=1_{s\leq\tau}\theta(ds)$ and
$h_{i}(t)=1_{t\leq\tau}\int_{[t,\tau]}\theta(ds)u^{i}_{t-s}=1_{t\leq\tau}\int_{[t,\tau]}\theta(ds)\phi^{i}(t-s)$
yields the dynamics in (3.20).

The objective criterion. The $i$-th bank chooses the strategy $u^{i}$ to
minimize the following objective criterion that involves the aggregated
average $\bar{x}=\frac{1}{N}\sum_{i=1}^{N}x^{i}$:

$\displaystyle\mathcal{J}^{i}_{sys}(u^{i},u^{-i})=\mathbb{E}\left[\int_{0}^{T}f^{sys}(x^{i}_{t},\bar{x}_{t},u^{i}_{t})dt+g^{sys}(x^{i}_{T},\bar{x}_{T})\right],$

with the running and terminal costs given by

$\displaystyle f^{sys}(x,\bar{x},u)$ $\displaystyle=\frac{u^{2}}{2}-\beta
u(\bar{x}-x)+\frac{\varepsilon}{2}(\bar{x}-x)^{2},$ $\displaystyle
g^{sys}(x,\bar{x})$ $\displaystyle=\frac{c}{2}(\bar{x}-x)^{2}.$

The running cost of borrowing/lending is given by $(u^{i})^{2}/2$, the
parameter $\beta>0$ controls the incentive to borrowing or lending depending
on the difference with the average capitalization level; the quadratic term in
$(\bar{x}-x)^{2}$ in the running and terminal costs penalize departure from
the average. The condition $\beta^{2}\leq\epsilon$ ensures convexity.

The correspondence with our Volterra game framework. Setting

$G(t):=\nu([0,t]),\quad t\geq 0,$

writing (3.20) in integral form, and applying Lemma 3.9 below, we get the
following Volterra representation for $x^{i}$:

$\displaystyle x_{t}^{i}=P^{i}_{t}+\int_{0}^{t}G(t-s)u^{i}_{s}ds$ (3.21)

with $P^{i}=x^{i}_{0}+\int_{0}^{\cdot}h_{i}(s)ds+\sigma_{i}W^{i}$.

Clearly, this corresponds to a Volterra game in the sense of Section 3.1 where
the controlled variables $X^{i}$ in (3.1) and $Y^{i}$ in (3.2) are given by
$(X^{i},Y^{i})=(x^{i},\bar{x})$ with $G_{1}={G_{2}}=G,\quad{G_{3}=0},\quad
R^{i}=\frac{1}{N}\sum_{j=1}^{N}P^{j},$ and where the objective criterion to be
maximized in (3.4) is defined in terms of $f^{Vol}$ as in (3.4) and $g^{Vol}$
as in (3.6) with the parameters $\displaystyle p$
$\displaystyle=\frac{1}{2},\quad Q=\frac{\epsilon}{2}\begin{pmatrix}1&-1\\\
-1&1\end{pmatrix},\quad q=\beta(-1,1)^{\top},\quad S=\frac{c}{\epsilon}Q,\quad
s^{i}=(0,0)^{\top}.$

######  Example 3.1.

We briefly point out some important examples for $G$ in this setting.

  * (i)

If $\nu(ds)=g(s)ds$ for some locally integrable function $g$, then
$G(t)=\int_{0}^{t}g(s)ds$, for all $t\geq 0$.

  * (ii)

If $\nu(ds)=\delta_{0}-\delta_{\tau}(ds)$ for some $\tau\geq 0$, then
$G(t)=1_{\\{t\geq 0\\}}-1_{\\{t\geq\tau\\}}$.

######  Remark 3.8 (Improvements with respect to the related literature).

In [10], the equilibrium to the above inter-bank lending and borrowing game
was characterised as a solution to a system of integral equations [10,
equations (41)-(45)]. Our main results not only provide novel explicit
operator formulas for the game, but also allow with no additional effort, non-
trivial realistic extensions of the game. First, we allow the inclusion of
lending and borrowing between the banks by introducing an interacting term

$\frac{1}{N}\sum_{j=1}^{N}\left(x^{j}_{t}-x^{i}_{t}\right),$

in the drift of $x^{i}$ in (3.20). This term represents the rate at which bank
$i$ borrows from or lends to bank $j$ in the spirit of the model in [9] but
with the addition of the delay feature, which was not considered [10]. Second,
we can include a common Brownian noise $W$ by updating $P^{i}$ to
$P^{i}=\int_{0}^{\cdot}h_{i}(s)ds+\sigma^{i}W^{i}+\tilde{\sigma}^{i}W$ in
(3.21). Third, we allow the possibility of adding a more general independent
and correlated noise which is not necessarily a semimartingale, where the
usual techniques that rely on Itô’s formula break down. Finally, we allow the
inclusion of delays in the mean-field interacting term in the drift, such
feature is discussed in the next section.

######  Lemma 3.9.

Let $\nu$ be a measure on ${\mathbb{R}}_{+}$ of locally finite total
variation. Set $G(t)=\nu([0,t])$, for all $t\geq 0$, then, for any locally
integrable measurable function $u$ it holds that

$\displaystyle\int_{[0,t]}G(t-s)u_{s}ds=\int_{[0,t]}\left(\int_{[0,l]}u_{l-r}\nu(dr)\right)dl,\quad
t\geq 0.$

###### Proof.

This is an application of Fubini’s theorem and the change of variables
formula:

$\displaystyle\int_{[0,t]}G(t-s)u_{s}ds$
$\displaystyle=\int_{[0,t]}\left(\int_{[0,t-s]}\nu(dr)\right)u_{s}ds$ (def of
$G$) $\displaystyle=\int_{[0,t]}\left(\int_{[0,t]}1_{\\{r+s\leq
t\\}}\nu(dr)\right)u_{s}ds$
$\displaystyle=\int_{[0,t]}\left(\int_{[0,t]}1_{\\{r+s\leq
t\\}}u_{s}ds\right)\nu(dr)$ (Fubini)
$\displaystyle=\int_{[0,t]}\left(\int_{[r,t+r]}1_{\\{l\leq
t\\}}u_{l-r}dl\right)\nu(dr)$
$\displaystyle=\int_{[0,t]}\left(\int_{[0,t]}1_{\\{r\leq l\leq
t\\}}u_{l-r}dl\right)\nu(dr)$
$\displaystyle=\int_{[0,t]}\left(\int_{[0,t]}1_{\\{r\leq l\leq
t\\}}u_{l-r}\nu(dr)\right)dl$ (Fubini)
$\displaystyle=\int_{[0,t]}\left(\int_{[0,l]}u_{l-r}\nu(dr)\right)dl,$

where we have done a change of variable $l=r+s$ in inner integral, in the
fourth inequality. ∎

###  3.4 Delay in the state: an advertising model with mean-field effect

Our next application is motivated by problems of stochastic optimal
advertising [33, 19, 6] with time-delay and persistence of advertising
expenditures, in the presence of competition between products or firms.

The model. We consider a company launching $N$ products such that the dynamics
for the advertising goodwill $x^{i}$ of the $i$-th product to be launched are
given by

$\displaystyle dx^{i}_{t}=\left(\beta
u^{i}_{t}+h_{i}(t)+\int_{[0,t]}\nu(ds)x^{i}_{t-s}+\int_{[0,t]}\mu(ds)\bar{x}_{t-s}\right)dt+\sigma_{i}dW^{i}_{t},\quad
x^{i}_{0}\in\mathbb{R}.\qquad$ (3.22)

Here $u^{i}$ is the intensity of advertising spending, $(W^{1},\ldots,W^{N})$
is a possibly correlated $N$-dimensional Brownian motion, $\beta\geq 0$ is a
constant advertising effectiveness factor. $h_{i}$ is a deterministic or
external stochastic factor for image deterioration in the absence of
advertising which can be common to several products. $\nu$ is the distribution
of the forgetting time, and $\mu$ is the distribution of a time-delayed
interactive competition among the products. Note that when $\nu$, $h_{i}$,
$\mu$, and $\sigma_{i}$ are identically zero, equation (3.22) reduces to the
classical model of Nerlove and Arrow [33].

######  Remark 3.10.

We note that the dynamics of $x^{i}$ are usually stated slightly differently
using the convention of delayed differential equations:

$\displaystyle dx^{i}_{t}=\left(\beta
u^{i}_{t}+\int_{[0,\tau]}\theta_{1}(ds)x^{i}_{t-s}+\int_{[0,\tau]}\theta_{2}(ds)\bar{x}_{t-s}\right)dt+\sigma_{i}dW^{i}_{t},$

for some fixed $\tau\geq 0$, with an initialization of the controlled
variables of the form

$\quad x^{i}_{s}=\phi^{i}(s),\quad\mbox{for }s\in[-\tau,0],$

for some specified functions $\phi^{i}$ and measures $\theta_{1}$ and
$\theta_{2}$. Setting $\nu(ds)=1_{s\leq\tau}\theta_{1}(ds)$,
$\mu(ds)=1_{s\leq\tau}\theta_{2}(ds)$ and

$\displaystyle h_{i}(t)$
$\displaystyle=1_{t\leq\tau}\int_{[t,\tau]}\left(\theta_{1}(ds)x^{i}_{t-s}+\theta_{2}(ds)\bar{x}_{t-s}\right)$
(3.23)
$\displaystyle=1_{t\leq\tau}\int_{[t,\tau]}\left(\theta(ds)\phi^{i}(t-s)+\theta_{2}(ds)\frac{1}{N}\sum_{j=1}^{N}\phi^{j}(t-s)\right),$
(3.24)

yields the dynamics (3.20).

The objective criterion. The following objective functional to be maximized
was presented in [19, Section 6],

$\displaystyle\mathcal{J}^{i}_{adv}(u^{i},u^{-i})=\mathbb{E}\left[-\int_{0}^{T}\lambda(u^{i}_{t})^{2}dt+\beta
x^{i}_{T}\right].$ (3.25)

The correspondence with our Volterra game framework. We start by showing that
the dynamics (3.22) can be written in the form (3.1). First, an application of
Lemma 3.9 shows that (3.22) can be written in the Volterra form

$\displaystyle
x^{i}_{t}=M^{i}_{t}+\int_{0}^{t}K(t-s)x^{i}_{s}ds+\int_{0}^{t}H(t-s)\bar{x}_{s}ds,$
(3.26)

with $M^{i}=\int_{0}^{\cdot}h_{i}(s)ds+\sigma_{i}W^{i}+\int_{0}^{\cdot}\beta
u^{i}_{s}ds$, $K(t)=\nu([0,t])$ and $H(t)=\mu([0,t])$. Then, an application of
Lemma 3.3 shows clearly that $x^{i}$ can be written in the form (3.1).
However, to ease notations, we will highlight the application of Lemma 3.3 for
the case $\nu=0$ and $h_{i}=0$, so that $K=0$:

Thanks to Remark 3.4 applied to (3.26) for the case $\nu=0$ and $h_{i}=0$, we
can deduce that the advertising problem with delay in the mean-field
interaction corresponds to a Volterra game where the controlled variables
$X^{i}$ in (3.1) and $Y^{i}$ in (3.2) are given by $(X^{i},Y^{i})=(x^{i},0)$
with $G_{2}(t,s)=\beta 1_{s\leq t},\quad
G_{3}(t,s)=\beta\int_{s}^{t}R^{H}(t,u)du,\quad G_{1}=0,$
$P^{i}=\sigma_{i}W^{i}+\int_{0}^{\cdot}R^{H}(\cdot,s)\frac{1}{N}\sum_{j=1}^{N}\sigma_{j}W^{j}_{s}ds,\quad
R^{i}=0,$ where $R^{H}$ is the resolvent of $H$, and where the objective
criterion in (3.4) is defined in terms of $f^{Vol}$ of (3.4) and $g^{Vol}$ of
(3.6) with $\displaystyle p$ $\displaystyle=\lambda,\quad
Q=0_{{\mathbb{R}}^{2\times 2}},\quad q=(0,0)^{\top},\quad
S=0_{{\mathbb{R}}^{2\times 2}},\quad s^{i}=(\beta,0)^{\top}.$

######  Remark 3.11 (Improvements with respect to the related literature).

To the best of our knowledge, the advertising multiplayer game above has not
been solved so far. A similar advertising game has been briefly presented in
[6, Section 2.2] in the context of certain Linear-Quadratic Stackelberg games.
In [20], a one-player game has been considered with $\mu$ and $\nu$ in (3.22)
absolutely continuous with respect the Lebesgue measure. Such control problem
has been characterized, under the particular setting of (3.25) in terms of a
system of integral equations, see [20, equations (30)-(32)]. Our framework
provides the first solution for a multiplayer advertising game with delay in
the individual state and in the mean-field state. What is more interesting, is
that our framework allows for the inclusion of delay in the control, common
and non-semimartingale noise which are not straightforward extensions usually.
As the authors point out in [20, Section 5]: the only technique that seems to
work when combining delay in the control and in the state is the framework of
viscosity solutions which does not provide enough regularity.

######  Remark 3.12.

Combined with the ideas in Section 3.3, we can also incorporate linear time
delays in the control $u^{i}$ and in the dynamics of $x^{i}$ in (3.22); see
Lemma 3.3 and Remark 3.4. Our framework allows for an immediate solution even
when the delays depend on a measure, which usually introduces technical
difficulties with the usual infinite dimensional lifts applied in the study of
controlled delayed equations; see, for instance, [19, Remark 1] and [20].

##  4 The Mean-Field Game

In this section we formulate and solve the corresponding asymptotic mean-field
game version of the finite-player game from Section 2 when the number of
players $N$ tends to infinity. We prove that the finite-player Nash
equilibrium strategies converge in the infinite population limit to the mean-
field game equilibrium and prove that the latter yields an approximate
equilibrium for the finite-player setup from Section 2. We first consider a
generic player in the mean-field game limit and then provide an equivalent
formulation in terms of infinitely many players. The main advantage of the
analysis in this section is that the mean-field limit provides both simpler
expressions to the equilibrium strategies and more relaxed conditions for
existence of an equilibrium, with respect to the finite-player game.

Let $(\Omega,\mathcal{F}_{T},\mathbb{F}=(\mathcal{F}_{t})_{0\leq t\leq
T},\mathbb{P})$ denote the filtered probability space from Section 2. Recall
that we use the notation $\mathbb{E}_{t}=\mathbb{E}[\cdot|\mathcal{F}_{t}]$ to
represent the conditional expectation with respect to $\mathcal{F}_{t}$.
Moreover, let $c$ denote an $\mathcal{F}_{T}$-measurable random variable and
let $\beta,\beta^{0}$ denote $\mathbb{F}$-progressively measurable processes
such that $\beta^{0}$ is independent of $\beta$. We set

$b:=\beta+\beta^{0}$ (4.1)

and denote by $\mathbb{F}^{0}:=(\mathcal{F}^{0}_{t})_{0\leq t\leq T}$ the
filtration generated by $\beta^{0}$ satisfying the usual conditions.

######  Assumption 4.1.

We assume that for all $t\in[0,T]$, $\mathcal{F}_{T}^{0}$ and
$\mathcal{F}_{t}$ are conditionally independent given $\mathcal{F}^{0}_{t}$.

######  Remark 4.2.

For instance, if we denote by $(\mathcal{F}^{\beta}_{t})_{t\leq T}$ the
filtration generated by the process $\beta$. Then, the filtration $\mathbb{F}$
defined by $\mathcal{F}_{t}:=\mathcal{F}^{\beta}_{t}\vee\mathcal{F}^{0}_{t}$
for all $t\leq T$, satisfies Assumption 4.1.

In addition to the set $\mathcal{U}$ introduced in (2.6) we also define the
subset of processes

$\mathcal{U}^{0}:=\left\\{v:\,\mathbb{F}^{0}\mbox{-progressively measurable
s.t. }\int_{0}^{T}\mathbb{E}[v_{s}^{2}]ds<\infty\right\\}\subset\mathcal{U}.$
(4.2)

The paradigm of the mean-field game limit of the finite-player game from
Section 2 is the following: First, we consider a generic player who seeks to
implement a strategy $v\in\mathcal{U}$ in order to maximize for a fixed
$\mu\in\mathcal{U}^{0}$ the objective function

$\displaystyle J(v;\mu):=$
$\displaystyle\,{\mathbb{E}}\left[-\langle\mu,\bm{A}_{1}\mu\rangle_{L^{2}}-\langle
v,\bm{A}_{2}v\rangle_{L^{2}}-\langle
v,(\bm{A}_{3}+\bm{A}^{*}_{3})\mu\rangle_{L^{2}}\right.$ (4.3)
$\displaystyle\quad\left.+\langle b,v\rangle_{L^{2}}+\langle
b^{0},\mu\rangle_{L^{2}}+c\right],$

which is akin to (2.9) with same operators $\bm{A}_{1},\bm{A}_{2},\bm{A}_{3}$
as well as $\mathbb{F}$-progressively measurable process $b^{0}$ as introduced
above in Section 2. Then, we determine the $\mathbb{F}^{0}$-measurable process
$\mu$ such that a mean-field game equilibrium with common noise $\beta^{0}$ is
obtained in the following sense.

######  Definition 4.3.

A pair $(\hat{v},\hat{\mu})\in\mathcal{U}\times\mathcal{U}^{0}$ is called a
mean-field game equilibrium if the control $\hat{v}$ solves the optimization
problem

$J(\hat{v};\hat{\mu})\rightarrow\max_{v\in\mathcal{U}}$ (4.4)

under the consistency condition

$\displaystyle{\mathbb{E}}[\hat{v}_{t}|\mathcal{F}^{0}_{T}]=\hat{\mu}_{t},\quad\Omega\times[0,T]\text{
almost everywhere.}$ (4.5)

######  Remark 4.4.

Note that due to Assumption 4.1, it holds in (4.5) that

$\displaystyle{\mathbb{E}}[\hat{v}_{t}|\mathcal{F}^{0}_{T}]={\mathbb{E}}[\hat{v}_{t}|\mathcal{F}^{0}_{t}]$

for all $t\in[0,T]$. Indeed, for all $\xi\in\mathcal{F}^{0}_{T}$ it follows
that

$\mathbb{E}[\xi\,{\mathbb{E}}[\hat{v}_{t}|\mathcal{F}^{0}_{t}]]=\mathbb{E}[{\mathbb{E}}[\xi|\mathcal{F}^{0}_{t}]\,{\mathbb{E}}[\hat{v}_{t}|\mathcal{F}^{0}_{t}]]=\mathbb{E}[\mathbb{E}[\xi\hat{v}_{t}|\mathcal{F}_{t}^{0}]]=\mathbb{E}[\xi\hat{v}_{t}],$

where the second equality is obtained from the conditional independence given
by Assumption 4.1.

######  Remark 4.5.

Observe that the process $b$ defined in (4.1) in the generic player’s
objective functional in (4.3) encodes two different sources of noise.
Specifically, $\beta^{0}$ can be interpreted as representing some common noise
affecting the whole system. $\beta$ represents the generic player’s
independent individual source of noise; as well as other random or
deterministic factors, which are idiosyncratic to the generic player. In
particular, the generic player in the mean-field game can be thought of as a
representative player chosen from a population of heterogeneous players who
are allowed to have their own $\beta$, which are not necessarily
(statistically) identical; cf. also Remark 2.5 in the finite-player game.

To state the mean-field game equilibrium, it is convenient to introduce the
following two solution maps $F$ and $G$ of two associated Fredholm equations:

$\begin{aligned} F(t,x):=&\left(({\rm
id}-\bm{\tilde{B}})^{-1}\tilde{a}(x)\right)(t)\\\ G(t,x):=&\left(({\rm
id}-\bm{\hat{B}})^{-1}\hat{a}(x)\right)(t)\end{aligned}\qquad\qquad(0\leq
t\leq T),$ (4.6)

where

$\displaystyle\tilde{a}_{t}(x)$
$\displaystyle:=\frac{1}{2\lambda}\left(x_{t}-\langle
1_{\\{t\leq\cdot\\}}\hat{A}_{2}(\cdot,t),\bm{\tilde{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}[x_{\cdot}]\rangle_{L^{2}}\right),$
(4.7) $\displaystyle\tilde{B}(t,s)$ $\displaystyle:=1_{\\{s\leq
t\\}}\frac{1}{2\lambda}\left(\langle
1_{\\{t\leq\cdot\\}}\hat{A}_{2}(\cdot,t),\bm{\tilde{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\hat{A}_{2}(\cdot,s)\rangle_{L^{2}}-\hat{A}_{2}(t,s)\right),$
$\displaystyle\bm{\tilde{D}}_{t}$ $\displaystyle:=2\lambda{\rm
id}+(\bm{\hat{A}}_{2})_{t}+(\bm{\hat{A}}^{*}_{2})_{t};$

and

$\displaystyle\hat{a}_{t}(x)$
$\displaystyle:=\frac{1}{2\lambda}\left(x_{t}-\langle
1_{\\{t\leq\cdot\\}}\hat{A}_{2}(\cdot,t),\bm{\hat{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}[x_{\cdot}]\rangle_{L^{2}}\right),$
(4.8) $\displaystyle\hat{B}(t,s):=$ $\displaystyle\,1_{\\{s\leq
t\\}}\frac{1}{2\lambda}\Big{(}\langle
1_{\\{t\leq\cdot\\}}(\hat{A}_{2}(\cdot,t)+{A}_{3}(\cdot,t)),\bm{\hat{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}(\hat{A}_{2}(\cdot,s)+{A}_{3}(\cdot,s))\rangle_{L^{2}}\Big{.}$
$\displaystyle\hskip
54.0pt\Big{.}-(\hat{A}_{2}(t,s)+\hat{A}_{3}(t,s))\Big{)},$
$\displaystyle\bm{\hat{D}}_{t}$ $\displaystyle:=2\lambda{\rm
id}+(\bm{\hat{A}}_{2})_{t}+(\bm{\hat{A}}^{*}_{2})_{t}+(\bm{A}_{3})_{t}+(\bm{A}^{*}_{3})_{t}.$

We are now ready to provide the solution to the mean-field game:

######  Theorem 4.6.

Assume that (2.2) as well as Assumptions 2.4 and 4.1 are satisfied. Then, the
unique mean-field game equilibrium
$(\hat{v},\hat{\mu})\in\mathcal{U}\times\mathcal{U}^{0}$ in the sense of
Definition 4.3 is given by

$\displaystyle\hat{v}_{t}$
$\displaystyle=F(t,b-\bm{A}_{3}(\hat{\mu})-\bm{A}^{*}_{3}({\mathbb{E}}_{\cdot}\hat{\mu})),$
(4.9) $\displaystyle\hat{\mu}_{t}$
$\displaystyle=G(t,\mathbb{E}[\beta]+\beta^{0}),$ (4.10)

for all $t\in[0,T]$, where $F$ and $G$ are defined in (4.6).

######  Remark 4.7.

To be more explicit, note that the process that appears in the right-hand side
of (4.9) is

$b_{s}-\bm{A}_{3}(\hat{\mu})(s)-\bm{A}^{*}_{3}({\mathbb{E}}_{s}\hat{\mu})(s)=b_{s}-\int_{0}^{s}A_{3}(s,r)\hat{\mu}_{r}dr-\int_{s}^{T}A_{3}(r,s){\mathbb{E}}_{s}[\hat{\mu}_{r}]dr,\quad
s\leq T.$

The proof of Theorem 4.6 is given in Section 7. We note that Remarks 2.12 and
3.2 remain valid for the expressions (4.9) and (4.10).

###  4.1 The Infinite-Player Game Formulation

It is also very natural to formulate the limiting mean-field game of the
finite-player game introduced in Section 2 as an infinite-player game when the
number of players $N$ is sent to infinity. We show below that the resulting
infinite-player mean-field game Nash equilibrium is equivalent to a generic
player’s mean-field game equilibrium derived above.

Specifically, consider a countable collection of controls
$(v^{i})_{i\in\mathbb{N}}\subset\mathcal{U}$ associated with infinitely many
agents indexed by $i\in\mathbb{N}$. Moreover, we let $\nu\in\mathcal{U}$
denote a stochastic process which represents the limit of the control
processes’ averages. Then, the infinite-player game version of each player
$i$’s objective from (2.9) is given by

$\displaystyle J^{i,\infty}(v^{i};\nu)=$
$\displaystyle\,{\mathbb{E}}\left[-\langle\nu,\bm{A}_{1}\nu\rangle_{L^{2}}-\langle
v^{i},\bm{A}_{2}v^{i}\rangle_{L^{2}}-\langle
v^{i},(\bm{A}_{3}+\bm{A}^{*}_{3})\nu\rangle_{L^{2}}\right.$ (4.11)
$\displaystyle\quad\left.+\langle b^{i},v^{i}\rangle_{L^{2}}+\langle
b^{0},\nu\rangle_{L^{2}}+c^{i}\right],$

with same operators $\bm{A}_{1},\bm{A}_{2},\bm{A}_{3}$, as well as
$\mathbb{F}$-progressively measurable processes
$b^{0},(b^{i})_{i\in\mathbb{N}}$ and $\mathcal{F}_{T}$-measurable
$(c^{i})_{i\in\mathbb{N}}$ as introduced above in Section 2.

The notion of a Nash equilibrium in this infinite-player mean-field game is to
first solve for a fixed $\nu\in\mathcal{U}$ simultaneously for each player
$i\in\mathbb{N}$ the optimal stochastic control problem

$J^{i,\infty}(v^{i};\nu)\rightarrow\max_{v^{i}\in\mathcal{U}}\qquad
i\in\mathbb{N},$ (4.12)

and then to determine the process $\nu$ in equilibrium in the following sense.

######  Definition 4.8.

A collection of processes
$(\hat{\nu},(\hat{v}^{i})_{i\in\mathbb{N}})\subset\mathcal{U}$ is called a
mean-field game Nash equilibrium with mean-field strategy $\hat{\nu}$ if for
all $i\in\mathbb{N}$ the control $\hat{v}^{i}$ solves the optimization problem

$J^{i,\infty}(v^{i};\hat{\nu})\rightarrow\max_{v^{i}\in\mathcal{U}}\qquad(i\in\mathbb{N})$
(4.13)

and $\hat{\nu}$ satisfies the consistency condition

$\lim_{N\rightarrow\infty}\sup_{0\leq t\leq
T}{\mathbb{E}}\left[\left(\frac{1}{N}\sum_{i=1}^{N}\hat{v}^{i}_{t}-\hat{\nu}_{t}\right)^{2}\right]=0.$
(4.14)

We make the following assumption on the players’ individual processes
$(b^{i})_{i\in\mathbb{N}}$ in (4.11):

######  Assumption 4.9.

We assume that there exists a unique $\mathbb{F}$-progressively measurable
process $b^{\infty}\in L^{2}(\Omega\times[0,T])$ and a bounded function
$h:\mathds{R}_{+}\mapsto\mathds{R}_{+}$ with $h(x)\rightarrow 0$ as
$x\rightarrow\infty$ such that

$\sup_{0\leq t\leq
T}{\mathbb{E}}\left[\left(\frac{1}{N}\sum_{i=1}^{N}b^{i}_{t}-b^{\infty}_{t}\right)^{2}\right]\leq
h(N),\quad\textrm{for all }N\geq 1.$ (4.15)

######  Remark 4.10.

The general Assumption 4.9 can be made precise in specific examples. For
instance, consider the portfolio liquidation price impact game from Section
3.2 above. Suppose that for each agent $i\in\mathbb{N}$ the initial inventory
is given by $x^{i}_{0}\in\mathbb{R}$ and their trading signal is given by
$P^{i}=B^{i}+B$ with idiosyncratic independent Brownian motions
$(B^{i})_{i\geq 1}$ which are also independent of a common noise Brownian
motion $B$. With this configuration, we observe,

$b^{i}_{t}=2(\varrho+\phi(T-t))x^{i}_{0}+\mathbb{E}_{t}\left[B^{i}_{t}-B^{i}_{T}\right]+\mathbb{E}_{t}\left[B_{t}-B_{T}\right]=2(\varrho+\phi(T-t))x^{i}_{0},\qquad
t\leq T.$ (4.16)

Therefore, the condition in (4.15) boils down to requiring that the limit
$x^{\infty}\in\mathbb{R}$ of the averages of all initial positions
$(x^{i}_{0})_{i\in\mathbb{N}}$ exists, i.e.,

$x^{\infty}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}x^{i}_{0}.$
(4.17)

Moreover, the process $b^{\infty}$ in (4.15) is then given by

$b^{\infty}_{t}=2(\varrho+\phi(T-t))x^{\infty},\qquad t\leq T.$ (4.18)

Akin to Theorem 4.6, the solution to the infinite-player mean-field game is
characterized as follows:

######  Theorem 4.11.

Assume that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. Then, the
unique solution $(\hat{\nu},(\hat{v}^{i})_{i\in\mathbb{N}})\subset\mathcal{U}$
of the infinite-player mean-field game in the sense of Definition 4.8 is given
by

$\displaystyle\hat{v}^{i}_{t}$
$\displaystyle=F(t,b^{i}-\bm{A}_{3}(\hat{\nu})+\bm{A}^{*}_{3}({\mathbb{E}}_{\cdot}\hat{\nu})),$
(4.19) $\displaystyle\hat{\nu}_{t}$ $\displaystyle=G(t,b^{\infty})$ (4.20)

for all $t\in[0,T]$, where $F$ and $G$ are defined in (4.6).

The proof of Theorem 4.11 is deferred to Section 7.

In light of Theorems 4.6 and 4.11, we readily obtain the following connection
between the mean-field game and the infinite-player game:

######  Remark 4.12.

Suppose it holds that

$b^{i}=\beta+\beta^{0},\qquad b^{\infty}=\mathbb{E}[\beta]+\beta^{0}.$

Then, the generic player’s mean-field game equilibrium strategy in the sense
of Definition 4.3 coincides with the $i$-th player’s strategy in the infinite-
player mean-field game Nash equilibrium in the sense of Definition 4.8.

######  Remark 4.13.

The Nash equilibrium of the infinite-player mean-field game in Theorem 4.11 is
comparable with the Nash equilibrium of the finite-player game presented in
Theorem 2.8. The main difference is that the associated Fredholm equations and
hence their solutions in (4.6), (4.7) and (4.8) become considerably simpler in
the infinite-player mean-field limit compared to their counterparts in the
finite-player game in (2.13), (2.14), (2.15) and (2.16).

###  4.2 Convergence and Approximation Results

The connection between the finite-player Nash equilibrium from Section 2 and
the infinite-player Nash equilibrium from Section 4.1 is established in the
next convergence theorem.

######  Theorem 4.14.

Assume that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. For any
$N\geq 2$, let $(\hat{u}^{i,N})_{i=1,\ldots,N}\subset\mathcal{U}$ denote the
Nash equilibrium strategies of the $N$-player game given in (2.15). Moreover,
let $(\hat{\nu},(\hat{v}^{i})_{i\in\mathbb{N}})\subset\mathcal{U}$ denote the
Nash equilibrium of the infinite-player game given in (4.19) and (4.20). Then,
there exists a constant $C>0$ such that

$\displaystyle\sup_{0\leq s\leq
T}\mathbb{E}\left[\left(\hat{\nu}_{s}-\frac{1}{N}\sum_{i=1}^{N}\hat{u}^{i,N}_{s}\right)^{2}\right]\leq
C\left(\frac{1}{N^{2}}\vee h(N)\right),\quad\textrm{for all }N\geq 1,$ (4.21)

as well as

$\displaystyle\sup_{0\leq s\leq
T}\mathbb{E}\left[\left(\hat{u}^{i,N}_{s}-\hat{v}^{i}_{s}\right)^{2}\right]\leq
C\left(\frac{1}{N^{2}}\vee h(N)\right),\quad\textrm{for all }N\geq 1,$ (4.22)

where $h$ is given in (4.15).

The proof of Theorem 4.14 can be found in Section 7.

In addition we obtain that the infinite-player Nash equilibrium from Section
4.1 provides an approximate Nash equilibrium for the finite-player game from
Section 2 in the sense of Theorem 4.17 below. To this end, we first recall the
definition of an $\varepsilon$-Nash equilibrium.

######  Definition 4.15.

Let $\mathcal{U}$ denote a class of admissible controls and fix
$\varepsilon>0$. A set of controls $\\{w^{j}\in\mathcal{U}:j=1,\ldots,N\\}$
forms an $\varepsilon$-Nash equilibrium with respect to a collection of
objective functionals $\\{J^{j}(\cdot,\cdot):j=1,\ldots,N\\}$ if it satisfies

$J^{j}(w^{j};w^{-j})\leq\sup_{w\in\mathcal{U}}J^{j}(w;w^{-j})\leq
J^{j}(w^{j};w^{-j})+\varepsilon,\quad\textrm{for all }j=1,\ldots,N.$

For any $u\in\mathcal{U}$ we introduce the following norm

$\|u\|_{2,T}=\left(\int_{0}^{T}{\mathbb{E}}[u_{t}^{2}]dt\right)^{1/2}.$

We have following approximation result.

######  Theorem 4.16.

Suppose that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. For any
$N\geq 2$, let $J^{i,N}$ be the performance functional of player
$i\in\\{1,\ldots,N\\}$ in the $N$-player game in (2.9). Moreover, let
$(\hat{v}^{i})_{i\in\mathbb{N}}$ denote the equilibrium strategies of the
mean-field game from Theorem 4.11 and let
$\hat{v}^{-i}=(\hat{v}^{1},\ldots,\hat{v}^{i-1},\hat{v}^{i+1},\ldots,\hat{v}^{N})$
for any $1\leq i\leq N$. Then, for all $u\in\mathcal{U}$ there exists a
constant $C>0$ independent of $N$ and $u$ such that for all
$i\in\\{1,\ldots,N\\}$ we have

$J^{i,N}(\hat{v}^{i};\hat{v}^{-i})\leq J^{i,N}(u;\hat{v}^{-i})\leq
J^{i,N}(\hat{v}^{i};\hat{v}^{-i})+C\|u\|^{2}_{2,T}(1\vee\|u\|^{2}_{2,T})(h(N)^{1/2}\vee
N^{-1}).$

The proof of Theorem 4.16 is deferred to Section 7.

From Theorem 4.16 and Theorem 4.14 it follows that we can define a subclass of
admissible strategies $\mathcal{U}_{b}\subset\mathcal{U}$ that includes
$\\{(\hat{u}^{1,N},\ldots,\hat{u}^{N,N}):N\in\mathbb{N}\\}$ from Theorem 2.8
and $(\hat{v}^{i})_{i\in\mathbb{N}}$ from Theorem 4.11 such that

$\sup_{u\in\mathcal{U}_{b}}\|u\|^{2}_{2,T}<\infty.$

Then the following corollary follows immediately from Theorem 4.16.

######  Corollary 4.17 ($\varepsilon$-Nash equilibrium).

Suppose that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. For any
$N\geq 2$, let $J^{i,N}$ be the performance functional of player
$i\in\\{1,\ldots,N\\}$ in the $N$-player game in (2.9). Moreover, let
$(\hat{v}^{i})_{i\in\mathbb{N}}$ denote the equilibrium strategies of the
mean-field game from Theorem 4.11 and let
$\hat{v}^{-i}=(\hat{v}^{1},\ldots,\hat{v}^{i-1},\hat{v}^{i+1},\ldots,\hat{v}^{N})$
for any $1\leq i\leq N$. Then, for all $u\in\mathcal{U}_{b}$ there exists a
constant $C>0$ independent of $N$ such that for all $u\in\mathcal{U}_{b}$ we
have

$J^{i,N}(\hat{v}^{i};\hat{v}^{-i})\leq\sup_{u\in\mathcal{U}_{b}}J^{i,N}(u;\hat{v}^{-i})\leq
J^{i,N}(\hat{v}^{i};\hat{v}^{-i})+O\left((h(N)^{1/2}\vee N^{-1})\right).$

##  5 Stochastic Fredholm equations

In this section we derive general results on linear stochastic Fredholm
equations of the second kind, which appear in the first order condition of the
$N$-player game and the mean-field game. Our results on the explicit solution
and the stability of the solution to this class of equations are central
ingredients in the proofs of the main results in Sections 2 and 4. All the
solutions to the equations considered in this section are referred to as
strong solutions.

Recall that for any $G\in\mathcal{G}$ we define

$G_{t}(s,r)=G(s,r)1_{\\{r\geq t\\}},$

and we denote by ${\bm{G}}_{t}$ the associated Volterra operator.

The following proposition derives the solution to stochastic Fredholm
equations of the second kind.

######  Proposition 5.1.

Let $K,L:[0,T]^{2}\to\mathbb{R}$ be two Volterra kernels in $\mathcal{G}$ such
that the operator $(\bm{K}+\bm{L}^{*})$ is self-adjoint and let
$(f_{t})_{0\leq t\leq T}$ be a progressively measurable process such that
$\int_{0}^{T}\mathbb{E}[f_{t}^{2}]dt<\infty$. Then, the linear stochastic
Fredholm equation

$v_{t}=f_{t}-\int_{0}^{t}K(t,r)v_{r}dr-\int_{t}^{T}L(r,t)\mathbb{E}_{t}v_{r}dr,\quad
t\in[0,T],$ (5.1)

admits a unique progressively measurable solution $(v_{t})_{0\leq t\leq T}$ in
$\mathcal{U}$ given by

$\displaystyle v_{t}=\left(({\rm id}-\bm{B})^{-1}a\right)(t),\quad 0\leq t\leq
T,$ (5.2)

where

$\displaystyle a_{t}$ $\displaystyle=f_{t}-\langle
1_{\\{t<\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{\\{t<\cdot\\}}{\mathbb{E}}_{t}f_{\cdot}\rangle_{L^{2}}$
(5.3) $\displaystyle B(t,s)$ $\displaystyle=1_{\\{s<t\\}}\left(\left\langle
1_{\\{t<\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}K(\cdot,s)\right\rangle_{L^{2}}-K(t,s)\right),$
${\bm{D}}_{t}={\rm id}+\bm{K}_{t}+\bm{L^{*}}_{t},$ (5.4)

and $\bm{B}$ is an integral operator induced by the kernel $B$.

The proof of Proposition 5.1 is postponed to the end of this section.

######  Remark 5.2.

Note that the assumption that $(\bm{K}+\bm{L}^{*})$ is self-adjoint in
Proposition 5.1 can be replaced by the assumption that the operators
${\bm{D}}_{t}$ are invertible for all $0\leq t\leq T$ and that (5.12) below is
satisfied.

We define operator norm as follows,

$\|\bm{G}\|_{\rm{op}}=\sup_{f\in
L^{2}([0,T],{\mathbb{R}})}\frac{\|\bm{G}f\|_{L^{2}}}{\|f\|_{L^{2}}},$ (5.5)

for an operator $\bm{G}$ from $L^{2}\left([0,T],{{\mathbb{R}}}\right)$ to
itself.

The following proposition derives a stability result for stochastic Fredholm
equations of the second kind.

######  Proposition 5.3.

Let $K^{N},L^{N},L,K:[0,T]^{2}\to\mathbb{R}$ be Volterra kernels in
$\mathcal{G}$ and let $f^{N},f$ be progressively measurable processes on the
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq
T},\mathbb{P})$, with sample paths in $L^{2}([0,T],\mathbb{R})$, for
$N=1,2,\ldots$ Assume further that there exist bounded functions
$h_{1},h_{2}:\mathbb{R}_{+}\mapsto\mathbb{R}_{+}$ such that
$\lim_{x\rightarrow\infty}h_{i}(x)=0$ and

$\displaystyle\sup_{t\leq T}\int_{0}^{T}\big{(}K^{N}(t,s)-K(t,s)\big{)}^{2}ds$
$\displaystyle\leq h_{1}(N),\quad\textrm{for all }N\geq 1,$ (5.6)
$\displaystyle\sup_{s\leq T}\int_{0}^{T}\big{(}L^{N}(t,s)-L(t,s)\big{)}^{2}dt$
$\displaystyle\leq h_{1}(N),\quad\textrm{for all }N\geq 1,$

and

$\sup_{t\leq T}{\mathbb{E}}\big{[}(f^{N}_{t}-f_{t})^{2}\big{]}\leq h_{2}(N).$
(5.7)

For each $N\geq 1$ let $v^{N}$ be the solution to (5.1) with
$(K^{N},L^{N},f^{N})$. Then there exists a constant $C>0$ such that

$\sup_{t\in[0,T]}{\mathbb{E}}\big{[}(v^{N}_{t}-v_{t})^{2}\big{]}\leq
C(h_{1}(N)+h_{2}(N)),\quad\textrm{for all }N\geq 1,$ (5.8)

where $v$ is the solution of (5.1) with $(K,L,f)$.

###### Proof.

Define $d^{N}_{t}=v^{N}_{t}-v_{t}$. Then from (5.26) it follows that

$\displaystyle d^{N}_{t}$
$\displaystyle=a^{N}_{t}-a_{t}+\int_{0}^{t}B^{N}(t,s)v^{N}_{s}ds-\int_{0}^{t}B(t,s)v_{s}ds$
(5.9)
$\displaystyle=a^{N}_{t}-a_{t}+\int_{0}^{t}B^{N}(t,s)d^{N}_{s}ds-\int_{0}^{t}(B(t,s)-B^{N}(t,s))v_{s}ds$

where $a$ and $B$ are as in (5.3) and

$\displaystyle a^{N}_{t}$ $\displaystyle=f^{N}_{t}-\langle
1_{\\{t<\cdot\\}}L^{N}(\cdot,t),({\bm{D}}^{N}_{t})^{-1}1_{\\{t<\cdot\\}}{\mathbb{E}}_{t}f^{N}_{\cdot}\rangle_{L^{2}}$
(5.10) $\displaystyle B^{N}(t,s)$
$\displaystyle=1_{\\{s<t\\}}\left(\left\langle
1_{\\{t\leq\cdot\\}}L^{N}(\cdot,t),({\bm{D}}_{t}^{N})^{-1}1_{\\{t\leq\cdot\\}}K^{N}(\cdot,s)\right\rangle_{L^{2}}-K^{N}(t,s)\right),$

and

${\bm{D}}^{N}_{t}=\text{id}+\bm{K}^{N}_{t}+(\bm{L}_{t}^{N})^{*}.$ (5.11)

We will need the following lemma. Recall that ${\bm{D}}_{t}$ was defined in
Proposition 5.1.

######  Lemma 5.4.

For any $0\leq t\leq T$ the operators ${\bm{D}}_{t}$,
$\\{{\bm{D}}^{N}_{t}\\}_{N\geq 1}$ are invertible and we have

$\sup_{t\leq T}\|\bm{D}^{-1}_{t}\|_{\rm{op}}+\sup_{N\geq 1}\sup_{t\leq
T}\|(\bm{D}^{N}_{t})^{-1}\|_{\rm{op}}<\infty.$ (5.12)

###### Proof.

Since by assumption
$\bm{K}_{t},\bm{L^{*}}_{t},\bm{K}^{N}_{t},(\bm{L}^{N}_{t})^{*}$ are
nonnegative definite operators in $\mathcal{G}$ and the operators
$(\bm{K}+\bm{L^{*}}),\\{(\bm{K}^{N}+\bm{(L^{N}})^{*})\\}_{N\geq 1}$ are self-
adjoint, the proof is similar to the proof of Lemma 7.2 in [2]. Indeed, by
following the proof therein we note that the uniform bound in $N$ on
$\|(\bm{D}^{N}_{t})^{-1}\|_{\rm{op}}$ follows since ${\bm{D}}^{N}_{t}$ in
(5.11) are positive definite uniformly in $N$, that is

$\langle f,{\bm{D}}^{N}_{t}f\rangle\geq\langle
f,\text{id}f\rangle\geq\|f\|,\quad\textrm{for all }f\in
L^{2}([0,T],\mathbb{R}).$

∎

From Lemma 5.4 and (5.6) it follows that

$\sup_{t\leq
T}\|{\bm{D}}^{N}_{t}-{\bm{D}}_{t}\|_{\textrm{op}}\leq(h_{1}(N))^{1/2},\quad\textrm{for
all }N\geq 1.$ (5.13)

We observe that

$(\bm{D}_{t}^{N})^{-1}-(\bm{D}_{t})^{-1}=(\bm{D}^{N}_{t})^{-1}\left(\bm{D}_{t}-\bm{D}^{N}_{t}\right)(\bm{D}_{t})^{-1}.$
(5.14)

By taking the operator norm on both sides of (5.14) and using Lemma 5.4 and
(5.13) it follows that

$\displaystyle\sup_{t\leq
T}\left\|(\bm{D}_{t}^{N})^{-1}-(\bm{D}_{t})^{-1}\right\|_{\rm op}$
$\displaystyle\leq C\sup_{t\leq T}\|\bm{D}_{t}^{N}-\bm{D}_{t}\|_{\rm op}$
(5.15) $\displaystyle\leq C(h_{1}(N))^{1/2},\quad\textrm{for all }N\geq 1.$

From (5.3), (5.6), (5.7), (5.10), (5.15), and several applications of Cauchy-
Schwartz inequality we get for all $N\geq 1$,

$\sup_{t\leq T}{\mathbb{E}}\big{[}(a^{N}_{t}-a_{t})^{2}\big{]}\leq
C(h_{1}(N)+h_{2}(N)),$ (5.16)

and

$\sup_{t\leq T}\int_{0}^{T}\big{(}B^{N}(t,s)-B(t,s)\big{)}^{2}ds\leq
Ch_{1}(N).$ (5.17)

Recall the notation

$\|v\|_{\mathcal{H}^{2}}={\mathbb{E}}\left[\int_{0}^{T}v_{s}^{2}ds\right].$

By plugging (5.16) and (5.17) into (5.9) and using Cauchy-Schwarz inequality,
we get for all $N$ sufficiently large,

$\displaystyle{\mathbb{E}}[(d^{N}_{t})^{2}]$ $\displaystyle\leq
C\Bigg{(}{\mathbb{E}}[(a^{N}_{t}-a_{t})^{2}]+{\mathbb{E}}\left[\left(\int_{0}^{t}B^{N}(t,s)d^{N}_{s}ds\right)^{2}\right]$
(5.18)
$\displaystyle\qquad+{\mathbb{E}}\left[\left(\int_{0}^{t}(B^{N}(t,s)-B(t,s))v_{s}ds\right)^{2}\right]\Bigg{)}$
$\displaystyle\leq
C\Bigg{(}h_{1}(N)+h_{2}(N)+{\mathbb{E}}\left[\int_{0}^{t}(d^{N}_{s})^{2}ds\right]\int_{0}^{t}(B^{N}(t,s))^{2}ds$
$\displaystyle\qquad+\|v_{s}\|_{\mathcal{H}^{2}}\int_{0}^{t}(B^{N}(t,s)-B(t,s))^{2}ds\Bigg{)}$
$\displaystyle\leq
C\left(h_{1}(N)+h_{2}(N)+h_{1}(N){\mathbb{E}}\left[\int_{0}^{t}(d^{N}_{s})^{2}ds\right]\right).$

Since $C$ above is not depending on $N$ and $t$ we get using Fubini’s theorem,

$\sup_{s\leq t}{\mathbb{E}}[(d^{N}_{s})^{2}]\leq
C\left(h_{1}(N)+h_{2}(N)+h_{1}(N)\int_{0}^{t}\sup_{r\leq
s}{\mathbb{E}}[(d^{N}_{r})^{2}]ds\right),\quad\textrm{for all }0\leq t\leq T.$
(5.19)

Since $h_{i}$ are bounded functions, from Gronwall lemma it follows that

$\sup_{s\in[0,T]}{\mathbb{E}}[(v^{N}_{s}-v_{s})^{2}]=\sup_{s\in[0,T]}{\mathbb{E}}[(d^{N}_{s})^{2}]\leq
C(T)(h_{1}(N)+h_{2}(N)),\quad\textrm{for all }N\geq 1.$ (5.20)

and this completes the proof. ∎

###### Proof of Proposition 5.1.

The proof of uniqueness is standard and it uses similar bounds as in the proof
of Lemma 5.4, hence it is omited.

We now derive the solution to (5.1). For fixed $t\in[0,T]$, we define the
process

$\displaystyle m_{t}(s):=1_{\\{t\leq s\\}}{\mathbb{E}}_{t}[v_{s}],\qquad
s\in[0,T].$

Taking conditional expectation $\mathbb{E}_{t}[\cdot]$ in the linear Volterra
equation in (5.1), multiplying by $1_{\\{t\leq s\\}}$ and using the tower
property of conditional expectation gives us

$\displaystyle m_{t}(s)=$ $\displaystyle\,1_{\\{t\leq
s\\}}{\mathbb{E}}_{t}f_{s}-1_{\\{t\leq
s\\}}\int_{0}^{t}K(s,r)v_{r}dr-1_{\\{t\leq
s\\}}\int_{t}^{s}K(s,r)\mathbb{E}_{t}v_{r}dr$ $\displaystyle\,-1_{\\{t\leq
s\\}}\int_{s}^{T}L(r,s){\mathbb{E}}_{t}v_{r}dr$ $\displaystyle=$
$\displaystyle\,f^{v}_{t}(s)\,-1_{\\{t\leq
s\\}}\int_{t}^{s}K(s,r)\mathbb{E}_{t}v_{r}dr-1_{\\{t\leq
s\\}}\int_{s}^{T}L(r,s){\mathbb{E}}_{t}v_{r}dr$ $\displaystyle=$
$\displaystyle\,f^{v}_{t}(s)\,-\int_{t}^{s}K_{t}(s,r)m_{t}(r)dr-\int_{s}^{T}L^{*}_{t}(s,r)m_{t}(r)dr$
$\displaystyle=$
$\displaystyle\,f^{v}_{t}(s)-\left(\bm{K}_{t}+\bm{L^{*}}_{t}\right)(m_{t})(s),\qquad
s\in[0,T],$ (5.21)

where

$f^{v}_{t}(s):=1_{\\{t\leq s\\}}{\mathbb{E}}_{t}f_{s}-1_{\\{t\leq
s\\}}\int_{0}^{t}K(s,r)v_{r}dr,$ (5.22)

and where we used the definition $K_{t}(s,r)=K(s,r)\mathds{1}_{r\geq t}$.

Recall that ${\bm{D}}_{t}$ was defined in (5.4). Using Lemma 5.4 we can invert
the equation (5) to get

$m_{t}(s)=({\bm{D}}_{t}^{-1}f^{v}_{t})(s),\quad s\in[0,T].$ (5.23)

Plugging (5.23) into (5.1) we obtain

$\displaystyle v_{t}=$
$\displaystyle\,f_{t}-\int_{0}^{t}K(t,r)v_{r}dr-\int_{t}^{T}L(r,t)({\bm{D}}_{t}^{-1}f^{v}_{t})(r)dr.$
(5.24)

Next, we focus on the third term on the right-hand side of (5.24) and re-
express it as a linear functional in $v$. Using (5.22) we get

$\displaystyle\int_{t}^{T}L(r,t)({\bm{D}}_{t}^{-1}f^{v}_{t})(r)dr$ (5.25)
$\displaystyle=\langle
1_{\\{t\leq\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}f^{v}_{t}\rangle_{L^{2}}$
$\displaystyle=-\left\langle
1_{\\{t\leq\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\int_{0}^{t}K(\cdot,r)v_{r}dr\right\rangle_{L^{2}}+\langle
1_{\\{t\leq\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{t}{\mathbb{E}}_{t}f_{\cdot}\rangle_{L^{2}}$
$\displaystyle=-\int_{0}^{t}\left\langle
1_{\\{t\leq\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}K(\cdot,r)\right\rangle_{L^{2}}v_{r}dr+\langle
1_{\\{t\leq\cdot\\}}L(\cdot,t),{\bm{D}}_{t}^{-1}1_{t}{\mathbb{E}}_{t}f_{\cdot}\rangle_{L^{2}}.$

Inserting the obtain expression back into (5.24) yields the following Fredholm
equation for $v$:

$\displaystyle v_{t}=a_{t}+\int_{0}^{t}B(t,s)v_{s}ds,$ (5.26)

with $a$ and $B$ as in (5.3).

It is left to argue that (5.26) has a solution. Note that (5.26) is a linear
Volterra equation which admits a solution for any fixed $\omega\in\Omega$,
whenever $a(\omega)\in L^{2}([0,T],\mathbb{R})$ and $B$ satisfies

$\sup_{t\leq T}\int_{0}^{T}B(t,s)^{2}ds<\infty.$

Indeed, the solution is given in terms of the resolvent $R^{B}$ of $B$, which
exists by virtue of Corollary 9.3.16 in [21] and satisfies

$\displaystyle\int_{0}^{T}\int_{0}^{T}|R^{B}(t,s)|dtds<\infty.$

In this case, the solution $u^{*}$ to (5.26) is given by

$\displaystyle u_{t}^{*}=a_{t}+\int_{0}^{t}R^{B}(t,s)a_{s}ds.$

Note that $R^{B}$ is the kernel induced by the operator given by

$\bm{R}^{B}=({\rm id}-\bm{B})^{-1}.$

One would still need to check that $u^{*}\in\mathcal{U}$ defined as in (2.6).
This follows from the following lemma.

######  Lemma 5.5.

Let $v$ be as in (5.1) and $a,B$ as in (5.3). Then the following hold:

  * (i)

$\mathbb{E}\left[\int_{0}^{T}a_{s}^{2}ds\right]<\infty$,

  * (ii)

$\sup_{t\leq T}\int_{0}^{T}B(t,s)^{2}ds<\infty,$

  * (iii)

$\mathbb{E}\left[\int_{0}^{T}(v_{s})^{2}ds\right]<\infty$.

The proof of Lemma 5.5 is similar to the proof of Lemma 7.1 in [2]; hence, it
is omitted.

It follows that the equation (5.26) admits a unique solution $v\in\mathcal{U}$
which is given by

$v_{t}=\left(({\rm id}-\bm{B})^{-1}a\right)(t),\quad 0\leq t\leq T,$

with $a$ given in (5.3) and $\bm{B}$ is the integral operator induced by the
kernel $B$ in (5.3).

∎

##  6 Proofs of Theorems 2.7 and 2.8

This section is dedicated to the proofs of Theorems 2.7 and 2.8. In order to
ease the notation we omit the upper case $N$ from $u^{N},\bar{u}^{N},u^{i,N}$
ect. throughout this section.

We first establish the strict concavity property of $u^{i}\mapsto
J^{i,N}(u^{i};u^{-i})$ which is crucial for the derivation of the Nash
equilibrium. In order to ease notations, we drop the $N$ superscript in this
section.

######  Lemma 6.1.

Let $i\in\\{1,\ldots,N\\}$. Then, under Assumption 2.4, for any
$u^{-i}\in\mathcal{U}^{N-1}$ fixed, the functional $u^{i}\mapsto
J^{i}(u^{i};u^{-i})$ in (2.9) is strictly concave in $u^{i}\in\mathcal{U}$.

###### Proof.

Fix $i\in\\{1,\ldots,N\\}$, let $u^{i}\in\mathcal{U}$ and
$u^{-i}\in\mathcal{U}^{N-1}$. First, using the notation from (2.7), we have

$\displaystyle\langle\bar{u},\bm{A}_{1}\bar{u}\rangle_{L^{2}}=$
$\displaystyle\langle\frac{u^{i}}{N}+\bar{u}^{-i},\bm{A}_{1}(\frac{u^{i}}{N}+\bar{u}^{-i})\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\langle\frac{u^{i}}{N},\bm{A}_{1}\frac{u^{i}}{N}\rangle_{L^{2}}+\langle\frac{u^{i}}{N},\bm{A}_{1}\bar{u}^{-i}\rangle_{L^{2}}+\langle\bar{u}^{-i},\bm{A}_{1}\frac{u^{i}}{N}\rangle_{L^{2}}+\langle\bar{u}^{-i},\bm{A}_{1}\bar{u}^{-i}\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\langle\frac{u^{i}}{N},\bm{A}_{1}\frac{u^{i}}{N}\rangle_{L^{2}}+\frac{1}{N}\langle
u^{i},(\bm{A}_{1}+\bm{A}_{1}^{*})\bar{u}^{-i}\rangle_{L^{2}}+\langle\bar{u}^{-i},\bm{A}_{1}\bar{u}^{-i}\rangle_{L^{2}}.$

Hence, we can rewrite the objective functional in (2.9) as follows

$\displaystyle J^{i}(u^{i};u^{-i})$ (6.1)
$\displaystyle={\mathbb{E}}\left[-\langle\bar{u},\bm{A}_{1}\bar{u}\rangle_{L^{2}}-\langle
u^{i},\bm{A}_{2}u^{i}\rangle_{L^{2}}-\langle
u^{i},(\bm{A}_{3}+\bm{A}_{3}^{*})\bar{u}\rangle_{L^{2}}+\langle
b^{i},u^{i}\rangle_{L^{2}}+\langle b^{0},\bar{u}\rangle_{L^{2}}+c^{i}\right]$
$\displaystyle={\mathbb{E}}\Big{[}-\langle
u^{i},\big{(}\frac{\bm{A}_{1}}{N^{2}}+\frac{\bm{A}_{3}+\bm{A}_{3}^{*}}{N}+\bm{A}_{2}\big{)}u^{i}\rangle_{L^{2}}-\langle
u^{i},(\bm{A}_{3}+\bm{A}_{3}^{*}+\frac{\bm{A}_{1}+\bm{A}_{1}^{*}}{N})\bar{u}^{-i}\rangle_{L^{2}}$
$\displaystyle\hskip
30.0pt-\langle\bar{u}^{-i},\bm{A}_{1}\bar{u}^{-i}\rangle_{L^{2}}+\langle
b^{i}+\frac{1}{N}b^{0},u^{i}\rangle_{L^{2}}+\langle
b^{0},\bar{u}^{-i}\rangle_{L^{2}}+c^{i}\Big{]}.$

Then, for any $w^{i}\in\mathcal{U}$ such that $w^{i}\neq u^{i}$,
$d\mathbb{P}\otimes dt$-a.e. on $\Omega\times[0,T]$, and for all
$\varepsilon\in(0,1)$, a direct computation yields

$\displaystyle J^{i}(\varepsilon
u^{i}+(1-\varepsilon)w^{i};u^{-i})-\varepsilon
J^{i}(u^{i};u^{-i})-(1-\varepsilon)J^{i}(w^{i};u^{-i})$ (6.2)
$\displaystyle=\varepsilon(1-\varepsilon){\mathbb{E}}\left[\langle
u^{i}-w^{i},\big{(}\frac{\bm{A}_{1}}{N^{2}}+\frac{\bm{A}_{3}+\bm{A}_{3}^{*}}{N}+\bm{\hat{A}_{2}}+\lambda{\rm
id}\big{)}(u^{i}-w^{i})\rangle_{L^{2}}\right]$
$\displaystyle=\varepsilon(1-\varepsilon)\left({\mathbb{E}}\left[\langle
u^{i}-w^{i},\big{(}\frac{\bm{A}_{1}}{N^{2}}+\frac{\bm{A}_{3}+\bm{A}_{3}^{*}}{N}+\bm{\hat{A}_{2}}\big{)}(u^{i}-w^{i})\rangle_{L^{2}}\right]+\lambda{\mathbb{E}}[\|u^{i}-w^{i}\|_{L^{2}}^{2}]\right)$
$\displaystyle>0,$

where we used the fact that $\|u^{i}-w^{i}\|_{L^{2}}^{2}>0$,
$\mathbb{P}$-a.s., Remark (2.3) and (2.8). ∎

Let $i\in\\{1,...,N\\}$ and $u^{-i}\in\mathcal{U}^{N-1}$. From Lemma 6.1, it
follows that, the map $J^{i}(\cdot;u^{-i}):u^{i}\mapsto J^{i}(u^{i};u^{-i})$
of player $i$’s best response to all other players’ fixed strategies $u^{-i}$
is strictly concave hence it admits a unique maximiser characterised by the
critical point at which the Gâteaux derivative

$\displaystyle\langle\nabla J^{i}(u^{i},u^{-i}),h\rangle=\lim_{\epsilon\to
0}\frac{J^{i}(u^{i}+\epsilon h,u^{-i})-J^{i}(u^{i},u^{-i})}{\epsilon},$

vanishes for all $h\in\mathcal{U}$. Therefore, a control $u^{i}\in\mathcal{U}$
minimizes (2.9) if and only if $u^{i}$ satisfies the first order condition

$\displaystyle\langle\nabla J^{i}(u^{i},u^{-i}),h\rangle=0,\quad
h\in\mathcal{U},$ (6.3)

(see Proposition 2.1 of [13, Chapter II]).

Using (6.1) and (2.11) we obtain by an application of Fubini’s theorem, that

$\displaystyle\langle\nabla J^{i}(u^{i},u^{-i}),h\rangle$ (6.4)
$\displaystyle=\mathbb{E}\left[\langle h,\,-2\lambda
u^{i}-(\bm{G}+\bm{G}^{*})(u^{i})-(\bm{H}+\bm{H}^{*})(\bar{u}^{-i})+b^{i}\rangle_{L^{2}}\right]$
(6.5) $\displaystyle=\int_{0}^{T}\mathbb{E}\left[h_{t}\left(\,-2\lambda
u^{i}_{t}-(\bm{G}+\bm{G}^{*})(u^{i})(t)-(\bm{H}+\bm{H}^{*})(\bar{u}^{-i})(t)+b^{i}_{t}\right)\right]dt,$
(6.6)

for any $h\in\mathcal{U}$. By conditioning on $\mathcal{F}_{t}$ and using the
tower property we get from (2.7), (6.3) and (6.4) the following first order
condition,

$\displaystyle-2\lambda
u^{i}_{t}-(\bm{G}+\bm{G}^{*})({\mathbb{E}}_{t}u^{i})(t)-(\bm{H}+\bm{H}^{*})({\mathbb{E}}_{t}\bar{u}^{-i})(t)+b^{i}_{t}$
(6.7) $\displaystyle\quad=\,-2\lambda
u^{i}_{t}-\left(\bm{G}+\bm{G}^{*}-\frac{1}{N}(\bm{H}+\bm{H}^{*})\right)({\mathbb{E}}_{t}u^{i})(t)-(\bm{H}+\bm{H}^{*})({\mathbb{E}}_{t}\bar{u})(t)+b^{i}_{t}$
$\displaystyle\quad=0,\quad d\mathbb{P}\otimes dt\textrm{-a.e.~{}on
}\Omega\times[0,T].$

It follows that for the Nash equilibrium the following must hold for all
$i\in\\{1,\ldots,N\\}$,

$\displaystyle 2\lambda u_{t}^{i}=$
$\displaystyle\,b^{i}_{t}-\int_{0}^{t}H(t,r)\bar{u}_{r}dr-\int_{t}^{T}H(r,t){\mathbb{E}}_{t}\bar{u}_{r}dr-\int_{0}^{t}\left(G(t,r)-\frac{1}{N}H(t,r)\right)u^{i}_{r}dr$
(6.8)
$\displaystyle-\int_{t}^{T}\left(G(r,t)-\frac{1}{N}H(r,t)\right){\mathbb{E}}_{t}u^{i}_{r}dr,\quad
d\mathbb{P}\otimes dt\textrm{-a.e.~{}on }\Omega\times[0,T].$

By summing over $i$ in (6.8), using (2.12) and scaling by $1/N$ we get the
following equation for $\bar{u}$:

$\displaystyle 2\lambda\bar{u}_{t}=$
$\displaystyle\,\bar{b}_{t}+\frac{1}{N}b^{0}_{t}-\int_{0}^{t}\left(H(t,r)+G(t,r)-\frac{1}{N}H(t,r)\right)\bar{u}_{r}dr$
(6.9)
$\displaystyle\,-\int_{t}^{T}\left(G(r,t)+H(r,t)-\frac{1}{N}H(r,t)\right){\mathbb{E}}_{t}\bar{u}_{r}dr$
$\displaystyle=$
$\displaystyle\,\bar{b}_{t}+\frac{1}{N}b^{0}_{t}-\int_{0}^{t}\left(\frac{N-1}{N}H(t,r)+G(t,r)\right)\bar{u}_{r}dr$
$\displaystyle\,-\int_{t}^{T}\left(\frac{N-1}{N}H(r,t)+G(r,t)\right){\mathbb{E}}_{t}\bar{u}_{r}dr,\quad
d\mathbb{P}\otimes dt\textrm{-a.e.~{}on }\Omega\times[0,T].$

After establishing (6.9) we are now ready to prove Theorem 2.7.

###### Proof of Theorem 2.7.

We now turn into the derivation of (2.13). We apply Proposition 5.1 to (6.9)
for $\bar{u}$ instead of $v$. Specifically we have

$f_{t}=\frac{1}{2\lambda}\bar{b}_{t}+\frac{1}{N}b^{0}_{t},\quad
K=L=\frac{1}{2\lambda}\left(\frac{N-1}{N}H+G\right).$ (6.10)

In this case we have that ${\bm{D}}_{t}$ in (5.4) is of the form
$\frac{1}{2\lambda}\overline{\bm{D}}_{t}$ with

$\overline{\bm{D}}_{t}=2\lambda{\rm id}+\bm{K}_{t}+\bm{L^{*}}_{t}=2\lambda{\rm
id}+\frac{N-1}{N}(\bm{H}_{t}+\bm{H}_{t}^{*})+\bm{G}_{t}+\bm{G}^{*}_{t}.$
(6.11)

Note that $\bm{K}+\bm{L}^{*}=\bm{K}+\bm{K}^{*}$ is clearly self-adjoint. From
Assumption 2.4 and (2.11) it follows that $\bm{G},\bm{H}\in\mathcal{G}$,
therefore $\bm{K},\bm{L}\in\mathcal{G}$. Hence, together with (2.2) it follows
that the assumptions of Proposition 5.1 are satisfied and we have

$\displaystyle\bar{u}_{t}=\left(({\rm
id}-\overline{\bm{B}})^{-1}\overline{a}\right)(t),\quad 0\leq t\leq T.$ (6.12)

where

$\displaystyle\overline{a}_{t}$
$\displaystyle:=\frac{1}{2\lambda}\left(\bar{b}_{t}+\frac{1}{N}b^{0}_{t}-\left\langle
1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,t),\overline{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}\left[\bar{b}_{\cdot}+\frac{1}{N}b^{0}_{\cdot}\right]\right\rangle_{L^{2}}\right),$
(6.13) $\displaystyle\overline{B}(t,s)$
$\displaystyle:=1_{\\{s<t\\}}\frac{1}{2\lambda}\bigg{(}\left\langle
1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,t),\overline{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\overline{K}(\cdot,s)\right\rangle_{L^{2}}-\overline{K}(t,s)\bigg{)},$

with $\overline{K}:=2\lambda K$ for $K$ as in (6.10). This agrees with (2.13)
and completes the proof. ∎

###### Proof of Theorem 2.8.

Given the solution $\bar{u}$ from Theorem 2.7, we can now continue with
solving the first order condition (6.8) separately for each $u^{i}$,
$i\in\\{1,\ldots,N\\}$ and hence derive a Nash equilibrium to the game (2.7).

We apply Proposition 5.1 to (6.8) for $u^{i}$ instead of $v$. Specifically we
have

$\displaystyle f_{t}$
$\displaystyle=\frac{1}{2\lambda}\left({\mathbb{E}}_{t}\left[b^{i}_{t}+\frac{1}{N}b^{0}_{t}\right]-\int_{0}^{t}H(t,r)\bar{u}_{r}dr-\int_{t}^{T}H(r,t){\mathbb{E}}_{t}\bar{u}_{r}dr\right),$
(6.14) $\displaystyle L$
$\displaystyle=K=\frac{1}{2\lambda}\left(G-\frac{1}{N}H\right).$

From (6.14), (2.11) and (2.8) it follows that $\bm{L}$ and $\bm{K}$ in
Proposition 5.1 are given by

$\displaystyle\bm{L}=\bm{K}$
$\displaystyle=\bm{\hat{A}}_{2}+\frac{\bm{A}_{3}}{N}.$ (6.15)

Hence $(\bm{K}+\bm{L}^{*})$ is trivially self adjoint. By Assumption 2.4 it it
follows that $\bm{\hat{A}}_{2},\bm{A}_{3}\in\mathcal{G}$, therefore
$\bm{K},\bm{L}\in\mathcal{G}$. Hence the assumptions of Proposition 5.1 are
satisfied. Note that ${\bm{D}}_{t}$ in Proposition 5.1 is of the form
$\frac{1}{2\lambda}\widetilde{\bm{D}}_{t}$ with

$\displaystyle\widetilde{\bm{D}}_{t}$ $\displaystyle:=2\lambda{\rm
id}+\bm{K}_{t}+\bm{L^{*}}_{t}$ (6.16) $\displaystyle=2\lambda{\rm
id}+\bm{\hat{A}}_{2}+\bm{\hat{A}}^{*}_{2}+\frac{\bm{A}_{3}+\bm{A}_{3}^{*}}{N}.$

Using Proposition 5.1 we obtain for all $i\in\\{1,\ldots,N\\}$ the
representation

$\displaystyle\hat{u}^{i}_{t}=\left(({\rm
id}-\bm{B})^{-1}a^{i}\right)(t),\quad 0\leq t\leq T$

where

$\displaystyle a^{i}_{t}$
$\displaystyle:=\frac{1}{2\lambda}\bigg{(}{\mathbb{E}}_{t}\left[b^{i}_{t}+\frac{1}{N}b^{0}_{t}-((\bm{H}+\bm{H}^{*})\bar{u})(t)\right]$
(6.17)
$\displaystyle\qquad\qquad\qquad-\left\langle{1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,t)},\widetilde{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}{\mathbb{E}}_{t}\left[b^{i}_{\cdot}+\frac{1}{N}b^{0}_{\cdot}-((\bm{H}+\bm{H}^{*})\bar{u})(\cdot)\right]\right\rangle_{L^{2}}\bigg{)},$
$\displaystyle B(t,s)$
$\displaystyle:=1_{\\{s<t\\}}\frac{1}{2\lambda}\bigg{(}\left\langle
1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,t),\widetilde{\bm{D}}_{t}^{-1}1_{\\{t\leq\cdot\\}}\hat{K}(\cdot,s)\right\rangle_{L^{2}}-\hat{K}(t,s)\bigg{)},$

where $\hat{K}=\left(G-\frac{1}{N}H\right)$. This agrees with (2.15) and
(2.16).

Next, we argue for uniqueness of the Nash equilibrium. Assume that
$(w^{1},\ldots,w^{N})$ in $\mathcal{U}^{N}$ is another Nash equilibrium. Then,
by Definition 2.6 it holds that

$\quad J^{i}(w^{i};w^{-i})\geq J^{i}(v;w^{-i}),\quad\textrm{for all
}v\in\mathcal{U},\,i\in\\{1,\ldots,N\\}.$ (6.18)

In particular, for a fixed $i\in\\{1,\ldots,N\\}$, $w^{i}$ maximizes the
strictly concave functional $w\to J^{i}(w;w^{-i})$ (recall Lemma 6.1). This
means that (6.3) is satisfied with $w$ in place of $u$. It follows that
$\bar{w}$ satisfies the equation (6.9) with $\bar{u}$ replaced by $\bar{w}$.
By uniqueness of the solution to the Fredholm equation (6.9), see Proposition
5.1, we get $\bar{w}=\bar{u}$, $d\mathbb{P}\otimes dt-a.e.$ on
$\Omega\times[0,T]$. Similarly, we obtain $w^{i}=v^{i}$ from the uniqueness of
solutions to the Fredholm equation (6.8) where we replaced $u$ by $w$. This
concludes the proof of uniqueness. ∎

##  7 Proofs of the results from Section 4

Similar to the finite-player game in Section 2, we have the following strict
concavity result.

######  Lemma 7.1.

Under Assumption 2.4, for any $\mu\in\mathcal{U}^{0}$, the functional
$v\mapsto J(v;\mu)$ in (4.3) is strictly concave in $v\in\mathcal{U}$.

###### Proof.

This follows similar as in the proof of Lemma 6.1 above. Let $v\in\mathcal{U}$
and $\mu\in\mathcal{U}^{0}$. Then, for any $w\in\mathcal{U}$ such that $w\neq
v$ $d\mathbb{P}\otimes dt$-a.e. on $\Omega\times[0,T]$ and for all
$\varepsilon\in(0,1)$ a direct computation, combined with the definition of
$\bm{A}_{2}$ in (2.8), shows that

$\displaystyle J(\varepsilon v+(1-\varepsilon)w;\mu)-\varepsilon
J(v;\mu)-(1-\varepsilon)J(w;\mu)$ (7.1)
$\displaystyle=\varepsilon(1-\varepsilon){\mathbb{E}}\left[\langle
v-w,\bm{A}_{2}(v-w)\rangle_{L^{2}}\right]$
$\displaystyle=\varepsilon(1-\varepsilon)\Big{(}{\mathbb{E}}\left[\langle
v-w,\bm{\hat{A}}_{2}(v-w)\rangle_{L^{2}}\right]+\lambda{\mathbb{E}}[\|v-w\|_{L^{2}}^{2}]\Big{)}$
$\displaystyle>0,$

since ${\mathbb{E}}[\|v-w\|_{L^{2}}^{2}]>0$ and the operator
$\bm{\hat{A}}_{2}$ is nonnegative definite; recall Remark 2.3. ∎

We are now ready to proof Theorem 4.6.

###### Proof of Theorem 4.6.

We start with rewriting the cost functional $J$ in (4.3) as

$\displaystyle J(v;\mu)=$
$\displaystyle\,{\mathbb{E}}\left[-\langle\mu,\bm{A}_{1}\mu\rangle_{L^{2}}-\langle
v,\bm{A}_{2}v\rangle_{L^{2}}-\langle
v,(\bm{A}_{3}+\bm{A}_{3}^{*})\mu\rangle_{L^{2}}\right.$
$\displaystyle\left.\quad\,+\langle b,v\rangle_{L^{2}}+\langle
b^{0},\mu\rangle_{L^{2}}+c\right]$ $\displaystyle=$
$\displaystyle\,{\mathbb{E}}\left[-\langle
v,\bm{A}_{2}v\rangle_{L^{2}}+\langle\tilde{b},v\rangle_{L^{2}}+\hat{c}\right],$

where

$\displaystyle\tilde{b}$ $\displaystyle=b-(\bm{A}_{3}+\bm{A}_{3}^{*})\mu,$
(7.2) $\displaystyle\tilde{c}$ $\displaystyle=\langle
b^{0},\mu\rangle_{L^{2}}+c-\langle\mu,\bm{A}_{1}\mu\rangle_{L^{2}}.$

By the strict concavity obtained in Lemma 7.1 and similarly to the proof of
Theorem 2.7 in the finite-player game, observe that for a fixed process
$\mu\in\mathcal{U}^{0}$ the first order condition for the generic player’s
best response is equivalent to

$\displaystyle 0$
$\displaystyle=-(\bm{A}_{2}+\bm{A}_{2}^{*})({\mathbb{E}}_{t}v)(t)+{\mathbb{E}}_{t}\tilde{b}_{t}$
$\displaystyle=-2\lambda
v_{t}-(\bm{\hat{A}}_{2})(v)(t)-(\bm{\hat{A}}^{*}_{2})({\mathbb{E}}_{t}v)(t)+{\mathbb{E}}_{t}\tilde{b}_{t}$
(7.3) $\displaystyle=-2\lambda
v_{t}-\int_{0}^{t}\hat{A}_{2}(t,s)v_{s}ds-\int_{t}^{T}\hat{A}_{2}(s,t){\mathbb{E}}_{t}v_{s}ds+{\mathbb{E}}_{t}\tilde{b}_{t},\quad\textrm{for
all }t\in[0,T],$

where we used the fact that $\hat{A}_{2}$ is a Volterra kernel. In other
words, $\hat{v}$ is optimal in (4.4) (with $\mu\in\mathcal{U}^{0}$ fixed) if
and only if $\hat{v}$ satisfies

$\displaystyle 2\lambda\hat{v}_{s}=$
$\displaystyle\,{b}_{s}-\int_{0}^{s}A_{3}(s,r)\mu_{r}dr-\int_{s}^{T}A_{3}(r,s){\mathbb{E}}_{s}\mu_{r}dr$
(7.4)
$\displaystyle\,-\int_{0}^{s}\hat{A}_{2}(s,r)\hat{v}_{r}dr-\int_{s}^{T}\hat{A}_{2}(r,s){\mathbb{E}}_{s}\hat{v}_{r}dr.$

Taking conditional expectation with respect to ${\mathcal{F}^{0}_{T}}$, using
the independence between $\beta$ and $\beta^{0}$, the fact that $\mu$ is
$\mathbb{F}^{0}$-progressively measurable and (4.1), we get

$\displaystyle 2\lambda\mathbb{E}[{\hat{v}}_{s}|\mathcal{F}^{0}_{T}]=$
$\displaystyle\,\beta^{0}_{s}+\mathbb{E}[\beta_{s}]-\int_{0}^{s}A_{3}(s,r)\mu_{r}dr-\int_{s}^{T}A_{3}(r,s){\mathbb{E}}_{s}[\mu_{r}]dr$
(7.5)
$\displaystyle-\int_{0}^{s}\hat{A}_{2}(s,r)\mathbb{E}[{\hat{v}}_{r}|\mathcal{F}^{0}_{T}]dr-\int_{s}^{T}\hat{A}_{2}(r,s)\mathbb{E}_{s}[\mathbb{E}[{\hat{v}}_{r}|\mathcal{F}^{0}_{T}]]dr.$
(7.6)

Together with the consistency condition in (4.5), this suggests a candidate
equation for $\hat{\mu}$ given by

$\displaystyle 2\lambda\hat{\mu}_{s}=$
$\displaystyle\,\beta^{0}_{s}{+\mathbb{E}[\beta_{s}]}-\int_{0}^{s}\left(A_{3}(s,r)+\hat{A}_{2}(s,r)\right)\hat{\mu}_{r}dr$
(7.7)
$\displaystyle-\int_{s}^{T}\left(A_{3}(r,s)+\hat{A}_{2}(r,s)\right)\mathbb{E}_{s}[{\hat{\mu}}_{r}]dr.$
(7.8)

An application of Proposition 5.1 yields the existence of a solution
$\hat{\mu}$ to (7.7) given by (4.10). In particular,
$\hat{\mu}\in\mathcal{U}^{0}$ as desired. Moreover, inserting $\hat{\mu}$ into
equation (7.4) and applying once more Proposition 5.1 yields that the
corresponding $\hat{v}$ is of the form (4.9) and belongs to $\mathcal{U}$.
This proves that $(\hat{v},\hat{\mu})$ satisfy the requirement (4.4). To
justify that the consistency condition (4.5) is satisfied, simply observe that
equation (7.6) with $\hat{\mu}$ in place of $\mu$ shows that
$(\mathbb{E}[\hat{v}_{s}|\mathcal{F}^{0}_{T}])_{s\leq T}$ solves the same
equation as $\hat{\mu}$ in (7.7). By uniqueness of the solution to the
Fredholm equation, we deduce the consistency condition (4.5) is indeed
satisfied. Uniqueness of the mean-field game equilibrium follows from the
uniqueness of the corresponding stochastic Fredholm equations that
characterize the optimum, along the lines of the uniqueness proof of Theorem
2.8. ∎

Next, we address the proof of Theorem 4.11 in the infinite-player game
formulation in Section 4.1. First, analogous to Lemmas 7.1, we have the
following:

######  Lemma 7.2.

Let $i\in\mathbb{N}$. Under Assumption 2.4, for any $\nu\in\mathcal{U}$, the
functional $v^{i}\mapsto J^{i,\infty}(v^{i};\nu)$ in (4.11) is strictly
concave in $v^{i}\in\mathcal{U}$.

###### Proof.

This follows as in the proof of Lemma 7.1 above. ∎

We are now ready to prove Theorem 4.11

###### Proof of Theorem 4.11.

Similar to equation (7.4) in the proof of Theorem 4.6, we obtain that for
fixed $\nu\in\mathcal{U}$ the first order condition for player $i$’s best
response is given by

$\displaystyle 2\lambda{v^{i}}_{s}=$
$\displaystyle\,{b}^{i}_{s}-\int_{0}^{s}A_{3}(s,r)\nu_{r}dr-\int_{s}^{T}A_{3}(r,s){\mathbb{E}}_{s}[\nu_{r}]dr$
(7.9)
$\displaystyle-\int_{0}^{s}\hat{A}_{2}(s,r)v^{i}_{r}dr-\int_{s}^{T}\hat{A}_{2}(r,s){\mathbb{E}}_{s}v^{i}_{r}dr.$
(7.10)

Solving (7.9) for $v^{i}$ (with $\nu\in\mathcal{U}$ fixed) is an application
of Proposition 5.1 and we get for all $i\in\mathbb{N}$ the representation

$v_{t}^{i}=F(t,b^{i}-\bm{A}_{3}(\nu)-\bm{A}^{*}_{3}({\mathbb{E}}_{\cdot}\nu)).$
(7.11)

Next, we claim that the infinite-player Nash equilibrium’s mean-field strategy
$\hat{\nu}$ must satisfy the linear Volterra equation

$\displaystyle 2\lambda\hat{\nu}_{s}=$ $\displaystyle
b^{\infty}_{s}-\int_{0}^{s}\left(\hat{A}_{2}(s,r)+{A}_{3}(s,r)\right)\hat{\nu}_{r}dr$
(7.12)
$\displaystyle\,-\int_{s}^{T}(\hat{A}_{2}(r,s)+A_{3}(r,s)){\mathbb{E}}_{s}[\hat{\nu}_{r}]dr,$

where $b^{\infty}$ denotes the limit from Assumption 4.9. By virtue of
Proposition 5.1, the solution to (7.12) is given by

$\hat{\nu}_{t}=G(t,b^{\infty}).$ (7.13)

Then, plugging $\hat{\nu}$ in (7.13) back into (7.11) above will give us the
desired mean-field game equilibrium strategies in (4.19) and (4.20) in the
sense of Definition 4.8, which we denote with
$(\hat{\nu},\hat{v}^{i})_{i\in\mathbb{N}}\subset\mathcal{U}$.

In order to justify our claim, it remains to show that the consistency
condition in (4.14) is satisfied by $\hat{\nu}$ characterized by (7.12). To
this end, we define for all $N\in\mathbb{N}$ the averages

$\hat{\nu}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\hat{v}^{i}.$ (7.14)

Note that taking the average over $i$ in (7.9) (with $\hat{v}^{i}$ and
$\hat{\nu}$) yields that $\hat{\nu}^{N}$ solves the Volterra equation

$\displaystyle 2\lambda{\hat{\nu}^{N}}_{s}=$
$\displaystyle\,\frac{1}{N}\sum_{i=1}^{N}b^{i}_{s}-\int_{0}^{s}A_{3}(s,r)\hat{\nu}_{r}dr-\int_{s}^{T}A_{3}(r,s){\mathbb{E}}_{s}[\hat{\nu}_{r}]dr$
(7.15)
$\displaystyle\,-\int_{0}^{s}\hat{A}_{2}(s,r){\hat{\nu}^{N}}_{r}dr-\int_{s}^{T}\hat{A}_{2}(r,s){\mathbb{E}}_{s}{\hat{\nu}^{N}}_{r}dr.$
(7.16)

Hence, together with Assumption 4.9, the convergence in the consistency
condition in (4.14) follows from an application of the stability property of
Fredholm equations derived in Proposition 5.3 to equations (7.15) and (7.12).
Uniqueness of the infinite-player game equilibrium follows from the uniqueness
of the corresponding stochastic Fredholm equations that characterize the
optimum, along the lines of the uniqueness proof of Theorem 2.8. ∎

We finish with providing the proofs of Section 4.2 and start with the
convergence result in Theorem 4.14.

###### Proof of Theorem 4.14.

We only prove (4.21) as the proof of the convergence in (4.22) follows the
same lines. From (6.9) above we get that

$\bar{u}^{N}_{t}:=\frac{1}{N}\sum_{i=1}^{N}\hat{u}^{i,N}_{t}$

satisfies the Volterra equation in (5.1) with

$f^{N}_{t}=\frac{1}{2\lambda}\bar{b}_{t},\quad
K^{N}=L^{N}=\frac{1}{2\lambda}\left(\frac{N-1}{N}H+G\right).$ (7.17)

From (7.12) it follows that $\hat{\nu}$ satisfies (5.1) with

$f_{t}=\frac{1}{2\lambda}b^{\infty}_{t},\quad
K=L=\frac{1}{2\lambda}\left(\hat{A}_{2}+A_{3}\right).$ (7.18)

Emphasizing the dependence of $\bm{G},\bm{H}$ defined in (2.11) on $N$ we
write $\bm{G}^{N},\bm{H}^{N}$. From (2.11) it follows that there exists $C>0$
such that

$\displaystyle\|\bm{G}^{N}-\bm{\hat{A}}_{2}\|_{\textrm{op}}$
$\displaystyle\leq C(N^{-1}),\quad\textrm{for all }N\geq 1,$ (7.19)
$\displaystyle\|\bm{H}^{N}-\bm{A}_{3}\|_{\textrm{op}}$ $\displaystyle\leq
C(N^{-1}),\quad\textrm{for all }N\geq 1.$

Moreover, from (7.17), (7.18) and (7.19) we get that

$\displaystyle\sup_{0\leq t\leq
T}\int_{0}^{T}\big{(}K^{N}(t,s)-K(t,s)\big{)}^{2}ds$ $\displaystyle\leq
C(N^{-2}),\quad\textrm{for all }N\geq 1,$ (7.20) $\displaystyle\sup_{0\leq
s\leq T}\int_{0}^{T}\big{(}L^{N}(t,s)-L(t,s)\big{)}^{2}dt$ $\displaystyle\leq
C(N^{-2}),\quad\textrm{for all }N\geq 1,$

and Assumption 4.9 implies

$\displaystyle\sup_{t\leq
T}{\mathbb{E}}\big{[}(f^{N}_{t}-f_{t})^{2}\big{]}\leq h(N),\quad\textrm{for
all }N\geq 1.$ (7.21)

Therefore, the convergence rate in (4.21) follows by an application of
Proposition 5.3. ∎

Next, in order to prove Theorem 4.16 the first ingredient is following
auxiliary result.

######  Lemma 7.3.

Assume that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. Let
$(\hat{\nu},(\hat{v}^{i})_{i\in\mathbb{N}})$ be the equilibrium strategies of
the mean-field game in the sense of Definition 4.8. Then we have

$\sup_{0\leq t\leq
T}{\mathbb{E}}\left[\left(\frac{1}{N}\sum_{i=1}^{N}\hat{v}_{t}^{i}-\hat{\nu}_{t}\right)^{2}\right]=O\left(h(N)\right),$
(7.22)

where $h$ is given in (4.15).

###### Proof.

From (7.9) and (7.12) it follows that $\hat{\nu}$ satisfies (5.1) with

$f_{t}=\frac{1}{2\lambda}b^{\infty}_{t},\quad
K=L=\frac{1}{2\lambda}\left(\hat{A}_{2}+A_{3}\right),$ (7.23)

and $\frac{1}{N}\sum_{i=1}^{N}\hat{v}_{t}^{i}$ satisfies (5.1) with

$f^{N}_{t}=\frac{1}{2\lambda}\frac{1}{N}\sum_{i=1}^{N}b^{i}_{t},\quad
K=L=\frac{1}{2\lambda}\left(\hat{A}_{2}+A_{3}\right).$ (7.24)

By applying Proposition 5.3 using (4.15) we get (7.22). ∎

The second ingredient is Lemma 7.4 below which provides a bound on the
difference between the performance functional $J^{i,\infty}$ of the mean-field
game in (4.11) and the $N$-player game’s performance functional $J^{i,N}$ in
(2.9).

######  Lemma 7.4.

Suppose that (2.2) as well as Assumptions 2.4 and 4.9 are satisfied. Let
$(\hat{\nu},(\hat{v}^{i})_{i\in\mathbb{N}})$ be the equilibrium strategies of
the mean-field game in the sense of Definition 4.8. For any $N\geq 2$ and
$1\leq i\leq N$, define
$\hat{v}^{-i}=(\hat{v}^{1},\ldots,\hat{v}^{i-1},\hat{v}^{i+1},\ldots,\hat{v}^{N})$.
Then, there exists a constant $C>0$ independent of $N$ such that for all
$u\in\mathcal{U}$ and $i\in\mathbb{N}$ we have

$\left|J^{i,N}(u,\hat{v}^{-i})-{J}^{i,\infty}(u,\hat{\nu})\right|\leq
C\|u\|_{2,T}(1+\|u\|_{2,T})\left(h(N)\vee N^{-1}\right).$

###### Proof.

Let $u\in\mathcal{U}$. From (2.9) and (4.11) we get

$\displaystyle\left|J^{i,N}(u;\hat{v}^{-i})-J^{i,\infty}(u;\hat{\nu})\right|$
(7.25)
$\displaystyle\leq\left|{\mathbb{E}}\left[\left\langle\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right),\bm{A}_{1}\left(\frac{1}{N}\big{(}\sum_{j\not=i}\hat{v}^{j}+u\big{)}\right)\right\rangle_{L^{2}}-\langle\hat{\nu},\bm{A}_{1}\hat{\nu}\rangle_{L^{2}}\right]\right|$
$\displaystyle\quad+\left|{\mathbb{E}}\left[\left\langle
u,(\bm{A}_{3}+\bm{A}_{3}^{*})\left(\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right)\right\rangle_{L^{2}}\right]\right|$
$\displaystyle\quad+\left|{\mathbb{E}}\left[\left\langle
b^{0},\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right\rangle_{L^{2}}\right]\right|$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

Using Fubini’s Theorem, Jensen and Hölder inequalities we get for the second
term $I_{2}$ in (7.25):

$\displaystyle I_{2}$ $\displaystyle=\left|{\mathbb{E}}\left[\left\langle
u,(\bm{A}_{3}+\bm{A}_{3}^{*})\left(\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right)\right\rangle_{L^{2}}\right]\right|$
(7.26) $\displaystyle\leq
C\|u\|_{2,T}\|\bm{A}_{3}+\bm{A}_{3}^{*}\|_{op}\left\|\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right\|_{2,T}$
$\displaystyle\leq
C\|u\|_{2,T}\|\bm{A}_{3}+\bm{A}_{3}^{*}\|_{op}\left(\left\|\frac{1}{N}\sum_{j=1}^{N}\hat{v}^{j}-\hat{\nu}\right\|_{2,T}+\frac{1}{N}\left\|\hat{v}^{i}-u\right\|_{2,T}\right)$
$\displaystyle\leq C(h(N)^{1/2}\vee
N^{-1})\|u\|_{2,T}\|\bm{A}_{3}+\bm{A}_{3}^{*}\|_{op}\left(1+\|\hat{v}^{i}\|_{2,T}+\|u\|_{2,T}\right)$
$\displaystyle\leq C(h(N)^{1/2}\vee N^{-1})(1+\|u\|_{2,T})\|u\|_{2,T},$

where we used Lemma 7.3 and the fact that $\hat{v}^{i}\in\mathcal{U}$ in the
last two inequalities. Similarly, using once more Fubini’s Theorem, Jensen and
Hölder inequalities we get for the third $I_{3}$ in (7.25):

$\displaystyle I_{3}$ $\displaystyle=\left|{\mathbb{E}}\left[\left\langle
b^{0},\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right\rangle_{L^{2}}\right]\right|$
(7.27)
$\displaystyle\leq\|b^{0}\|_{2,T}\left\|\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu}\right\|_{2,T}$
$\displaystyle\leq\|b^{0}\|_{2,T}\left(\left\|\frac{1}{N}\sum_{j=1}^{N}\hat{v}^{j}-\hat{\nu}\right\|_{2,T}+\frac{1}{N}\left\|\hat{v}^{i}-u\right\|_{2,T}\right)$
$\displaystyle\leq C(h(N)^{1/2}\vee
N^{-1})\|b^{0}\|_{2,T}\left(1+\|\hat{v}^{i}\|_{2,T}+\|u\|_{2,T}\right)\|u\|_{2,T}$
$\displaystyle\leq C(h(N)^{1/2}\vee
N^{-1})\|b^{0}\|_{2,T}\left(1+\|u\|_{2,T}\right)\|u\|_{2,T},$

where we used again Lemma 7.3 and the fact that $\hat{v}^{i}\in\mathcal{U}$ in
the last two inequalities.

Lastly, for the first term $I_{1}$ in (7.25) we obtain

$\displaystyle I_{1}=$
$\displaystyle\,\left|{\mathbb{E}}\left[\left\langle\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right),\bm{A}_{1}\left(\frac{1}{N}\big{(}\sum_{j\not=i}\hat{v}^{j}+u\big{)}\right)\right\rangle_{L^{2}}-\langle\hat{\nu},\bm{A}_{1}\hat{\nu}\rangle_{L^{2}}\right]\right|$
(7.28) $\displaystyle\leq$
$\displaystyle\,\left|{\mathbb{E}}\left[\left\langle\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right),\bm{A}_{1}\left(\frac{1}{N}\big{(}\sum_{j\not=i}\hat{v}^{j}+u\big{)}-\hat{\nu}\right)\right\rangle_{L^{2}}\right]\right|$
$\displaystyle+\left|{\mathbb{E}}\left[\left\langle\frac{1}{N}\left(\sum_{j\not=i}\hat{v}^{j}+u\right)-\hat{\nu},\bm{A}_{1}\hat{\nu}\right\rangle_{L^{2}}\right]\right|.$

Once can repeat similar steps as in (7.26) and (7.27) by using Lemma 7.3 in
order to get

$\displaystyle I_{3}$ $\displaystyle\leq C(h(N)^{1/2}\vee
N^{-1})\left(1+\|u\|_{2,T}\right)\|u\|_{2,T}.$ (7.29)

Then, from (7.26), (7.27) and (7.29) we get the desired upper bound in (7.25).
∎

We are now ready to prove Theorem 4.16.

###### Proof of Theorem 4.16.

First, note that the inequality

$J^{i,N}(\hat{v}^{i};\hat{v}^{-i})\leq\sup_{u\in\mathcal{U}}J^{i,N}(u;\hat{v}^{-i})$

holds trivially by the definition of the supremum. Next, by virtue of Lemma
7.4 we get for any $u\in\mathcal{U}$ the upper bound

$\displaystyle J^{i,N}(u;\hat{v}^{-i})$ $\displaystyle\leq
J^{i,\infty}(u,\hat{\nu})+C(h(N)^{1/2}\vee N^{-1})\|u\|_{2,T}(1+\|u\|_{2,T})$
(7.30) $\displaystyle\leq J^{i,\infty}(\hat{v}^{i},\hat{\nu})+C(h(N)^{1/2}\vee
N^{-1})\|u\|_{2,T}(1+\|u\|_{2,T}),$

where $C>0$ is a constant not depending on $u$ or $N$ and we used
$J^{i,\infty}(\hat{u}^{i},\hat{\nu})=\sup_{u\in\mathcal{U}}J^{i,\infty}(u,\hat{\nu})$
in the second inequality. Moreover, using Lemma 7.4 again we get for some
constant $\tilde{C}>0$ the upper bound

$\displaystyle J^{i,\infty}(\hat{v}^{i},\hat{\nu})$ $\displaystyle\leq
J^{i,N}(\hat{v}^{i},\hat{v}^{-i})+C(h(N)^{1/2}\vee
N^{-1})\|\hat{v}^{i}\|_{2,T}(1+\|\hat{v}^{i}\|_{2,T})$ (7.31)
$\displaystyle\leq J^{i,N}(\hat{v}^{i},\hat{v}^{-i})+\tilde{C}(h(N)^{1/2}\vee
N^{-1}),$

where we used Lemma 7.3 and the fact that $\hat{v}^{i}\in\mathcal{U}$ in the
second inequality. Finally, using (7.31) to bound the right hand side of
(7.30), we get

$J^{i,N}(u;\hat{v}^{-i})\leq
J^{i,N}(\hat{v}^{i},\hat{v}^{-i})+C(h(N)^{1/2}\vee
N^{-1})\|u\|^{2}_{2,T}(1\vee\|u\|^{2}_{2,T})$

for all $u\in\mathcal{U}$ which completes the proof. ∎

## Acknowledgments

We would like to thank Fabrice Djete for a number of useful comments and
suggestions that significantly improved this paper.

## References

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