\section{Introduction}\label{sec:introduction}
\noindent Mean-field games (MFGs for short) are, loosely speaking, limits of symmetric stochastic differential games with a large number of players, where each one of them interacts with the average behaviour of his/her competitors. They were introduced in the seminal papers by \citet{lasry-lions2006a,lasry-lions2006b,lasry-lions2007} and, simultaneously, by \citet{huang2006}. An increasing stream of research has been flourishing since then, producing theoretical results as well as a wide range of applications in many fields such as economics, finance, crowd dynamics and social sciences in general. For an excellent presentation of the theory we refer to the lecture notes by \citet{cardaliaguet} and the two-volume monograph by \citet{carmonadelarue}.\medskip
	
\textit{Motivation.}\quad In most of the literature on MFGs, all players stay in the game until the end of the period, while in many applications, especially in economics and finance, it is natural to have a mechanism deciding when some player has to leave. Such a mechanism can be modelled by introducing an absorbing boundary for the state space as in \citet{campi}, which is the starting of our study (other related references will be discussed later in detail). Therein, existence of solutions of the MFG and of approximate Nash equilibria for the $N$-player games were provided under some boundedness assumptions on the coefficients and without including the effect of past absorption on the survivors behaviour. The present paper continues the investigation of this kind of games, with two main extensions: \begin{itemize}
\item[(i)] We introduce a direct dependence on past absorptions in the drift of the Stochastic Differential Equations (SDEs) describing the evolution of the players' states, so that the proportion of the absorbed players have an effect on the future time evolution of the survivors. This feature was not present in \citet{campi}, where the empirical measure of the survivors was re-normalized at each time. Such a dependence on past absorption is also included in the costs.  
\item[(ii)] We allow both the drift and the cost functional of the players to have linear growth, hence they are not necessarily bounded unlike in \citet{campi}. Moreover, the set of absorbing states $\mathcal O$ can be unbounded. Dropping the boundedness of the game data increases the flexibility of our setting, which can include more realistic dynamics from the viewpoint of applications (for more details, see later in this introduction). 
\end{itemize}

\indent To be more precise, the purpose of this paper is to study $N$-player games and related MFGs in the presence of an absorbing set (i.e. a player is eliminated from the game once his/her private state process leaves a given open set $\mathcal{O} \in \mathbb{R}^{d}$), and where the vector of private states $\textbf{X}^{N} = (X^{N,1}, \ldots, X^{N,N})$ is assumed to evolve according to:
\begin{eqnarray} 
X^{N,i}_t = X^{N,i}_0 &+& \int_{0}^{t}\round{u^{N,i}\round{s,\textbf{X}^N} + \bar{b}\round{s, X^{N,i}_s, L^N_s, \int_{\mathbb{R}^d}w(y)\mu^N_s(dy)}}\,ds\nonumber\\					    &+& \sigma W^{N,i}_t, \,\,\, t \in \brackets{0,T},\, i \in \braces{1,\ldots ,N},
\label{eq:U:SDE}
\end{eqnarray}
\noindent where $\textbf{\textit{u}}^N =(u^{N,1},\ldots ,u^{N,N})$ is a vector of feedback strategies, $W^{N,1},\ldots,W^{N,N}$ are independent $d$-dimensional Wiener processes defined on some filtered probability space, $\sigma$ is some (non-degenerate) diffusion matrix, and $\bar{b}$, $w$ are given deterministic functions.  Additionally, $\mu^{N}_t$ is the (random) empirical \textit{sub-probability} measure on $\mathbb{R}^{d}$ representing the (empirical) density of survivors at time $t \in [0,T]$, and $L_{t}^{N}$ is a counting process representing the fraction of players who left $\mathcal{O}$ by time $t\in[0,T]$:
\begin{eqnarray}
\mu_t^N\round{\cdot} \doteq \frac{1}{N}\sum_{i = 1}^{N} \delta_{X_t^{N,i}}\round{\cdot}\mathbf{1}_{[0,\tau^{X^{N,i}})}\round{t}\quad\text{and}\quad L_t^{N} \doteq \frac{1}{N}\sum_{i = 1}^{N} \mathbf{1}_{[\tau^{X^{N,i}},\infty)}\round{t},
\nonumber
\end{eqnarray}
\noindent where $\tau^{X^{N,i}} \doteq \inf\{t \in[0,T] : X^{N,i}_t \not\in \mathcal{O}\}$ is the time of first exit of $X^{N,i}$ from $\mathcal{O}$, with the convention $\inf\emptyset = \infty$. Each player evaluates a strategy vector $\textbf{\textit{u}}^N = (u^{N,1},\ldots ,u^{N,N})$ according to his/her expected costs over a random time horizon:
\begin{eqnarray}
J^{N,i}\round{\textbf{\textit{u}}^N} \doteq \mathbb{E}\Biggl[\int_{0}^{\tau^{N,i}}\bar{f}\round{s,X^{N,i}_s,L_s^N,\int_{\mathbb{R}^d}w(y)\mu^N_s(dy),u^{N,i}(s,\textbf{X}^N)}ds\nonumber\\
+F\round{\tau^{N,i},X^{N,i}_{\tau^{N,i}}}\Biggr].
\label{eq:U:Cost}
\end{eqnarray}
\noindent In Eq.\eqref{eq:U:Cost}, $\textbf{X}^N$ is the $N$-player dynamics under $\textbf{\textit{u}}^N$ and $\tau^{N,i} \doteq \tau^{X^{N,i}} \wedge T$.  In the present work, we are interested in drifts $\bar{b}$ and cost functional $\bar{f}$ with sub-linear growth, hence possibly unbounded. Further details on the setting with all the technical assumptions will be given in Section \ref{sec:preliminaries}.\\
\indent As we have anticipated before, the dynamics above is (also) motivated by economic models for corporate finance, systemic risk, and asset allocation. For instance, we can interpret players as firms whose values are represented by the state variables $X^{N,i}$ for $i\in \{1,\ldots,N\}$. Each company is affected by the fraction of both defaulted and non-defaulted firms through the processes $L^{N}$ and $\mu^{N}$ respectively, and takes strategic decisions accordingly. Moreover, sub-linearity of the drift allows to include a mean-reversion term representing the trend to follow the crowd. A possible application is the pricing of portfolio credit derivatives where the pricing depends upon the so called distance-to-default of the assets in the portfolio (\citet{hambly2017stochastic}). Alternatively, each player can be interpreted as a bank, whose monetary reserve evolves according to the stochastic dynamics in Eq.\eqref{eq:U:SDE} where the drift depends on both the rate of interbank borrowing/lending and on a controlled borrowing/lending to a central bank, as in \citet{carmona2015mean}. However, in \citet{carmona2015mean} no absorbing boundary conditions and no dependence on the loss process are considered. The latter features could be incorporated in the model by introducing absorbing boundary conditions at the default level, similarly to \citet{hambly2017stochastic}. This would enable to study the impact of defaults on systemic risk and stability of the financial system described by the game. Last but not least, the proposed set-up allows for a Brownian motion with an Ornstein–Uhlenbeck type drift modelling for the private state, a model that has been used (for instance) for the notion of flocking to default in the financial literature (\citet{fouque2013systemic}). However, in the present paper we focus on the mathematical properties of the proposed family of games and we leave the applications for future research.\medskip

\noindent\textit{Main results.}\quad The main contributions of the paper can be summarized as follows:
\begin{itemize}
\item We introduce the MFG with smooth dependence on past absorptions, i.e. the limit model corresponding to the above $N$-player games as $N$ tends to infinity. For a solution of the MFG, the empirical densities of survivors $\mu_t^N$  are replaced by a flow of sub-probability measures. See Definition \ref{def:MFGsol}.
\item We prove existence of a relaxed feedback solution of the MFG with smooth dependence on past absorptions and, under an additional assumption, we show that the optimal feedback strategies are in strict form; see Theorem \ref{teo:existenceRelFeedSolMFG}, Proposition \ref{prop:existenceRelFeedlSolMFG} and Proposition \ref{prop:existenceStrictFeedlSolMFG}. Additionally, we show  that there exist  relaxed and strict feedback solutions that are Markovian up to the exit time; see Lemma \ref{lem:existencemarkovSolMFG}.
\item We show that if we have a feedback solution of the MFG (either relaxed or strict) we can construct a sequence of approximate Nash equilibria for the corresponding $N$-player games with a vanishing approximation error as $N \rightarrow \infty$; see Theorem \ref{teo:approxNashRel} and Corollary \ref{cor:approxNash}.
\end{itemize}

\noindent The proof of the existence of feedback solutions of the MFG is inspired by the work of \citet{lacker} in the sense that we proceed by approximation. We construct a sequence of approximating MFGs with bounded drift and cost functional to which we can apply the results of \citet{campi}. Then, we prove convergence of the solutions of these approximating MFGs to a solution of the original one. Nonetheless, the procedure in \citet{lacker} cannot be applied directly to our case mainly due to the history dependence and the discontinuities of the coefficients induced by past absorptions. In particular, a different instance of the mimicking result of \citet{brunick} applies to our case.\\ 
\indent The proof of the existence of approximate Nash equilibria for the $N$-player game is based on weak convergence arguments and controlled martingale problems. The use of martingale problems in proving convergence to the  McKean-Vlasov limit and propagation of chaos for weakly interacting systems goes back to \citet{funaki1984certain}, \citet{oelschlager1984martingale} and \citet{meleard1996asymptotic}. We observe that, whereas standard results prove convergence in law of the empirical measures, in the present paper we follow the approach of \citet{lacker2018} to obtain a strong form of propagation of chaos with possibly unbounded and path-dependent drift. We show that the empirical measures converge in a stronger topology (the $\tau$-topology), a result that enables us to take the limit as $N \rightarrow \infty$ even if strategies are (only) progressively measurable functionals of the state trajectories. In particular, in our framework continuity of the MFG solution with respect to the state variable is no longer feasible due to the possibly unboundedness of the coefficients, whereas the proof of \citet{campi} relies strongly on this property.

\medskip

\noindent \textit{Related literature.}\quad We have already discussed the paper of \citet{campi}, so here we focus on some other recent contributions in the literature of mean-field models and games related to ours. First, we cite the works of \citet{giesecke2013default} and \citet{giesecke2015large} where a point process model of correlated defaults timing in a portfolio of firms is introduced. A mean-reverting jump-diffusion dynamics that depends on the default rate in the pool describes the intensity with which a firm defaults. Every time one firm defaults a jump occurs in the default rate of the pool and hence in the intensities of the survivors.  \citet{giesecke2013default} prove a LLN for the default rate as the number $N$ of firms goes to infinity. 

Motivated by modelling the contagion effect are the works of \citet{ hambly2017stochastic}, \citet{hambly2018mckean} and \citet{hambly2018spde} too. In the first work the authors prove a LLN for the empirical measure of a system of finitely many (uncontrolled) diffusions on the half-line, absorbed when they hit zero and correlated through the proportion of the processes that have been absorbed. In \citet{hambly2018mckean} extended their model by including a positive feedback mechanism when the particles hit the barrier, thus modelling contagious blow-ups. A mathematical complement to the previous work is provided in \citet{ledger2018uniqueness}. More recently,  \citet{hambly2018spde} have proposed a general model for systemic (or macroscopic) events. By working on a set-up similar to \citet{hambly2017stochastic}, they interpret the diffusions as distances-to-default of financial institutions and model the correlation effect through a common source of noise and a form of mean-reversion in the drift.  A form of endogenous contagion mechanism is also considered. 

On the side of applications to economics, \citet{chan2015bertrand} and \citet{chan2017fracking} study oligopolistic models with exhaustible resources formulated as MFGs with absorption at zero. In their model they keep track of the fraction of active players at each time. However, this fraction appears in the objective functions but not in the state variable.  

Two more papers are those by \citet{delarue2015global,delarue2015particle}, where a particle system approach is used to study the mathematical properties of an integrate-and-fire model from neurology. The particles' dynamics have some resetting mechanism which activates as soon as some particle hits a given threshold.

We conclude this review with the two very recent papers by \citet{nadtochiy2018mean,nadtochiy2019particle}. The first one focuses on the cascade effect in a new interbank mean-field model with defaults and a contagion effect modelled via a singular interaction through hitting times. The second one investigates the associated mean-field game also including more general dynamics and connection structures.

\medskip

\noindent \textit{Structure of the paper.}\quad In Section \ref{sec:preliminaries} we introduce the notation and present both the $N$-player and the mean-field dynamics along with the main assumptions. Section \ref{sec:mfg} contains results on the existence of feedback solutions of the MFG. In Section \ref{sec:Nplayer} we construct approximate Nash equilibria for the $N$-player game from solutions of the limit problem. The technical results used in the paper can be found in the Appendix \ref{app:Appendix}.
\section{Preliminaries and assumptions}\label{sec:preliminaries}
\noindent In the present section, we provide the definitions of the different spaces of trajectories and measures used in the paper along with the corresponding topologies, distances and notions of convergence.  We describe the setting of $N$-player games with smooth dependence on past absorptions and the corresponding MFG by stating the main assumptions. In particular, we give the definition of $\epsilon$-Nash equilibrium for the $N$-player game and that of solution of the MFG. We conclude the section by extending the notion of control to that of relaxed control and adapt the definitions of Nash equilibrium and MFG solution accordingly.\\

\noindent \textit{Spaces of trajectories.}\quad Let $d\in\mathbb{N}$. We denote by $\mathcal{O}\subset\mathbb{R}^d$ an open subset of $\mathbb{R}^d$ representing the space of the players' private states and by $\mathcal{X}\doteq C([0,T];\mathbb{R}^d)$ the space of $\mathbb{R}^d$-valued continuous trajectories on the time interval $[0,T]$, $T<\infty$. The space $\mathbb{R}^d$ is equipped with the standard Euclidean norm, always indicated by $|\cdot|$,  while $\mathcal{X}$ with the sup-norm, denoted by $\infnorm{\cdot}$, which makes $\mathcal{X}$ separable and complete. We use the notation $\| \cdot\|_{\infty,t}$ whenever the sup-norm is computed over the time interval $[0,t]$, $t < T$. Besides, we denote with $\mathcal{X}^N\doteq C([0,T];\mathbb{R}^{d\times N})$ the space of $N$-dimensional vectors of continuous trajectories and identify it with $\mathcal{X}^{\times N}$.\\

\noindent \textit{Spaces of measures.} We use flows of probability and sub-probability measures to describe the density of players and its time evolution in $\mathcal{O}$. For $E$ a Polish space, let $\mathcal{M}_f(E)$ denote  the space of finite Borel measures on $E$, $\mathcal{P}(E)$ the space of Borel probability measures on $E$ and $\mathcal{M}_{\leq 1}(E)$ the space of Borel sub-probability measures on $E$, i.e. measures $\mu \in \mathcal{M}_f(E)$ such that $\mu(E)\leq 1$. These spaces are endowed with the weak convergence of measures (\citet{billingsley}). We define by $\Upsilon_{\mathcal{P}}^{T}(E)$ (resp. by $\Upsilon_{\leq 1}^{T}(E)$) the spaces of measurable flows of probability (resp. sub-probability) measures on $E$, i.e. the space of Borel measurable maps $\pi$ (resp. $\mu$) from the time interval $[0,T]$ to $\mathcal{P}(E)$ (resp. $\mathcal{M}_{\leq 1}(E)$). Wherever possible without confusion, we use $\Upsilon_{\mathcal{P}}^{T}$ (resp. $\Upsilon_{\leq 1}^{T}$) when $E=\mathbb{R}^d$. We denote by $\mathcal{P}_1(E)$ and by $\mathcal{M}_{\leq 1,1}(E)$ the following subsets of $\mathcal{P}(E)$ and $\mathcal{M}_{\le 1}(E)$:
\begin{eqnarray} 
\mathcal{P}_1\rbr{E}&\doteq &\cbr{\pi\in\mathcal{P}\rbr{E}:\int_E d_E(x,x_0)\pi(dx) <\infty\text{ for some }x_0\in E},\nonumber\\
\mathcal{M}_{\leq 1,1}\rbr{E}&\doteq &\cbr{\mu\in\mathcal{M}_{\leq 1}\rbr{E}:\int_E d_E(x,x_0)\mu(dx) <\infty\text{ for some }x_0\in E}.
\nonumber
\label{eq:firstMom}
\end{eqnarray}
\noindent We endow $\mathcal{P}_1(E)$ with the $1$-Wasserstein distance $W_1$:
\begin{eqnarray} 
W_1(\mu,\nu)\doteq\inf_{\pi\in\Pi(\mu,\nu)}\int_{E\times E}d_E\rbr{x,y}d\pi(x,y)=\sup_{f\in\text{Lip}_1\rbr{E;\mathbb{R}}}\int_{E}f(x)d(\mu-\nu)(x),
\label{eq:was1Dist}
\end{eqnarray}
\noindent where $\Pi(\mu,\nu) \subset \mathcal{P}_1(E\times E)$ represents the set of probability measures with given marginals $\mu$ and $\nu$, and $\text{Lip}_1(E;\mathbb{R})$ the set of Lipschitz functions on $E$ with unitary Lipschitz constant. The second equality in Eq.\eqref{eq:was1Dist} is due to the Kantorovich-Rubinstein Theorem  (see, for instance, Theorem 6.1.1 in \citet{ambrosiogigli}). Notice that $(\mathcal{P}_1(E),W_1)$  is a separable and complete metric space whenever $(E,d_E)$ is separable and complete. Finally, let $\Upsilon_{\mathcal{P},1}^{T}(E)$ (resp. $\Upsilon_{\leq 1,1}^{T}(E)$) denote the space of measurable flows of probability measures in $\mathcal{P}_1(E)$ (resp. in $\mathcal{M}_{\leq 1,1}(E)$). Again,  wherever possible without confusion, we use  $\Upsilon_{\mathcal{P},1}^{T}$ and $\Upsilon_{\leq 1,1}^{T}$ when $E=\mathbb{R}^d$.\\


\medskip
\noindent Now, let $\mathcal{O} \subset \mathbb{R}^{d}$ be a non-empty open set, the set of non-absorbing states and let $\Gamma\subset\mathbb{R}^{d}$ the set of control actions. Consider $w: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d_0}$, $d_0 \in \mathbb{N}$, and
\begin{eqnarray} 
&&\bar{b}:\brackets{0,T} \times \mathbb{R}^{d} \times \mathbb{R}^{d_0} \rightarrow \mathbb{R}^{d},\quad\quad\quad\quad\quad\,\,\, \sigma \in \mathbb{R}^{d \times d},\nonumber\\
&& \bar{f}:\brackets{0,T} \times \mathbb{R}^{d} \times  \mathbb{R}^{d_0} \times \Gamma \rightarrow [0,\infty),\quad\quad F:\brackets{0,T} \times  \mathbb{R}^{d} \rightarrow [0,\infty)
\nonumber
\end{eqnarray}
as in Eq.\eqref{eq:U:SDE} and \eqref{eq:U:Cost}. We make the following assumptions:
\begin{itemize}
\item[$\round{\mathrm{H1}}$] $w$  is continuous and with sub-linear growth, i.e. $|w(x)| \leq C(1+|x|)$ for all $x \in \mathbb{R}^d$ and for some constant $C>0$.
\item[$\round{\mathrm{H 2}}$] $\bar{b}$, the drift, is jointly continuous and satisfies the following uniform Lipschitz continuity: 
\begin{eqnarray} 
\sup_{t\in[0,T]}\modl{\bar{b}\rbr{t,x,l,m}-\bar{b}\rbr{t,x',l',m'}}\leq L\rbr{\modl{x-x'}+\modl{l-l'}+\modl{m-m'}}
\nonumber
\label{eq:driftLip}
\end{eqnarray}
\noindent for all $(x,l,m),(x',l',m')\in\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}$. Moreover it has sub-linear growth, i.e.
\begin{eqnarray} 
\modl{\bar{b}\rbr{t,x,l,m}}\leq C\rbr{1+\modl{x}+\modl{m}}\quad \forall (t,x,l,m)\in[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}.
\nonumber
\label{eq:driftLin}
\end{eqnarray}
\item[$\round{\mathrm{H3}}$] $\bar{f}$, the running cost, and $F$, the terminal cost, are jointly continuous with sub-linear growth:
\begin{eqnarray}
\modl{\bar{f}(t,x,l,m,u)}&\leq & C\rbr{1+\modl{x}+\modl{m}}\nonumber \\
\modl{F(t,x)}&\leq & C\rbr{1+\modl{x}}
\nonumber
\end{eqnarray}
\noindent for all $(t,x,l,m,u)\in[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\Gamma$.
\item[$\round{\mathrm{H4}}$] The set $\mathcal{O}$ is open, convex and strictly included in $\mathbb{R}^d$ with $\mathcal{C}^2$-boundary, i.e. $\partial\mathcal{O}$ is the graph of a $\mathcal{C}^2$ function. Alternatively, $\mathcal{O}=(0, \infty)^{\times d}$ is also allowed.
\item[$\round{\mathrm{H5}}$] The set $\Gamma\subset\mathbb{R}^d$ is compact and convex.
\item[$\round{\mathrm{H6}}$] $\nu\in\mathcal{P}(\mathbb{R}^d)$, the initial distribution, satisfies $\int_{\mathbb{R}^d}\text{e}^{\lambda|x|^2}\nu(dx)<\infty$ for some $\lambda >0$.
\item[$\round{\mathrm{H7}}$] $\sigma\in\mathbb{R}^{d\times d}$, the diffusion matrix, is positive definite (full rank).
\end{itemize}
\noindent \textit{The $N$-player dynamics.}  Let $N \in \mathbb{N}$ be the number of players. We assume that the players' private states evolve according to the following system of $N$ $d$-dimensional SDEs: for $t \in \brackets{0,T}$, $i \in \braces{1,\ldots ,N}$,
\begin{eqnarray} 
X^{N,i}_t = X^{N,i}_0 + \int_{0}^{t}\round{u^{N,i}\round{s,\textbf{X}^N} + \bar{b}\round{s, X^{N,i}_s, L^N_s, m_s^{N}}}\,ds + \sigma W^{N,i}_t,
\label{eq:Ngame}
\end{eqnarray}
\noindent where $X^{N,i}_0\overset{d}{\sim}\nu$, $\textbf{W}^N\doteq (W^{N,1}, \ldots,W^{N,N})$ is an $N$-dimensional vector of independent $d$-dimensional Wiener processes and  $\textbf{X}^N\doteq(X^{N,1},\ldots,X^{N,N})$ denotes the vector of all players' private states. Moreover, the mean-field interaction terms $L^N$ and $m^N$, called respectively loss process and empirical average process, are defined as:
\begin{eqnarray} 
L^N_t\doteq \frac{1}{N}\sum_{i = 1}^{N} \mathbf{1}_{[\tau^{X^{N,i}},\infty)}\round{t}\quad\text{and}\quad m^N_t\doteq \frac{1}{N}\sum_{i = 1}^{N} w\round{X^{N,i}_t}\mathbf{1}_{[0,\tau^{X^{N,i}})}\round{t},
\nonumber
\end{eqnarray}
\noindent for $t\in[0,T]$, with $\tau^{X^{N,i}} \doteq \inf\{t \in[0,T]\,:\,X^{N,i}_t \not\in \mathcal{O}\}$ the first exit time of $X^{N,i}$ from the set $\mathcal{O}$. We can introduce the random empirical \emph{sub-probability measure} $\mu_t ^N$, defined as:
\begin{eqnarray}
\mu_t^N\round{\cdot} \doteq \frac{1}{N}\sum_{i = 1}^{N} \delta_{X_t^{N,i}}\round{\cdot}\mathbf{1}_{[0,\tau^{X^{N,i}})}\round{t},\quad t\in[0,T],
\nonumber
\end{eqnarray}
so that, for future use, the empirical average process can also be expressed as
\[ m_s^{N} \doteq \int_{\mathbb{R}^d}w(y)\mu^N_s(dy).\]
Solutions of the previous SDEs are intended in the weak sense on some filtered probability space $(\Omega^N,\mathcal{F}^N,(\mathcal{F}^N_t)_{t\in[0,T]},\mathbb{P}^N)$ satisfying the usual conditions (see Remark \ref{rem:existenceUniqWeak} below). 

In this game, the players implement feedback strategies $\textbf{\textit{u}}^N =(u^{N,1},\ldots,u^{N,N})$  belonging to $\mathcal{U}_{N}^{N}$, the set of all vectors $\textbf{\textit{u}}^N$, whose entries $u^{N,i}$, $i \in \braces{1, \ldots, N}$, belong to $\mathcal{U}_{1}^{N}$, the set of all progressively measurable functions $u:[0,T]\times \mathcal{X}^N\rightarrow\Gamma$.  Player $i$ evaluates a strategy vector $\textbf{\textit{u}}^N\in\mathcal{U}^N_N$ according to his/her expected costs over a random time horizon:
\begin{eqnarray}
J^{N,i}\round{\textbf{\textit{u}}^N} \doteq \mathbb{E}\Biggl[\int_{0}^{\tau^{N,i}}\bar{f}\round{s,X^{N,i}_s,L_s^N,m^N_s,u^{N,i}(s,\textbf{X}^N)}ds+F\round{\tau^{N,i},X^{N,i}_{\tau^{N,i}}}\Biggr],
\label{eq:Ncost}
\end{eqnarray}
\noindent where $\textbf{X}^N$ is the $N$-player dynamics under $\textbf{\textit{u}}^N$ and $\tau^{N,i} \doteq \tau^{X^{N,i}} \wedge T$. Our aim is the construction of approximate Nash equilibria for the $N$-player game from a solution of the limit problem. In the next definition, we use the standard notation $[u^{N,-i},v]$ to indicate a strategy vector equal to $\textbf{\textit{u}}^N$ for all players but $i$-th, who deviates by playing $v\in\mathcal{U}^N_1$ instead. 
\begin{definition}[\textit{$\epsilon$-Nash equilibrium}] Let $\epsilon \geq 0$.  A strategy profile $\textbf{\textit{u}}^N\in\mathcal{U}_N^N$ is named $\epsilon$-Nash equilibrium for the $N$-player game if for every $i\in\{1,\ldots,N\}$ and for any deviation $v\in\mathcal{U}^N_1$ we have:
\begin{eqnarray} 
J^{N,i}(\textbf{\textit{u}}^N)\leq J^{N,i}\round{\brackets{u^{N,-i},v}}+\epsilon.
\nonumber
\label{eq:epsilonNash}
\end{eqnarray}
\noindent A strategy profile $\textbf{\textit{u}}^N\in\mathcal{U}_N^N$ is a Nash equilibrium for the $N$-player game if it is an $\epsilon$-Nash equilibrium with $\epsilon=0$.
\end{definition}


\medskip
\noindent \textit{The mean-field dynamics.} Let $\mu \in \Upsilon_{\leq 1,1}^{T}$ be a flow of sub-probability measures. Given a feedback strategy $u$ and a flow of measures $\mu \in \Upsilon_{\leq 1,1}^{T}$, the representative player's state evolves according to the following equation:
\begin{eqnarray} 
X_t = X_0 + \int_{0}^{t}\round{u\round{s,X} + \bar{b}\round{s, X_s, L\round{s;\mu}, m\round{s;\mu}}}\,ds + \sigma W_t,\quad t \in \brackets{0,T},
\label{eq:MFG}
\end{eqnarray}
where $X$ is a $d$-dimensional stochastic process starting at $X_0 \overset{d}{\sim} \nu \in \mathcal{P}(\mathbb{R}^d)$ and $W$ is a $d$-dimensional Wiener process on some filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$. Again, solutions of \eqref{eq:MFG} are intended in the weak sense (see Remark \ref{rem:existenceUniqWeak}). 

Moreover, $L:[0,T] \times \mathcal{M}_{\leq 1}(\mathcal{X})\rightarrow [0,1]$ and $m:[0,T]\times\mathcal{M}_{\leq 1}(\mathcal{X}) \rightarrow \mathbb{R}^{d_0}$ denote the following deterministic functions:
\begin{eqnarray} 
L\round{t;\theta}&\doteq &1-\int_{\mathcal{X}}\mathbf{1}_{[0,\tau(\varphi))}(t)\theta(d\varphi),\nonumber \\
m\round{t;\theta}&\doteq &\int_{\mathcal{X}}w\round{\varphi(t)}\mathbf{1}_{[0,\tau(\varphi))}(t)\theta(d\varphi),
\nonumber
\end{eqnarray}
for all $(t,\varphi)\in[0,T]\times\mathcal{X}$ and where $\tau(\varphi) \doteq\inf\{t \in[0,T]\,:\, \varphi(t) \not\in \mathcal{O}\}$, $\varphi \in \mathcal{X}$. Let $\mathcal{U}_{fb}$ denote the set of all feedback strategies:
\begin{eqnarray}
\mathcal{U}_{fb} \doteq \braces{u:\brackets{0,T}\times\mathcal{X} \rightarrow \Gamma : \text{\,$u$ is progressively measurable}}.
\nonumber 
\end{eqnarray}
The cost associated with a strategy $u \in \mathcal{U}_{fb}$, a flow of measure $\mu$, and an initial distribution $\nu \in\mathcal{P}(\mathbb{R}^d)$ with support in $\mathcal{O}$ is given by (we omit, for the sake of simplicity, the explicit dependence of $J^{\mu}(u)$ from $\nu$):
\begin{eqnarray}
J^{\mu}\round{u} \doteq \mathbb{E}\Biggl[\int_{0}^{\tau}\bar{f}\round{s,X_s,L\round{s;\mu},m\round{s;\mu},u\round{s,X}}ds+F\round{\tau,X_{\tau}}\Biggr]
\label{eq:MFGcost}
\end{eqnarray}
\noindent where $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\in[0,T]},\mathbb{P}, W, X)$ is a solution of \eqref{eq:MFG} under $u$ with initial distribution $\nu$, and $\tau \doteq \tau^{X} \wedge T$ the random time horizon. We define
\[V^{\mu}\doteq \inf_{u \in \mathcal{U}_{fb}} J^{\mu}(u),\] 
and we make the following 
additional assumption on the initial conditions $X^{N,i}_0$, $i\in\{1,\ldots,N\}$ and $X_0$ :
\begin{itemize}
\item[$\round{\mathrm{H8}}$] The initial conditions $X^{N,i}_0$, $i\in\{1,\ldots,N\}$, and $X_0$ have distribution $\nu\in\mathcal{P}(\mathbb{R}^d)$.
\end{itemize}
\begin{rmk}\label{rem:existenceUniqWeak} Both existence and uniqueness in law of weak solutions of Eq.s \eqref{eq:Ngame} and \eqref{eq:MFG} are guaranteed by Lemma \ref{lem:benevs}, and by Proposition 5.3.6, Remark 5.3.8 and Proposition 5.3.10 in \citet{karaztas} (see our Lemma \ref{lem:existenceUniquenessWeak}). Precisely, they can be proved by means of Girsanov's theorem and Ben\v{e}s' condition \cite{benevs}.
\end{rmk}
\medskip

\noindent The notion of solution we consider for the MFG is the following:
\begin{definition}[\textit{Feedback MFG solution}]\label{def:MFGsol} A feedback solution of the MFG is a triplet $(\nu,u,\mu)$ such that:
\begin{itemize}
\item[($i$)] $\nu\in\mathcal{P}(\mathbb{R}^d)$ with $\text{supp}(\nu)\subset\mathcal{O}$, $u\in\mathcal{U}_{fb}$ and $\mu\in\Upsilon_{\leq 1,1}^{T}$.
\item[($ii$)] Strategy $u$ is optimal for $\mu$, i.e. $V^{\mu} = J^{\mu}(u)$.
\item[($iii$)] Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P},X,W)$ is a weak solution of Eq.\eqref{eq:MFG} with flow of sub-probability measures $\mu$, strategy $u$ and initial condition $\nu$. Then 
\[\mu_t(\cdot)=\mathbb{P}(\{X_t \in \cdot\}\cap\{ \tau^X > t\}), \quad t\in[0,T].\]
\end{itemize}
\end{definition}
\begin{rmk}[\textit{Feedback and open-loop controls}]\label{rem:karoui} Feedback controls induce stochastic open-loop controls, i.e. tuples $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P},X,u,W)$ that are weak solutions of: 
\begin{eqnarray} 
X_t = X_0 + \int_{0}^{t}\round{u_s + \bar{b}\round{s, X_s, L\round{s;\mu}, m\round{s;\mu}}}\,ds + \sigma W_t\quad t \in \brackets{0,T},
\label{eq:MFG2}
\nonumber 
\end{eqnarray}
\noindent where $u$ is a progressively measurable $\Gamma$-valued stochastic process. As a consequence, the computation of the infimum of $J^{\mu}(\cdot)$ over the class of stochastic open-loop controls would imply a lower value for $V^{\mu}$. However, thanks to Proposition 2.6 in \citet{karoui}, the two minimization problems are equivalent from the point of view of the value function.
\end{rmk}


\noindent \textit{The canonical space.} We will often work on the canonical filtered probability space, denoted by $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$ and defined on $\mathcal{X}$ as follows. Set $\Omega\doteq\mathcal{X}$, let $\xi$ be an $\mathbb{R}^d$-valued random variable with law $\nu\in\mathcal{P}(\mathbb{R}^d)$ and let $W$ be a $d$-dimensional Wiener process on $\mathcal{X}$ independent of $\xi$. Define $\mathcal{W}^{\nu}\in\mathcal{P}(\mathcal{X})$ as the law of $\xi+\sigma W$. Set $\mathcal{F}$ as the $\mathcal{W}^{\nu}$-completion of the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{X})$ and $(\mathcal{F})_{t\in[0,T]}$ as the $\mathcal{W}^{\nu}$-augmentation of the filtration generated by the canonical process $\hat{X}$ on $\mathcal{X}$, i.e. $\hat{X}(\varphi)\doteq \varphi(t)$ for all $(t,\varphi)\in[0,T]\times\mathcal{X}$. In particular, $(\mathcal{F})_{t\in[0,T]}$ satisfies the usual conditions. Finally set $\mathbb{P}\doteq\mathcal{W}^{\nu}$ and $W\doteq\sigma^{-1}(\xi-\hat{X})$, which is a Wiener process on $\mathcal{X}$. Where no confusion is possible, we will drop the hat for simplicity and write $X$ in place of $\hat{X}$.\\ 

\noindent \textit{Relaxed controls.} In what follows we use relaxed controls (see \citet{karoui} for a precise definition), which allow us to view progressively measurable controls with values on a compact set $\Gamma$ as elements of the space of probability measures on $\Gamma$. The latter space is compact and endowed with the weak convergence of measures. We introduce the space $\mathcal{V}$ of relaxed controls:
\begin{eqnarray} 
\mathcal{V}\doteq \cbr{q\in \mathcal{M}_f\round{[0,T]\times \Gamma}:q(dt,d\gamma)=dt q_t (d \gamma),\,t\mapsto q_t\in \mathcal{P}(\Gamma)\,\text{Borel measurable}},
\nonumber
\end{eqnarray}
i.e. the space of finite positive measures on $[0,T] \times \Gamma$ with Lebesgue time marginal. With a slight abuse of notation, we denote with $\hat{\Lambda}$ both the identity map and the canonical process on $\mathcal{V}$ (where no confusion is possible, we drop the hat and write $\Lambda$ in place of $\hat{\Lambda}$). Precisely, a single-player relaxed control is a $\mathcal{V}$-valued random variable $\Lambda$ such that $(\Lambda_t)_{t\in[0,T]}$ is a progressively measurable $\mathcal{P}(\Gamma)$-valued stochastic process. We say that $\Lambda$ is a feedback control if there exists a progressively measurable functional $\lambda:[0,T]\times\mathcal{X}\rightarrow\mathcal{P}(\mathcal{X})$ such that $\Lambda_t\doteq\lambda(t,X)$ for all $t\in[0,T]$, with $X$ denoting the player's dynamics. Moreover, we say that $\Lambda$ is a strict and feedback control if there exists $u\in\mathcal{U}_{fb}$ such that $\lambda(t,X)\doteq\delta_{u(t,X)}$ for all $t\in[0,T]$.\\
\indent We rewrite the dynamics and the cost functional for both the $N$-player game (\eqref{eq:Ngame} and \eqref{eq:Ncost}) and the MFG (\eqref{eq:MFG}) and \eqref{eq:MFGcost}) by using the notion of relaxed controls. To this end,  for all $(t,\varphi,\theta,u)\in[0,T]\times\mathcal{X}\times\mathcal{P}(\mathcal{X})\times\Gamma$ we define 
a more general drift function $b:[0,T]\times\mathcal{X}\times\mathcal{M}_{\leq 1}(\mathcal{X})\times\Gamma\rightarrow\mathbb{R}^d$ and running cost function $f:[0,T]\times\mathcal{X}\times\mathcal{M}_{\leq 1}(\mathcal{X})\times\Gamma\rightarrow[0,+\infty)$:
\begin{eqnarray} 
b\round{t,\varphi,\theta,u}&\doteq& u+\bar{b}\round{t,\varphi(t),L(t;\theta),m(t;\theta)}, \label{eq:driftCostRel}\\
f\round{s,\varphi,\theta,u}&\doteq& \bar{f}\round{t,\varphi(t),L(t;\theta),m(t;\theta),u}.
\nonumber
\end{eqnarray}
Moreover, we define as $\widetilde{\mathcal{U}}^N_N$ and $\widetilde{\mathcal{U}}^N_1$ the sets of relaxed feedback $N$-player and single-player strategies for the $N$-player game. We call $\widetilde{\mathcal{U}}_{fb}$ the set of relaxed feedback controls for the MFG. With this notation, the relaxed $N$-player dynamics and costs are defined by:
\begin{eqnarray} 
X^{N,i}_t &=& X^{N,i}_0 + \int_{[0,t]\times\Gamma}b\round{s, X^{N,i},\mu^{N},u}\lambda^{N,i}\round{s,X^{N,i}}(du)ds+\sigma W^{N,i}_t, \label{eq:NgameRel}\\
J^{N,i}\round{\boldsymbol{\lambda}^N} &=&\mathbb{E}\Biggl[\int_{[0,\tau^{N,i}]\times\Gamma}f\round{s, X^{N,i},\mu^{N},u}\lambda^{N,i}\round{s,X^{N,i}}(du)ds+F\round{\tau^{N,i},X^{N,i}_{\tau^{N,i}}}\Biggr],
\nonumber
\label{eq:costNgameRel}
\end{eqnarray}
\noindent $t\in[0,T]$, $i\in\{1,\ldots,N\}$, $\boldsymbol{\lambda}^N\in\widetilde{\mathcal{U}}^N_N$ and $\lambda^{N,i}\in\widetilde{\mathcal{U}}^N_1$ for all $i\in\{1,\ldots,N\}$. The MFG reads as:
\begin{eqnarray} 
X_t &=& X_0 + \int_{[0,t]\times\Gamma}b\round{s,X,\mu,u}\lambda\round{s,X}(du)ds+ \sigma W_t, \label{eq:MFGRel}\\
J^{\mu}\round{\lambda}&=&\mathbb{E}\Biggl[\int_{[0,\tau]\times\Gamma}f\round{s,X,\mu,u}\lambda\round{s,X}(du)ds+F\round{\tau,X_{\tau}}\Biggr]
\label{eq:costMFGRel}
\nonumber 
\end{eqnarray}
\noindent where $t\in[0,T]$ and $\lambda\in\widetilde{\mathcal{U}}_{fb}$.\\ 
\noindent In accordance, we give the notion of $\epsilon$-Nash equilibrium and feedback solution of the MFG by using relaxed controls.
\begin{definition}[\textit{Relaxed $\epsilon$-Nash equilibrium}]\label{def:epsilonNashRel} A strategy profile $\boldsymbol{\lambda}^N\in\widetilde{\mathcal{U}}_N^N$ is an $\epsilon$-Nash equilibrium for the $N$-player game if for every $i\in\{1,\ldots ,N\}$ and for any single-player strategy $\beta\in\widetilde{\mathcal{U}}^N_1$
\begin{eqnarray} 
J^{N,i}(\boldsymbol{\lambda}^N)\leq J^{N,i}\round{\brackets{\boldsymbol{\lambda}^{N,-i},\beta}}+\epsilon.
\nonumber
\label{eq:epsilonNashRel}
\end{eqnarray}
\noindent  A strategy profile $\boldsymbol{\lambda}^N\in \widetilde{\mathcal{U}}_N^N$ is a Nash equilibrium for the $N$-player game if it is an $\epsilon$-Nash equilibrium with $\epsilon=0$.
\end{definition}
\begin{definition}[\textit{Relaxed feedback MFG solution}]\label{def:MFGsolRel} A relaxed feedback solution of the MFG is a triplet $(\nu,\lambda,\mu)$ such that:
\begin{itemize}
\item[($i$)] $\nu\in\mathcal{P}(\mathbb{R}^d)$ with $\text{supp}(\nu)\subset\mathcal{O}$, $\lambda\in\tilde{\mathcal{U}}_{fb}$ and $\mu\in\Upsilon_{\leq 1,1}^{T}$.
\item[($ii$)] $\lambda$ is optimal, i.e. $V^{\mu} = J^{\mu}(\lambda)$.
\item[($iii$)] Let $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q},X,W)$ be a weak solution of Eq.\eqref{eq:MFGRel} with flow of sub-probability measures $\mu$, control $\lambda$ and initial condition $\nu$. Then 
\[\mu_t(\cdot)=\mathbb{Q}(\{X_t \in \cdot\}\cap\{ \tau^X > t\}), \quad t\in[0,T].\]
\end{itemize}
\end{definition}
\begin{rmk}[\textit{Feedback and open-loop relaxed controls}]\label{rem:karoui2} Feedback relaxed controls induce relaxed stochastic open-loop controls, i.e. tuples  $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q},X,\Lambda,W)$ that are weak solutions of:
\begin{eqnarray} 
X_t &=& X_0 + \int_{[0,t]\times\Gamma}b\round{s,X,\mu,u}\Lambda_s(du)ds+ \sigma W_t,\quad t\in[0,T]
\nonumber 
\end{eqnarray}
\noindent where $\Lambda$ is a progressively measurable $\mathcal{P}(\Gamma)$-valued stochastic process. A similar argument as in Remark  \ref{rem:karoui}  holds in this case as well.
\end{rmk}
\noindent\textit{The extended canonical probability space.} When dealing with relaxed controls we work on an extended version of the canonical probability space to $\mathcal{X}\times\mathcal{V}$. Set $\tilde{\Omega}\doteq \mathcal{X}\times\mathcal{V}$, let $\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]}$ be the canonical $\sigma$-algebra and filtration on $\mathcal{X}$ whereas let $\mathcal{G},(\mathcal{G}_t)_{t\in[0,T]}$ be the Borel $\sigma$-algebra and the filtration generated by the canonical process $\hat{\Lambda}$ on $\mathcal{V}$. Set $\tilde{\mathcal{F}}_t\doteq \mathcal{F}_t\otimes\mathcal{G}_t$ for all $t\in[0,T]$ and $\tilde{\mathcal{F}}\doteq \mathcal{F}\otimes \mathcal{G}$. \\

\noindent\textit{Laws of processes with sub-linear drift.} As said, in this paper we deal with Brownian-driven stochastic processes with sub-linear drift (H2). Their laws belong to $\mathcal{P}(\mathcal{X})$ or, more precisely, to a subset $\mathcal{Q}_{\nu,K,C}$ that we define in the following way. $\mathcal{Q}_{\nu,K,C}\subset\mathcal{P}(\mathcal{X})$ is the space of all laws $\theta\in\mathcal{P}(\mathcal{X})$ of continuous stochastic processes $X$ defined on some filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$ with Wiener process $W$ and with the following dynamics:
\begin{eqnarray} 
X_t=\xi+\int_0^t u_s+v\round{s,X}ds+\sigma W_t \quad t\in[0,T],
\nonumber
\end{eqnarray}
\noindent where $\xi\overset{d}{\sim}\nu$, $u$ is a progressively measurable process bounded by some constant $K>0$, $v:[0,T]\times\mathcal{X}\rightarrow\mathbb{R}^d$ is a progressively measurable functional such that $|v(t,\varphi)|\leq C(1+|\varphi(t)|)$ for some constant $C>0$ and for all $(t,\varphi)\in[0,T]\times\mathcal{X}$. Notice that laws of processes of the form 
\begin{eqnarray} 
X_t=\xi+\int_{[0,t]\times\Gamma}\round{u+v\round{s,X}}\Lambda_s(du)ds+\sigma W_t\quad t\in[0,T],
\nonumber
\end{eqnarray}
\noindent with $\xi$ and $v$ as above belong to $\mathcal{Q}_{\nu,|\Gamma|,C}\subset\mathcal{P}(\mathcal{X})$ where $|\Gamma|$ denotes the diameter of $\Gamma$.
\section{Existence of solutions of the mean-field game}\label{sec:mfg}
\noindent Throughout this section hypotheses (H1)-(H8) are in force. Under these and some additional assumptions we show that both a relaxed and a strict feedback solution of the MFG exist; see Theorem \ref{teo:existenceRelFeedSolMFG}, Proposition \ref{prop:existenceRelFeedlSolMFG} and Proposition \ref{prop:existenceStrictFeedlSolMFG}. In addition, we guarantee the existence of a feedback solution of the MFG with Markovian feedback strategy up to the exit time; see Lemma \ref{lem:existencemarkovSolMFG}.
\begin{theorem}[\textit{Existence of relaxed and strict feedback MFG solutions}]\label{teo:existenceRelFeedSolMFG} Under assumptions (H1)-(H8) and (C1), there exists a relaxed feedback solution $(\nu,\lambda,\mu)$ of the MFG. Moreover, under the additional assumption (C2), there exists a strict feedback solution $(\nu,u,\mu)$ of the MFG.
\end{theorem}

\noindent To prove Theorem \ref{teo:existenceRelFeedSolMFG} we proceed by approximation in the sense that -- in the spirit of \citet{lacker} -- we construct a sequence of approximating MFGs with bounded drift and cost functional for which 
the existence of a solution is guaranteed by an extension of the results in \citet{campi}; see Subsection \ref{subsec:approximatingMFG}. Then, we prove convergence of the solutions of these approximating MFGs to a solution of the original MFG by interpreting the approximating solutions as relaxed controls; see Subsection \ref{subsec:convergence} (see \citet{karoui} for details on relaxed controls). 

Before proceeding, we have to ensure the well-posedness of the game, i.e. we have to ensure that the private state $X$ of the representative agent remains in $\mathcal{O}$ up to time $T$ with some positive probability. This is the content of the following Lemma.
\begin{lemma} \label{lem:exitO} Grant assumptions (H1)-(H8). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P},X,W)$ be a weak solution of Eq.\eqref{eq:MFG}. Then $\mathbb{P}(\tau^X>t)>0$ for all $t\in[0,T]$. 
\end{lemma}
\begin{proof}\noindent Set $b_t\doteq u(t,X)+\bar{b}(t,X_t,L(t;\mu),m(t;\mu))$ for $t\in[0,T]$, and define $Z \doteq (Z_t)_{t\in[0,T]}$ as:
\begin{eqnarray}
Z_t\doteq\mathcal{E}_t\round{-\int_0^{\cdot}\sigma^{-1}b_sdW_s},\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent By Lemma \ref{lem:benevs}, $Z$ is a true martingale. Define $\mathbb{Q}$ by $\frac{d\mathbb{Q}}{d\mathbb{P}}\doteq Z_T$. By Girsanov's theorem $\widetilde{W}_t\doteq W_t+\int_0^t\sigma^{-1}b_sds$, $t\in[0,T]$, is a $\mathbb{Q}$-Wiener process, and under $\mathbb{Q}$ the process $X$ has law $\mathcal{W}^{\nu}$. As a consequence of the law of iterated logarithms, any Wiener process remains in an open set (e.g., $\mathcal{O}\subset \mathbb{R}^d$) for a finite time ($T<\infty$) with strictly positive probability. Therefore $\mathbb{Q}(\tau^X>T)>0$ and thus $\mathbb{P}(\tau^X>T)>0$.
\end{proof}
\subsection{Approximating MFGs}
\label{subsec:approximatingMFG}
\noindent First, we define the approximating MFGs. Second, we prove the existence of feedback solutions for these approximating models by extending the existence result of \citet{campi}. In particular, with respect to the previous authors, we include the loss process $L$ in the drift and let the density of players $\mu$ be a flow of sub-probability measures, without renormalizing it.
\medskip

\noindent \textit{The approximating models.}\quad Let $(K_n)_{n\in\mathbb{N}}\subset\mathbb{R}_+$ be an increasing sequence such that $K_n\nearrow+\infty$. The $n^{\rm th}$ approximating MFG model, denoted by  MFG($n$), is obtained using the following truncation procedure:
\begin{itemize}
\item[$\round{\mathbf{T}_n}$] $w^n(x)=w(x)$ where $|w(x)|\leq K_n$ while it is continuously truncated at level $K_n$, i.e. $|w^n(x)|= K_n$, otherwise. Similarly for the drift and the costs $b^n$,$f^n$, $F^n$.
\end{itemize}
\noindent Notice that neither the action space $\Gamma$ (as is already compact) nor the possibly unbounded open set $\mathcal O$ of non-absorbing states are truncated. Moreover, in the MFG($n$) the representative player's state evolves according to the following SDE:
\begin{eqnarray} 
X_t = X_0 + \int_{0}^{t}\round{u\round{s,X} + \bar{b}^n\round{s, X_s, L\round{s;\mu}, m^n\round{s;\mu}}}\,ds + \sigma W_t
\label{eq:MFGn} 
\end{eqnarray}
\noindent for $t\in[0,T]$ and where now $m^n:[0,T]\times\mathcal{P}(\mathcal{X})\rightarrow\mathbb{R}^d$ is defined as the flow of the average of $w^n$. Still $u\in\mathcal{U}_{fb}$ and $\mu\in\Upsilon^T_{\leq 1,1}$. The associated cost functional is:
\begin{eqnarray}
J^{n,\mu}\round{u} \doteq \mathbb{E}\Biggl[\int_{0}^{\tau}f^n\round{s,X_s,L\round{s;\mu},m^n\round{s;\mu},u\round{s,X}}ds+F^n\round{\tau,X_{\tau}}\Biggr].
\label{eq:MFGcostn}
\end{eqnarray}
\noindent We associate to the MFG($n$)s the following Hamiltonians: 
\begin{eqnarray}
h^n(t,x,l,m,z,u)&\doteq& f^n(t,x,l,m,u) + z\,\sigma^{-1}\,b^n(t,x,l,m,u)\nonumber\\
H^n(t,x,l,m,z)&\doteq&\inf_{\substack{u\in\Gamma}} h^n(t,x,l,m,z,u)
\label{eq:hamiltonian}
\nonumber 
\end{eqnarray}
and the set of minimizers: 
\begin{eqnarray} 
A^n(t,x,l,m,z) &\doteq &\left\{u\in\Gamma\,:\,h^n(t,x,l,m,z,u)=H^n(t,x,l,m,z)\right\}
\label{eq:minimizers} 
\nonumber
\end{eqnarray}
\noindent for $(t,x,l,m,z)\in[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\mathbb{R}^{d}$. We make the following additional assumption:
\begin{itemize} 
\item[$\round{\mathrm{C1}}$] For each $n\in\mathbb{N}$, $A^n(t,x,l,m,z)$ is convex for all $(t,x,l,m,z)\in[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\mathbb{R}^{d}$.
\end{itemize}
\begin{rmk}
\noindent Assumption (C1) is common in control theory and it is crucial in order to apply the Kakutani-Fan-Glicksberg fixed point theorem (\citet{aliprantis}, Corollary 17.55). It usually involves convexity or quasi-convexity of the Hamiltonians. In our case it is trivially satisfied if, for instance, the running cost $f$ is bounded and convex in the control variable $u\in\Gamma$. Indeed in this case, due to the flexibility in the choice of the truncation procedure, it is sufficient to choose $K^n\geq\infnorm{f}$ for all $n\in\mathbb{N}$ so that $f^n=f$ for all $n\in\mathbb{N}$. Then convexity is preserved by adding any sub-linear term.
\end{rmk}
\begin{theorem}[\textit{Existence of solutions of MFG($n$)}]\label{teo:existenceSolMfgn} Under assumptions (H1)-(H8) and (C1) there exists a feedback solution $(\nu,u^n,\mu^n)$ of MFG($n$).
\end{theorem}
\begin{proof} The proof follows similar steps to those in Section 6 of \citet{campi}, therefore we only sketch the relevant passages. The main difference with \cite{campi} is that, due to assumption (C1), we have to deal with set-valued maps, hence to apply a version of Kakutani's fixed point theorem instead of Brouwer's. We use the version proposed by \citet{carmonaweak}, Proposition 7.4, which is in turn based on the results of \citet{cellina}. Other adjustments are due to the presence of the loss process $L$ in the drift, the fact that $\mu$ is a flow of sub-probability measures (instead of probability measures) and that $\mathcal{O}$ can be unbounded.
\\
\indent Fix $n\in\mathbb{N}$. The proof is based on the construction of a suitable map $\Psi:\mathcal{P}(\mathcal{X})\times\mathbb{U}\rightarrow \mathcal{P}(\mathcal{X})$ on an appropriate compact and convex subset of $\mathcal{P}(\mathcal{X})$, where $\mathbb{U}$ is the space of progressively measurable $\Gamma$-valued stochastic processes. The fixed points of $\Psi$ then provide MFG($n$) solutions. More in detail, let us consider
\[ \Psi : \mathcal{Q}_{\nu,K_n,0}\times\mathbb{U} \ni (\theta,u) \mapsto \mathbb{P}^{\theta,u} \circ X^{-1} \in \mathcal{Q}_{\nu,K_n,0},\]
\noindent where the probability measure $\mathbb{P}^{\theta,u}$ is defined as follows. Let $(\theta,u)\in\mathcal{Q}_{\nu,K_n,0}\times\mathbb{U}$ and let $\mu^{\theta}\in\Upsilon_{\leq 1}^{T}$ be defined as $\mu_t^{\theta}(\cdot)\doteq\theta(\{X_t\in\cdot \}\cap\{ \tau^{X}>t\})$ for all $t\in[0,T]$, where $X$ is the canonical process on $\mathcal{X}$. Define $(\Omega,\mathcal{F}^{u},(\mathcal{F}^{u}_t)_{t\in[0,T]},\mathbb{P}^{\theta,u},X,W^{u})$ as the weak solution of
\begin{eqnarray} 
X_t = X_0 + \int_{0}^{t}\round{u_s + \bar{b}^n\round{s, X_s, L(s;\mu^{\theta}), m^n(s;\mu^{\theta})}}\,ds + \sigma W^{u}_t,\quad t\in[0,T],
\nonumber 
\end{eqnarray}
\noindent on the canonical space $(\Omega\doteq \mathcal{X},\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$. Moreover, for $\theta\in \mathcal{Q}_{\nu,K_n,0}$ we call $u^{\theta}$ an optimal control for the cost
\begin{eqnarray}
J^{n,\mu^{\theta}}\round{u} \doteq \mathbb{E}^{\mathbb{P}^{\theta,u}}\Biggl[\int_{0}^{\tau}f^n\round{s,X_s,L(s;\mu^{\theta}),m^n(s;\mu^{\theta}),u_s}ds+F^n\round{\tau,X_{\tau}}\Biggr].
\nonumber
\end{eqnarray}
\noindent Such optimal controls $u^{\theta}$ can be constructed by standard BSDE techniques as in \cite{campi}, Section 6.1 (by means of Theorem 3.4 in \citet{darling}, in order to deal with random terminal times). Under assumption (C1) optimal controls $u^{\theta}$ are in general not unique. Indeed
\begin{eqnarray} 
A^{n}(\theta)\doteq \cbr{u^\theta \in\mathbb{U}: u^\theta \in\mathbb{U}\in A^{n}(\cdot,X_{\cdot},L(\cdot;\mu^{\theta}), m^n(\cdot;\mu^{\theta}),Z^{\theta}_{\cdot}),\,\mathcal{L}_T\otimes\mathbb{P}-a.s.}
\nonumber
\end{eqnarray}
\noindent provides an entire set of optimal controls, where $Z^{\theta}$ is part of the the solution of the associated BSDE and $\mathcal{L}_T$ denotes the Lebesgue measure on $[0,T]$. Moreover, by measurable selection there exists a measurable function $\hat{u}^{n,\theta}:[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\mathbb{R}^{d}\rightarrow \Gamma$ such that
\begin{eqnarray} 
\hat{u}^{n,\theta}(t,x,l,m,z)\in A^n(t,x,l,m,z),\quad (t,x,l,m,z)\in[0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\mathbb{R}^{d}.
\nonumber
\end{eqnarray}
\noindent Hence $\hat{u}^{n,\theta}(t,X_t,L(t;\mu^{\theta}), m^n(t;\mu^{\theta}),Z^{\theta}_t)$, for $t\in[0,T]$, is a progressively measurable control process that can be written in feedback form. Indeed, $Z^{\theta}$, being part of the solution of the associated BSDE, is progressively measurable for the canonical filtration, hence $Z^{\theta}_t=\zeta^{\theta}(t,X)$ for some progressively measurable functional $\zeta^{\theta}:[0,T]\times\mathcal{X}\rightarrow\mathbb{R}^d$ and for $t\in[0,T]$.\\
\indent Now, a fixed point for the map $\Psi$ is a probability measure $\theta\in\mathcal{Q}_{\nu,K^n,0}$ such that $\theta\in\Psi(\theta,A(\theta))$. Existence is provided by Proposition 7.4 in \cite{carmonaweak} so to conclude the proof it suffices to check that all the required assumptions are satisfied in our case.
The set $\mathcal{Q}_{\nu,K^n,0}\subset\mathcal{P}(\mathcal{X})$ is a (weakly) compact, convex and metrizable subset of $C_b^*(\mathcal{X})$, the dual of the space of bounded and continuous functions on $\mathcal{X}$, which is a locally convex topological vector space with the weak* topology (that induces the weak convergence of measures on $\mathcal{P}(\mathcal{X})$). We endow the vector space $\mathbb{U}$ with the norm $\nor{\cdot}_{\mathbb{U}}$ defined as $\nor{u}_{\mathbb{U}}\doteq\mathbb{E}[\int_0^T|u_t|dt]$. As a consequence of Berge's maximum theorem (see, e.g., Theorem 17.31 in \citet{aliprantis}) and of assumption (C1) the set-valued map $A^n:\mathcal{Q}_{\nu,K^n,0}\rightarrow\mathbb{U}$ is upper hemicontinuous and has non-empty convex and closed values (see the proof of Lemma 7.11 in \cite{carmonaweak}). Finally, by following the proof in Section 6.2 of \cite{campi} and that of Theorem 3.5 in \cite{carmonaweak}, we obtain that $\Psi$ is continuous. For this step, Lemma 6.2 in \cite{campi} has to be replaced with our Lemma \ref{lem:regularity}, which guarantees almost sure continuity of $L$ and $m$.\\
\indent Therefore, Proposition 7.4 in \cite{carmonaweak} applies, yielding the existence of a feedback solution of MFG($n$).
\end{proof}

\noindent\textit{A-priori estimates.}\quad We ensure now that the moments up to any order $\alpha$ of the state process  remain bounded uniformly in $n$. Such estimates will be very useful when relaxing the truncation by letting $n \to \infty$.
\begin{lemma}[\textit{A-priori estimates}]\label{lem:alphaEstimate} Grant assumptions (H1)-(H8) and (C1). Consider feedback solutions $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ and $(\nu,u,\mu)$ of the MFG(n)'s and of the MFG, respectively. 

Let $(\Omega^n,\mathcal{F}^n,(\mathcal{F}^n_t)_{t\in[0,T]},\mathbb{P}^n,X^n,W^n)_{n\in\mathbb{N}}$ be a sequence of weak solutions of the SDEs \eqref{eq:MFGn} and $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P},X,W)$ a weak solution of the SDE \eqref{eq:MFG}. Then for any $\alpha\geq 1$,
\begin{eqnarray} 
\sup_{n\in\mathbb{N}}\mathbb{E}^{\mathbb{P}^n}\brackets{\infnorm{X^n}^{\alpha}}\leq K\round{\alpha}\quad\text{and}\quad\mathbb{E}^{\mathbb{P}}\brackets{\infnorm{X}^{\alpha}}\leq K\round{\alpha},
\nonumber
\label{eq:alphaEstimate}
\end{eqnarray} 
where $K(\alpha)<\infty$ is a positive constant independent of $n$.
\end{lemma}
\begin{proof}\noindent For the sake of readability, we omit the dependence on $\mathbb{P}$ in the expectations below.  Let $\alpha\geq 1$ and let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P},X,W)$ be a weak solution of the SDE in Eq.\eqref{eq:MFG}. Then, using the linear growth of the drift we get:
\begin{eqnarray} 
\mathbb{E}\brackets{\infnormt{X}^{\alpha}}&\leq &C(\alpha)\Bigl(\mathbb{E}\brackets{\modl{\xi}^{\alpha}}+\modl{\sigma}^{\alpha}\mathbb{E}\brackets{\infnormt{W}^{\alpha}}+C(\alpha,T,|\Gamma|)\Bigr)+\nonumber\\
&+&C'(\alpha)C\int_0^t\round{1+\mathbb{E}\brackets{\infnorms{X}^{\alpha}}+\infnorms{m(\cdot\,;\mu)}^{\alpha}}ds
\nonumber
\end{eqnarray}
\noindent for all $t\in[0,T]$ and for some constants $C(\alpha),C'(\alpha),C(\alpha,T,|\Gamma|)>0$. By assumption (H6), $\mathbb{E}[|\xi|^{\alpha}]<\infty$ for all $\alpha\geq 1$. In addition, $\infnormt{m(\cdot\,;\mu)}^{\alpha}\leq C''(\alpha)C(1+\mathbb{E}[\infnormt{X}^{\alpha}])$ for all $t\in[0,T]$ and for some constant $C''(\alpha)>0$. Therefore, we conclude by using Gr\"onwall's lemma. The same estimates hold also for any weak solution of Eq.\eqref{eq:MFGn}.
\end{proof}

\noindent Let $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ and $(\nu,u,\mu)$ be feedback solutions of the MFG($n$)s and of the MFG. We now ensure that the corresponding mean and loss processes are continuous in time.
\begin{lemma}[\textit{Continuity of the mean and loss processes}]\label{lem:contMeanLoss} Let $\theta\in\mathcal{Q}_{\nu,K,C}$. Define $\mu(\cdot)\doteq \theta(\{X_t\in\cdot\}\cap\{\tau^X>t\})$, where $X$ is the identity process on $\mathcal X$. Then, the functions $m(\cdot\,;\mu):[0,T]\rightarrow\mathbb{R}^{d_0}$ and $L(\cdot\,;\mu):[0,T]\rightarrow[0,1]$ are continuous.
\end{lemma}
\begin{proof}\noindent We prove the continuity only for $m(\cdot\,;\mu)$ since a similar reasoning applies to $L(\cdot\,;\mu)$ as well.  By Lemma \ref{lem:regularity} the mapping $t\mapsto \mathbf{1}_{[0,\tau^X)}(t)$, where $X$ is the identity on $\mathcal{X}$, is $\theta$-a.s. continuous and bounded. The function $w$ is continuous and integrable with respect to $\mu_t$ because of its linear growth and of the a-priori estimates. Moreover $X$ has continuous trajectories. Now let $(t_n)_{n\in\mathbb{N}}\subset[0,T]$ converge to some $t\in[0,T]$. Then $w\round{X_{t_n}}\mathbf{1}_{[0,\tau^X)}(t_n)\rightarrow w\round{X_{t}}\mathbf{1}_{[0,\tau^X)}(t)$ $\theta$-a.s. and
\begin{eqnarray} 
\mathbb{E}^{\theta}\brackets{w\round{X_{t_n}}\mathbf{1}_{[0,\tau^X)}(t_n)}\underset{n\rightarrow\infty}{\longrightarrow} \mathbb{E}^{\theta}\brackets{w\round{X_t}\mathbf{1}_{[0,\tau^X)}(t)},
\nonumber
\end{eqnarray}
by the dominated convergence theorem.
\end{proof}

\subsection{Convergence of the approximating MFGs}
\label{subsec:convergence}
\noindent Let $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ be a sequence of feedback solutions of the approximating MFGs introduced in the previous Subsection \ref{subsec:approximatingMFG}, whose existence is guaranteed by Theorem \ref{teo:existenceSolMfgn}.  In addition, let $(\Omega^n,\mathcal{F}^n,(\mathcal{F}^n_t)_{t\in[0,T]},\mathbb{P}^n,X^n,W^n)_{n\in\mathbb{N}}$ be a sequence of weak solutions of the SDEs in Eq.\eqref{eq:MFGn} associated to $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$. Let $\theta^n$ be defined as $\theta^n \doteq \mathbb{P}^n\circ (X^n)^{-1}$ for each $n\in\mathbb{N}$. 

To prove the convergence of the approximating MFGs we proceed in the following way. First, we show that there exists a subsequence of $(\theta^n)_{n \in \mathbb{N}}$, say $(\theta^{n_k})_{n_k \in \mathbb{N}}$, that converges in  $\mathcal{P}_1(\mathcal{X})$ to some limit $\theta\in\mathcal{P}_1(\mathcal{X})$. To prove this, we interpret $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ as relaxed feedback solutions, $(\nu,\lambda^n,\mu^n)_{n\in\mathbb{N}}$. Second, we show that also the sequence of the corresponding extended laws $(\Theta^n)_{n\in\mathbb{N}}\subset\mathcal{P}(\mathcal{X}\times\mathcal{V})$  converges in $\mathcal{P}_1(\mathcal{X}\times\mathcal{V})$ to some limit $\Theta\in\mathcal{P}_1(\mathcal{X}\times\mathcal{V})$. Here, a key role is played by Proposition \ref{prop:absContLimit} which ensures the absolute continuity of the limit measures with respect to $\mathcal{W}^{\nu}$. In this way, we are able to deal with the weak convergence of measures even in the presence of the discontinuities induced by the first exit time of the state processes from the set $\mathcal{O}$. Finally, we characterize the limit points by means of the martingale problem of Stroock and Varadhan (see \citet{stroock1969,stroock}).
\begin{lemma}[\textit{Tightness}]\label{lem:tightness}$(\theta^n)_{n\in\mathbb{N}}$ is tight in $\mathcal{P}(\mathcal{X})$.
\end{lemma}
\begin{proof}\noindent Let $t_1,t_2\in[0,T]$ with $t_1<t_2$.  First, notice that the laws $\theta^n_0\doteq\mathbb{P}^n\circ(X^n_0)^{-1}$ are trivially tight since they are all equal to the same law $\nu$. Second, by standard estimates and by using Lemma \ref{lem:alphaEstimate} we have for all $n\in\mathbb{N}$ that:
\begin{eqnarray}
\mathbb{E}^{\mathbb{P}^n}\brackets{\modl{X^n_{t_2}-X^n_{t_1}}}\leq \modl{t_2-t_1}\round{\modl{\Gamma}+C\round{1+K+C(1+K)}} +\modl{t_2-t_1}^{1/2}\modl{\sigma}C_T^{W},
\nonumber
\end{eqnarray}
\noindent where we set $C_T^{W}\doteq \mathbb{E}^{\mathbb{P}}[\|W\|_{\infty,T}]$ with $W$ a $\mathbb{P}$ Wiener process. We conclude by applying the Kolmogorov-Chentsov criterion (see, e.g., Corollary 14.9 in \citet{kallenberg}).
\end{proof}
\begin{corollary}[\textit{Relative compactness}]\label{cor:relComp}$(\theta^n)_{n\in\mathbb{N}}$ is relatively compact in $\mathcal{P}(\mathcal{X})$.
\end{corollary}
\begin{proof}\noindent This is a consequence of Lemma \ref{lem:tightness} and of Prohorov's Theorem.
\end{proof}
\noindent Now, let $\theta\in\mathcal{P}(\mathcal{X})$ be a limit point for $(\theta^n)_{n\in\mathbb{N}}$ and let $(\theta^{n_k})_{n_k\in\mathbb{N}}$ be a subsequence of $(\theta^n)_{n\in\mathbb{N}}$ such that 
$\theta^{n_k}\overset{w}{\rightharpoonup}\theta$ as $n_k\rightarrow\infty$. With a slight abuse of notation, in what follows we identify $(\theta^{n_k})_{n_k\in\mathbb{N}}$ with $(\theta^n)_{n\in\mathbb{N}}$. We now show that the latter convergence is actually stronger by proving that $(\theta^n)_{n\in\mathbb{N}}$ converges to $\theta$ in the 1-Wasserstein distance. 
 
\begin{lemma}[\textit{Convergence in the 1-Wasserstein distance}]\label{lem:conv1Wass}Let $(\theta^n)_{n\in\mathbb{N}}$  be as above. Then
$W_1(\theta^n,\theta)\rightarrow 0$ and $\theta\in\mathcal{P}_1(\mathcal{X})$.  
\end{lemma}
\begin{proof}\noindent Notice that by Lemma \ref{lem:alphaEstimate} $(\theta^n)_{n\in\mathbb{N}}\subset\mathcal{P}_1(\mathcal{X})$. To prove convergence in the 1-Wasserstein distance, we have to show that (see, for instance, Theorem 7.12.ii in \citet{villani})
\begin{eqnarray} 
\lim_{R\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbb{E}^{\mathbb{P}^n}\brackets{\infnorm{X}\mathbf{1}_{\cbr{\infnorm{X}\geq R}}}=0.
\nonumber
\end{eqnarray}
\noindent Set $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. Then, for any  $\epsilon>0$ by Young's and Markov's inequalities, and by Lemma  \ref{lem:alphaEstimate} we have:
\begin{eqnarray} 
\mathbb{E}^{\mathbb{P}^n}\brackets{\infnorm{X}\mathbf{1}_{\cbr{\infnorm{X}\geq R}}}
&\leq &\epsilon^{p}\frac{\mathbb{E}^{\mathbb{P}^n}\brackets{\infnorm{X}^{p}}}{p}+\dfrac{\mathbb{P}^n\cbr{\infnorm{X}\geq R}}{\epsilon^{q}q}\nonumber\\
&\leq &\epsilon^{p}\frac{K(p)}{p}+\dfrac{K}{\epsilon^{q}q R}\, ,
\nonumber
\end{eqnarray}
\noindent for some positive constants $K(p)$ and $K$ independent of $n\in\mathbb{N}$.  The conclusion immediately follow thanks to the fact that convergence in the 1-Wasserstein distance preserves the finiteness of the first moment.
\end{proof}
\begin{proposition}[\textit{Absolute continuity of limit measures}]\label{prop:absContLimit}Let $\theta,(\theta^n)_{n\in\mathbb{N}}\subset\mathcal{P}_1(\mathcal{X})$ be as in Lemma \ref{lem:conv1Wass}. Then $\theta$ is absolutely continuous with respect to $\mathcal{W}^{\nu}$, i.e. $\theta\ll\mathcal{W}^{\nu}$.
\end{proposition}
\begin{proof} By construction $\theta^n\ll\mathcal{W}^{\nu}$ for all $n\in\mathbb{N}$, hence we have to make sure that the absolute continuity is also preserved at the limit. For doing so, we are going to apply Theorem VI.4.13 in \citet{jacod}. In particular, we have to verify that all assumptions therein are fulfilled, which in our setting are reduced to the following properties:
\begin{itemize}
\item[(i)] the contiguity of the sequence of $\theta^n$ with respect to the Wiener measure $\mathcal{W}^{\nu}$, i.e. for any sequence of measurable sets $B_n$ with $\mathcal{W}^{\nu}(B_n) \to 0$ we have $\theta^n (B_n) \to 0$ as $n \to \infty$ (see, e.g., Definition V.1.1 in \cite{jacod});
\item[(ii)] the tightness of the sequence of $\mathcal{W}^{\nu}$-martingales $(M^n)_{n\in\mathbb{N}}$, where each $M^n=(M^n_t)_{t\in[0,T]}$ is defined as:
\begin{eqnarray} 
M^n_t\doteq\mathcal{E}_t\round{\int_0^{\cdot}\sigma^{-1}\round{u^n(s,X)+\bar{b}^n(s,X_s,L(s;\theta^n),m^n(s;\theta^n))}dW_s},\quad t\in[0,T].
\nonumber
\end{eqnarray}
\end{itemize}
\noindent In order to check property (i), we first show that the sequence of Radon-Nikodym derivatives $(\frac{d\theta^n}{d\mathcal{W}^{\nu}})_{n\in\mathbb{N}}$ is uniformly integrable under $\mathcal{W}^{\nu}$. This is a consequence of the following bound 
\begin{eqnarray}
\sup_{n\in\mathbb{N}}\mathbb{E}^{\mathcal{W}^{\nu}}\left[\left(\frac{d\theta^n}{d\mathcal{W}^{\nu}}\right)^p\right]<\infty, \quad p\in[1,\infty), \label{supRNder}
\end{eqnarray}
which follows from Corollary \ref{cor:expmartMoments} and by the fact that, by bounds in \citet{grigelionis} and by Lemma \ref{lem:alphaEstimate} all upper bounds are uniform in $n\in\mathbb{N}$.  

Now, property (i) can be obtained as follows: for all sequences of measurable sets $B_n$ with $\mathcal{W}^{\nu}(B_n) \to 0$, we have
\[ \theta^n (B_n) = \mathbb E^{\mathcal{W}^{\nu}} \left[  \frac{d\theta^n}{d\mathcal{W}^{\nu}} \mathbf 1_{B_n}\right] \to 0, \quad n \to \infty,\]
by an application of dominated convergence theorem due to the bound \eqref{supRNder}. Hence the sequence of measures $\theta^n$ is contiguous to $\mathcal{W}^{\nu}$.\medskip

\noindent Property (ii) follows from Aldous criterion (Condition VI.4.4 in \cite{jacod}) that is:
\begin{eqnarray} 
\lim_{\delta\rightarrow 0}\underset{n\rightarrow\infty}{\lim\sup}\sup_{\tau\leq\sigma\leq\tau+\delta}\mathcal{W}^{\nu}\round{\modl{M^n_{\sigma}-M^n_{\tau}}\geq r}=0
\label{eq:jacod}
\end{eqnarray}
\noindent for all $r>0$ and where $\tau$ and $\sigma$ are stopping times bounded by $T$. As a consequence, we will also have the tightness property for the pair $(X,M^n)_{n\in\mathbb{N}}$ under the measure $\mathcal{W}^{\nu}$. By Theorem VI.4.13 in \cite{jacod} it is sufficient to check the tightness property for the corresponding quadratic variation processes:
\begin{eqnarray} 
\langle M^n\rangle_t=\int_0^t \modl{\sigma^{-1}\round{u^n(s,X)+\bar{b}^n(s,X_s,L(s;\theta^n),m^n(s;\theta^n))}M^n_s}^2ds,\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent First, by Markov's inequality $\mathcal{W}^{\nu}(|\langle M^n\rangle_{\sigma}-\langle M^n\rangle_{\tau}|\geq r)\leq\frac{1}{r}\mathbb{E}^{\mathcal{W}^{\nu}}[|M^n_{\sigma}-M^n_{\tau}|]$. Then,  by Young's inequality for all $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$ we have:
\begin{eqnarray} 
&&\mathbb{E}^{\mathcal{W}^{\nu}}\brackets{\modl{\langle M^n\rangle_{\sigma}-\langle M^n\rangle_{\tau}}}\leq\nonumber\\
&&\leq \mathbb{E}^{\mathcal{W}^{\nu}}\brackets{\int_{\tau}^{(\tau+\delta)\wedge T} \modl{\sigma^{-1}}^2\modl{\round{u^n(s,X)+\bar{b}^n(s,X_s,L(s;\theta^n),m^n(s;\theta^n))}}^2\modl{M^n_s}^2ds}\nonumber\\
&&\leq \frac{1}{p}\modl{\sigma^{-1}}^2\int_{\tau}^{(\tau+\delta)\wedge T} \mathbb{E}^{\mathcal{W}^{\nu}}\brackets{\modl{\round{u^n(s,X)+\bar{b}^n(s,X_s,L(s;\theta^n),m^n(s;\theta^n))}}^{2p}}ds\nonumber\\
&&\quad +\frac{1}{q}\modl{\sigma^{-1}}^2\int_{\tau}^{(\tau+\delta)\wedge T} \mathbb{E}^{\mathcal{W}^{\nu}}\brackets{\modl{M^n_s}^{2q}}ds\nonumber\\
&&\leq \round{\frac{K(p)}{p}+\frac{K(q)}{q}}\modl{\sigma^{-1}}^2\round{(\tau+\delta)\wedge T-\tau}
\nonumber
\end{eqnarray}
\noindent for some positive constants $K(p)$ and $K(q)>0$ independent of $n \in \mathbb{N}$. Notice that the last inequality is a consequence of Lemma \ref{lem:alphaEstimate} and Property (i). Therefore, Aldous criterion in Eq.\eqref{eq:jacod} is satisfied.\medskip

\noindent After checking properties (i) and (ii) above, we can at last apply Theorem X.3.3 in \cite{jacod}, yielding that tightness of $(\mathcal{W}^{\nu}\circ(X,M^n)^{-1})_{n\in\mathbb{N}}$ implies the tightness of $(\theta^n\circ(X,M^n)^{-1})_{n\in\mathbb{N}}$. In particular, if $(\mathcal{W}^{\nu}\circ(X, M^n)^{-1})_{n\in\mathbb{N}}$ weakly converges to some $\Theta'$ in $\mathcal{P}(\mathcal{X}\times \mathcal X)$ then $(\theta^n\circ(X,M^n)^{-1})_{n\in\mathbb{N}}$ weakly converges to some other $\Theta''\ll\Theta'$ in $\mathcal{P}(\mathcal{X}\times \mathcal X)$, and the same holds true for their first marginals on $\mathcal{X}$. Therefore, we can conclude that $\theta\ll\mathcal{W}^{\nu}$.
\end{proof}
\noindent \textit{Compactification method.}\quad So far, we have established the convergence of the laws $(\theta^n)_{n\in\mathbb{N}}$ to some limit law $\theta$ in the 1-Wasserstein distance. Now, in order to prove the convergence of the approximating feedback solutions $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ to some feedback MFG solution $(\nu,u,\mu)$, we need to show that the sequence of optimal controls $(u^n)_{n\in\mathbb{N}}$ converges to a control $u$, which is optimal for the limit game. To do this, we interpret the sequence of strict        
feedback solutions $(\nu,u^n,\mu^n)_{n\in\mathbb{N}}$ as a sequence of relaxed feedback solutions  $(\nu,\lambda^n,\mu^n)_{n\in\mathbb{N}}$, by defining $\lambda^n:[0,T]\times\mathcal{X}\rightarrow\mathcal{P}(\Gamma)$ as $\lambda^n(t,\varphi)\doteq\delta_{u^n(t,\varphi)}$ for all $(t,\varphi)\in[0,T]\times\mathcal{X}$ and for all $n\in\mathbb{N}$. Furthermore, we identify each $\lambda^n$ with a stochastic relaxed control $\Lambda^n$.  We then fix a sequence of associated weak solutions $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q}^n,X^n,W^n)$ of Eq.\eqref{eq:MFGRel} and set $\Theta^n\doteq\mathbb{Q}^n\circ (X^n,\Lambda^n)^{-1}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ for all $n\in\mathbb{N}$. Finally, we associate to each MFG($n$) and to the limit MFG a martingale problem (\citet{stroock1969,stroock}) and show that the limit points $\Theta\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ of $(\Theta^n)_{n\in\mathbb{N}}$ solve the limit relaxed martingale problem. We start with the following lemma.
\begin{lemma}[\textit{Relative compactness}]\label{cor:relCompRel} $(\Theta^n)_{n\in\mathbb{N}}$ is relatively compact in $\mathcal{P}(\mathcal{X}\times\mathcal{V})$.
\end{lemma}
\begin{proof}\noindent This is a consequence of the relative compactness of $(\theta^n)_{n\in\mathbb{N}}$ in $\mathcal{P}(\mathcal{X})$ in Lemma \ref{cor:relComp} and of the compactness of $\mathcal{V}$, and hence of $\mathcal{P}(\mathcal{V})$.
\end{proof}
\noindent Now, let $\Theta\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ be a limit point for $(\Theta^n)_{n\in\mathbb{N}}$ and let $(\Theta^{n_k})_{n_k\in\mathbb{N}}$ be a subsequence of $(\Theta^n)_{n\in\mathbb{N}}$ such that $\Theta^{n_k}\overset{w}{\rightharpoonup}\Theta$ for $n_k\rightarrow\infty$. We now show that the latter convergence is actually stronger by proving that $(\Theta^n)_{n\in\mathbb{N}}$ converges to $\Theta$ in the 1-Wasserstein distance. Again, with a slight abuse of notation, we identify $(\Theta^{n_k})_{n_k\in\mathbb{N}}$ with $(\Theta^n)_{n\in\mathbb{N}}$.
\begin{lemma}[\textit{Convergence in the 1-Wasserstein distance}]\label{lem:conv1WassRel} Let $(\Theta^n)_{n\in\mathbb{N}}$ be as above. Then $W_1(\Theta^n,\Theta)\rightarrow 0$ and $\Theta\in\mathcal{P}_1(\mathcal{X}\times\mathcal{V})$.
\end{lemma}
\begin{proof}\noindent By Lemma \ref{lem:conv1Wass} $(\Theta^n\circ X^{-1})_{n\in\mathbb{N}}\subset\mathcal{P}_1(\mathcal{X})$ is relatively compact in $\mathcal{P}_1(\mathcal{X})$. Set now $|\theta|^{\alpha}\doteq\int_{E}|x|^{\alpha}\theta(dx)$ for all $\theta\in\mathcal{P}(E)$, $E$ a Polish space. Then it holds that:
\begin{eqnarray} 
\sup_{n\in\mathbb{N}}\mathbb{E}^{\mathbb{Q}^n}\brackets{\int_0^T\modl{\Lambda_t}^{\alpha}dt}\leq T\modl{\Gamma}^{\alpha}.
\nonumber
\end{eqnarray}
\noindent We conclude thanks to Proposition B.3 in \citet{lacker}.
\end{proof}
\begin{proposition}[\textit{Absolute continuity of limit measures}]\label{prop:absContLimitRel}Let $\Theta,(\Theta^n)_{n\in\mathbb{N}}\subset\mathcal{P}_1(\mathcal{X}\times\mathcal{V})$ be as in Lemma \ref{lem:conv1WassRel}. Then $\Theta\circ X^{-1}\ll\mathcal{W}^{\nu}$.
\end{proposition}
\begin{proof}
\noindent This is a consequence of Proposition \ref{prop:absContLimit}, the fact that by construction $\theta^n=\Theta^n\circ X^{-1}$ for all $n\in\mathbb{N}$, and the fact that weak convergence of the joint law implies weak convergence of the marginals.
\end{proof}

\noindent Finally, to characterize the limit points of $(\Theta^n)_{n\in\mathbb{N}}$, we associate to each approximating MFG($n$) and limit MFG a martingale problem, henceforth RM($n$) and RM respectively. Then, we show that limit points $\Theta$ are solutions of RM. 

We need some further preliminary notation for the definition of (approximating) martingale problem. Similarly to the definition of $b$ in Eq.\eqref{eq:driftCostRel} we set $b^n:[0,T]\times \mathcal{X}\times \mathcal{M}_{\leq 1}(\mathcal{X})\times\Gamma\rightarrow\mathbb{R}^d$ as 
\[ b^n(t,\varphi,\theta,u)\doteq u +\bar{b}^n(t,\varphi(t),L(t;\theta),m^n(t;\theta)).\]

\begin{definition}{\textit{The approximating martingale problems (RM($n$)).}}\label{def:relMartProbln} We say that $\widehat{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ is a solution of RM($n$) if for all $g\in \mathcal{C}^{2}_c(\mathbb{R}^d)$ the following process
\begin{eqnarray} 
M^{n,g}_t(\varphi,q;\widehat{\Theta})\doteq g(\varphi(t))-g(\varphi(0))-\int_{[0,t]\times\Gamma}b^n(s,\varphi,\hat{\theta},u)\nabla g(\varphi(s))q(ds,du)&&\nonumber\\
+\frac{1}{2}\int_0^t\sum_{i,j=1}^d\round{\sigma \sigma^T}_{i,j}\frac{\partial^2 g(\varphi(s))}{\partial x_i\partial x_j}ds,\quad t\in[0,T],&&
\nonumber 
\end{eqnarray}
\noindent is a $\widehat{\Theta}$-martingale, where $\hat{\theta}\doteq \widehat{\Theta}\circ X^{-1}$ and $X$ is the identity process on $\mathcal{X}$. \end{definition}
\noindent Observe that, by construction, each $\Theta^n$ solves RM($n$). In Proposition \ref{prop:approxMart}, instead, we characterize the limit points as solutions of the following relaxed martingale problem.
\begin{definition}{\textit{The limit martingale problem (RM).}}\label{def:relMartProbl} We say that $\widehat{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ is a solution of RM if for all $g\in \mathcal{C}^{2}_c(\mathbb{R}^d)$ the following process
\begin{eqnarray} 
M^{g}_t(\varphi,q;\widehat{\Theta})\doteq g(\varphi(t))-g(\varphi(0))-\int_{[0,t]\times\Gamma}b(s,\varphi,\hat{\theta},u)\nabla g(\varphi(s))q(ds,du)&&\nonumber\\
+\frac{1}{2}\int_0^t\sum_{i,j=1}^d\round{\sigma \sigma^T}_{i,j}\frac{\partial^2 g(\varphi(s))}{\partial x_i\partial x_j}ds,\quad t\in[0,T],&&
\nonumber 
\end{eqnarray}
\noindent is a $\widehat{\Theta}$-martingale, where $\hat{\theta}\doteq \widehat{\Theta}\circ X^{-1}$ and $X$ is the identity process on $\mathcal{X}$.
\end{definition}
\begin{rmk} \label{rem:martProble}  The martingale property in both RM($n$) and in RM is intended on $(\mathcal{X}\times\mathcal{V},\mathcal{B}(\mathcal{X}\times\mathcal{V}))$ with respect to the $\Theta$-augmentation of the canonical filtration made right continuous by a standard procedure. However, to conclude it is sufficient to check that the martingale property holds with respect to the canonical filtration on $\mathcal{X}\times\mathcal{V}$ (see, for instance, Problem 5.4.13 in \citet{karaztas}).
\end{rmk}
\noindent We now characterize the limit points via the martingale problems.
\begin{proposition}[\textit{Characterization of limit points via martingale problems}]\label{prop:approxMart} $\Theta$ solves RM as in Definition \ref{def:relMartProbl}.
\end{proposition}
\begin{proof}\noindent Fix $t_1,t_2\in[0,T]$, $t_1<t_2$, $g\in \mathcal{C}^2_c(\mathbb{R}^d)$ and $\psi\in\mathcal{C}_b(\mathcal{X}\times\mathcal{V})$ measurable with respect to $\mathcal{B}_{t_1}(\mathcal{X}\times\mathcal{V})$. Define $\Psi,\Psi^n:\mathcal{P}(\mathcal{X}\times\mathcal{V})\rightarrow\mathbb{R}$ as:
\begin{eqnarray} 
\Psi\round{\Theta';\Theta}&\doteq &\mathbb{E}^{\Theta'}\brackets{\psi\round{M^{g}_{t_2}(\cdot\,;\Theta)-M^{g}_{t_1}(\cdot\,;\Theta)}},\nonumber\\
\Psi^n\round{\Theta';\Theta}&\doteq &\mathbb{E}^{\Theta'}\brackets{\psi\round{M^{n,g}_{t_2}(\cdot\,;\Theta)-M^{n,g}_{t_1}(\cdot\,;\Theta)}},
\nonumber
\label{eq:psi}
\end{eqnarray}
\noindent for $\Theta',\Theta\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ and for all $n\in\mathbb{N}$. $\Psi^n(\Theta^n;\Theta^n)=0$ for all $n\in\mathbb{N}$ hence trivially $\Psi^n(\Theta^n;\Theta^n)\rightarrow 0$ as $n\rightarrow\infty$. We need to prove that $\Psi^n(\Theta^n;\Theta^n)\rightarrow\Psi(\Theta;\Theta)$ as $n\rightarrow\infty$. 
To do so, we show that both summands on the right-hand side of:
\begin{eqnarray} 
\modl{\Psi^n\round{\Theta^n;\Theta^n}-\Psi\round{\Theta;\Theta}}\leq \modl{\Psi^n\round{\Theta^n;\Theta^n}-\Psi\round{\Theta^n;\Theta}}+\modl{\Psi\round{\Theta^n;\Theta}-\Psi\round{\Theta;\Theta}}.
\nonumber
\end{eqnarray}
\noindent vanish as $n\rightarrow\infty$.\\
\indent\textit{Step 1.}\quad We show that $|\Psi^n(\Theta^n;\Theta^n)-\Psi(\Theta^n;\Theta)|\rightarrow 0$ as $n\rightarrow\infty$. Notice that $\Psi^n(\Theta^n;\Theta^n)-\Psi(\Theta^n;\Theta)=\mathbb{E}^{\Theta^n}[\psi\Delta M^{n,g}_{t_1,t_2}]$, where we set:
\begin{eqnarray} 
\Delta M^{n,g}_{t_1,t_2}(\varphi,q)\doteq\int_{[t_1,t_2]\times\Gamma}\round{b^n(t,\varphi,\theta^n,u)-b(t,\varphi,\theta,u)}q(dt,du),
\nonumber
\end{eqnarray}
\noindent for all $(\varphi,q)\in\mathcal{X}\times\mathcal{V}$. Using the Lipschitz continuity of the drifts and the truncation described in ($\mathbf{T}^n$), we have:
\begin{eqnarray} 
\modl{\psi\Delta M^{n,g}_{t_1,t_2}(\varphi,q)}&\leq &C_{\psi}C_{\nabla g}L\int_{t_1}^{t_2}\round{\modl{L(t;\theta^n)-L(t;\theta)}+\modl{m^n(t;\theta^n)-m(t;\theta)}}dt\nonumber\\
&+&C_{\psi}C_{\nabla g}\int_{t_1}^{t_2}\bar{b}(t,\varphi(t),L(t;\theta),m(t;\theta))\textbf{1}_{\{|\bar{b}|\geq K_n\}}dt,
\label{ineq-DeltaM}
\end{eqnarray}
\noindent for all $(\varphi,q)\in\mathcal{X}\times\mathcal{V} $ and where $C_{\psi},C_{\nabla g}>0$ are two uniform bounds for $|\psi|$ and $|\nabla g|$. Now, let us admit for the moment the following two continuity properties:
\begin{itemize} 
\item[($i$)]\quad $L(t;\theta^n)\rightarrow L(t;\theta)$ as $n\rightarrow\infty$,
\item[($ii$)]\quad $m^n(t;\theta^n)\rightarrow m(t;\theta)$ as $n\rightarrow\infty$,
\end{itemize}
that will be checked at the end of this proof. Properties (i) and (ii) above guarantee the convergence to zero (as $n \to \infty$) of the first summand in the right-hand side of inequality \eqref{ineq-DeltaM}.
Therefore, in order to perform this step of the proof, it is sufficient to show that:
\begin{eqnarray} 
\int_{\mathcal{X}\times\mathcal{V}}\int_{t_1}^{t_2}\bar{b}(t,\varphi(t),L(t;\theta),m(t;\theta))\textbf{1}_{\{|\bar{b}|\geq K_n\}}dt\Theta^n(d\varphi,dq)\underset{n\rightarrow\infty}{\longrightarrow} 0.
\nonumber
\end{eqnarray}
\noindent To do that, set $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. Then, for any $\epsilon>0$ by Young's and Markov's inequalities:
\begin{eqnarray} \mathbb{E}^{\Theta^n}\brackets{\int_{t_1}^{t_2}\bar{b}(t,\cdot,L(t;\theta),m(t;\theta))\textbf{1}_{\{|\bar{b}|\geq K_n\}}dt}\leq\round{\frac{\epsilon^{p}C^{p}K(p)}{p}+\frac{K}{K_n\epsilon^{q}q}}(t_2-t_1)
\label{eq:youngMarkov}
\end{eqnarray}
\noindent for some positive constants $K(p)$ and $K>0$ independent of $n\in\mathbb{N}$ (see Lemma \ref{lem:alphaEstimate}). 

To conclude this step of the proof, we need to check properties $(i)$ and $(ii)$. We denote by $D_{\mathbf{1}_{[0,\tau)}(t)}$ the set of discontinuity points of the map $\phi\mapsto\mathbf{1}_{[0,\tau(\phi))}(t)$ for $t\in[0,T]$.

\textit{Proof of ($i$).}\, By Corollary \ref{cor:relComp}, $\theta^n\overset{w}{\rightharpoonup}\theta$. Then:
\begin{eqnarray} 
L(t;\theta^n)-L(t;\theta)=-\int_{\mathcal{X}}\mathbf{1}_{[0,\tau(\phi))}(t)\round{\theta^n-\theta}(d\phi)\underset{n\rightarrow\infty}{\longrightarrow} 0
\nonumber
\end{eqnarray}
\noindent for all $t\in[0,T]$. This follows from the definition of weak convergence of measures and the fact that $\theta(D_{\mathbf{1}_{[0,\tau)}(t)})=0$ for all $t\in[0,T]$ by Proposition \ref{prop:absContLimit} and Lemma \ref{lem:regularity}.(d).\\
\indent\textit{Proof of ($ii$).}\, By Lemma \ref{lem:conv1Wass}, $W_1(\theta^n,\theta)\rightarrow 0$. In addition:
\begin{eqnarray} 
\modl{m^n(t;\theta^n)-m(t;\theta)}&\leq &\modl{\int_{\mathcal{X}}w(\phi(t))\mathbf{1}_{\{|w|\geq K_n\}}\mathbf{1}_{[0,\tau(\phi))}(t)\theta^n(d\phi)}\nonumber\\
&+&\modl{\int_{\mathcal{X}}w(\phi(t))\mathbf{1}_{[0,\tau(\phi))}(t)\round{\theta^n-\theta}(d\phi)}\underset{n\rightarrow\infty}{\longrightarrow} 0,
\nonumber
\end{eqnarray}
\noindent for all $t\in[0,T]$. Indeed, the first term converges to zero as $n\rightarrow\infty$ by applying Young and Markov's inequalities as in Eq.\eqref{eq:youngMarkov}. On the other hand, the vanishing of the second term as $n\rightarrow\infty$ is a consequence of the convergence in the 1-Wasserstein distance, Theorem 7.12.iv in \citet{villani}, and the fact that $\theta(D_{\mathbf{1}_{[0,\tau)}(t)})=0$ for all $t\in[0,T]$ by Proposition \ref{prop:absContLimit} and Lemma \ref{lem:regularity}.(d).\medskip

\indent\textit{Step 2.}\, The fact that $|\Psi(\Theta^n;\Theta)-\Psi(\Theta;\Theta)|\rightarrow 0$ as $n\rightarrow\infty$ is a consequence of $W_1(\Theta^n,\Theta)\rightarrow 0$ and of Theorem 7.12.iv in \citet{villani}.
\end{proof}
\begin{corollary}[\textit{Representation of limit points}]\label{cor:limitPointsRep} Let $\Theta$ be a solution of RM, as in Definition \ref{def:relMartProbl}. Then there exists a weak solution $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,t]},\mathbb{Q},X,\Lambda,W)$ of
\begin{eqnarray}
X_t &=& X_0 + \int_{[0,t]\times\Gamma}b\round{s,X,\theta,u}\Lambda_s(du)ds+ \sigma W_t,\quad t\in[0,T],
\label{eq:MFGRel2}
\end{eqnarray}
\noindent such that $\Theta$ is the law of $(X,\Lambda)$ under $\mathbb{Q}$ and $\theta=\Theta\circ X^{-1}$.
\end{corollary}
\begin{proof}
\noindent See, for instance, Proposition 5.4.6 and Corollary 5.4.8 in \citet{karaztas}.
\end{proof}

\subsection{Optimality of the limit points}
\noindent In this subsection, we show that any limit point $\Theta\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ of $(\Theta^n)_{n\in\mathbb{N}}$ is optimal according to the cost functional of the MFG. Moreover, we extend our notion of relaxed MFG solution further, in order to include controls that are in general not in feedback form. In this case we evaluate optimality according to the following cost functional:
\begin{eqnarray} 
\mathcal{J}^{\mu}\round{\Lambda}&\doteq &\mathbb{E}\Biggl[\int_{[0,\tau]\times\Gamma}f\round{s,X,\mu,u}\Lambda_s(du)ds+F\round{\tau,X_{\tau}}\Biggr]
\nonumber
\end{eqnarray}
\noindent for $\Lambda$ relaxed stochastic control and where $\tau \doteq \tau^{X} \wedge T$, subject to the following dynamics:
\begin{eqnarray}
X_t &=& X_0 + \int_{[0,t]\times\Gamma}b\round{s,X,\mu,u}\Lambda_s(du)ds+ \sigma W_t,\quad t\in[0,T].
\label{eq:MFGRel3}
\end{eqnarray}
\noindent We set $V^{\mu} =\inf_{\Lambda} J^{\mu}(\Lambda)$. The latter minimization is actually performed over the set of relaxed stochastic open-loop controls, i.e. over the tuples $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q},X,\Lambda,W)$ that are weak solutions of Eq.\eqref{eq:MFGRel2} and where $\Lambda$ is a progressively measurable $\mathcal{P}(\Gamma)$-valued stochastic process. Nonetheless, for simplicity of notation, we write $\Lambda$ also when we refer to tuples. Moreover, when working on the canonical space $\mathcal{X}\times\mathcal{V}$, where the canonical process $(X,\Lambda)$ is completely characterized by its law $\Theta$, we will simply identify relaxed stochastic controls with their laws and write $J^{\mu}(\Theta)$ in place of $J^{\mu}(\Lambda)$.
\begin{definition}[\textit{Relaxed MFG solution}]\label{def:MFGsolRel2} A relaxed solution of the MFG is a triplet $(\nu,\Lambda,\mu)$ such that:
\begin{itemize}
\item[($i$)] $\nu\in\mathcal{P}(\mathbb{R}^d)$ with $\text{supp}(\nu)\subset\mathcal{O}$, $\Lambda$ is a relaxed stochastic control and $\mu\in\Upsilon_{\leq 1,1}^{T}$.
\item[($ii$)] $\Lambda$ is optimal, i.e. $V^{\mu} = J^{\mu}(\Lambda)$.
\item[($iii$)] Let $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q},X,W)$ be a weak solution of Eq.\eqref{eq:MFGRel2} with flow of sub-probability measures $\mu$, stochastic control $\Lambda$ and initial condition $\nu$. Then 
\[ \mu_t(\cdot)=\mathbb{Q}(\{X_t \in \cdot\}\cap\{ \tau^X > t\}), \quad t\in[0,T].\]
\end{itemize}
\end{definition}
\begin{proposition}[\textit{Existence of relaxed MFG solutions}]\label{prop:existenceRelSolMFG} Grant assumptions (H1)-(H8) and (C1). Let $\Theta$ be a limit point of $(\Theta^n)_{n\in\mathbb{N}}$ in $\mathcal{P}_1(\mathcal{X}\times \mathcal{V})$. Set $\mu\in\Upsilon^T_{\leq 1,1}$ as
\begin{eqnarray} 
\mu_t\round{\cdot}\doteq\Theta\round{\cbr{X_t\in \cdot}\cap\cbr{\tau^{X}>t}}\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent Then $(\nu,\Theta,\mu)$ is a relaxed MFG solution according to Definition \ref{def:MFGsolRel2}.
\end{proposition}
\begin{proof}\noindent Property ($i$) in Definition \ref{def:MFGsolRel2} is satisfied by construction. Property ($iii$), instead, is a consequence of the fact that $\Theta$ is a solution of RM as in Definition \ref{def:relMartProbl}. To prove Property ($ii$), we proceed through the following steps:
\begin{itemize} 
\item[($j$)]\quad Let $\tilde{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ be a solution of RM. Then there exists a sequence of solutions $(\tilde{\Theta}^n)_{n\in\mathbb{N}}$ of RM($n$) such that $\lim_{n\rightarrow\infty}J^{\mu^n}(\tilde{\Theta}^n)=J^{\mu}(\tilde{\Theta})$.
\item[($jj$)]\quad $\lim_{n\rightarrow\infty}J^{\mu^n}(\Theta^n)=J^{\mu}(\Theta)$.
\item[($jjj$)]\quad $J^{\mu}(\Theta)\leq J^{\mu}(\tilde{\Theta})$ for any $\tilde{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ solution of RM.
\end{itemize}
\noindent The proof of $(j)-(jjj)$ largely follows that of Theorem 3.6 in \citet{lacker}. Therefore, in what follow we highlight only the main differences between the two results, which are due to the sub-linear growth of the drift and cost-functional, and to the history dependence and discontinuities induced in the coefficients by the presence of the first exit time from the set $\mathcal{O}$.\\
\indent\textit{Proof of ($j$).}\quad Let $\tilde{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ be a solution of RM. We construct a sequence of solutions $(\tilde{\Theta}^n)_{n\in\mathbb{N}}\subset\mathcal{P}(\mathcal{X}\times\mathcal{V})$ of RM($n$) such that $\tilde{\Theta}^n\overset{w}{\rightharpoonup}\tilde{\Theta}$. Toward this aim, let $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\tilde{\Theta},X,\Lambda,W)$ be a weak solution of Eq.\eqref{eq:MFGRel2} on the canonical space $\tilde{\Omega}=\mathcal{X}\times\mathcal{V}$. The existence of this solution is guaranteed by Corollary \ref{cor:limitPointsRep}. Now fix $\Lambda$ and let $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\tilde{\Theta},X^n,\Lambda,W)_{n\in\mathbb{N}}$ be a sequence of strong solutions of:
\begin{eqnarray} 
X^n_t=\xi+\int_{[0,t]\times\Gamma}b^n\round{s,X^n_s,L(s;\mu^n),m(s;\mu^n),u}\Lambda_s(du)ds+\sigma W_t,\quad t\in[0,T].
\nonumber
\end{eqnarray}
Set $\tilde{\Theta}^n\doteq\tilde{\Theta}\circ (X^n,\Lambda)^{-1}$ for each $n\in\mathbb{N}$. Notice $(\tilde{\Theta}^n)_{n\in\mathbb{N}}\subset\mathcal{P}_1(\mathcal{X}\times\mathcal{V})$. Moreover each $\tilde{\Theta}^n$ solves RM($n$) as in Definition \ref{def:relMartProbln}. We now show that:
\begin{eqnarray} 
\mathbb{E}^{\tilde{\Theta}}\brackets{\infnorm{X^n-X}}\underset{n\rightarrow\infty}{\longrightarrow} 0\quad\text{and}\quad W_1(\tilde{\Theta}^n,\tilde{\Theta})\underset{n\rightarrow\infty}{\longrightarrow} 0.
\label{eq:convL1}
\end{eqnarray}
Regarding the first limit, it is sufficient to note that:
\begin{eqnarray} 
\mathbb{E}^{\tilde{\Theta}}\brackets{\infnormt{X^n-X}}&\leq & L\int_0^t\mathbb{E}^{\tilde{\Theta}}\brackets{\infnorms{X^n-X}}ds+\mathbb{E}^{\tilde{\Theta}}\brackets{\int_{[0,t]\times\Gamma}\Delta b^n(s,u)\Lambda_s(du)ds},
\nonumber
\end{eqnarray}
\noindent where we set 
\[ \Delta b^n(t,u)\doteq |b^n(t,X_t,L(t;\mu^n),m^n(t;\mu^n),u)-b(t,X_t,L(t,\mu),m(t;u),u)|.\] 
The first term converges to zero by Gr\"onwall's Lemma, whereas the second by applying a similar argument as in the first step of the proof of Proposition \ref{prop:approxMart}. Regarding the second limit in Eq.\eqref{eq:convL1}, the first limit in the same equation implies convergence in probability, and therefore in law, of $X^n$ to $X$ (see, for instance, Theorem 13.7 in \citet{williams}). Thus, by an argument similar to that of Lemma \ref{lem:conv1WassRel} we can prove the convergence in the 1-Wasserstein distance. At this point, the convergence of the costs is a consequence of: i) the convergence in the 1-Wasserstein distance; ii) the sub-linearity of the running cost combined with Theorem 7.12.iv in \citet{villani}; iii) Lemma \ref{lem:regularity} applied to $\tilde{\Theta}\circ X^{-1}\in\mathcal{Q}_{\nu,|\Gamma|,C}$; iv) Lemma \ref{lem:regularityConvergence}. Indeed the running and the terminal costs have sub-linear growth and the set of their discontinuity points has measure zero with respect to the limit measure.\\
\indent\textit{Proof of ($jj$)}\quad This follows from an argument similar to the second part of $(j)$.\\
\indent\textit{Proof of ($jjj$)}\quad Let $\tilde{\Theta}\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ be a solution of RM and $(\tilde{\Theta}^n)_{n\in\mathbb{N}}\subset\mathcal{P}(\mathcal{X}\times\mathcal{V})$ an approximating sequence as in $(j)$. By the optimality of $\Theta^n$ we have
\begin{eqnarray} 
J^{n,\mu^n}\round{\Theta^n}\leq J^{n,\mu^n}\round{\tilde{\Theta}^n},
\nonumber
\end{eqnarray}
\noindent for all $n\in\mathbb{N}$. The optimality of $\Theta$ follows by taking the limit inferior for $n\rightarrow\infty$ in both sides of the inequality above.
\end{proof}
\subsection{Existence of solutions}
We conclude the proof of Theorem \ref{teo:existenceRelFeedSolMFG}, by proving the existence of a relaxed feedback MFG solution and, under additional convexity assumptions (\citet{filippov,haussmann}), the existence of a strict feedback MFG solution. In addition, we also prove existence of solutions that are Markovian up to the exit time.\\

\noindent\textit{Relaxed feedback MFG solutions}\quad The main mathematical tool here is the mimicking result of \citet{brunick}. We follow the procedure in \citet{lacker} but with modifications due the peculiarities of our model induced mainly by the presence of absorptions. We give more details in the proof below.
\begin{proposition}[\textit{Existence of relaxed feedback MFG solutions}]\label{prop:existenceRelFeedlSolMFG} Grant assumptions (H1)-(H8) and (C1). Let $(\nu,\Theta,\mu)$ be a relaxed MFG solution according to Definition \ref{def:MFGsolRel2}. Then there exist another relaxed MFG solution $(\nu,\Theta',\mu)$ and a progressively measurable functional $\lambda:[0,T]\times\mathcal{X}\rightarrow\mathcal{P}(\Gamma)$ such that $\Theta'((\varphi,q)\in\mathcal{X}\times\mathcal{V}:q_t=\lambda(t,\varphi))=1$ for Lebesgue-a.e. $t\in[0,T]$ and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$, i.e. $(\nu,\lambda,\mu)$ is a relaxed and feedback solution of the MFG according to Definition \ref{def:MFGsolRel}.
\end{proposition}
\begin{proof}
\noindent We adapt to our setting the proof of Theorem 3.7 in \citet{lacker}, by exploiting the mimicking result in Corollary 3.11 of \citet{brunick} instead of Corollary 3.7 as in \cite{lacker}. As a consequence, the mimicking process that we get is not Markovian as in Lacker. However, it has the same law of the original process. In particular, equality in law is needed due to the history dependence induced by the the presence of the first exit time from the set $\mathcal{O}$. 

We start with the construction of $\lambda$ by disintegration. Precisely, define $\eta\in\mathcal{P}([0,T]\times\mathcal{X}\times\Gamma)$ as:
\begin{eqnarray} 
\eta\round{I\times B\times G}&\doteq &\frac{1}{T}\mathbb{E}^{\Theta}\brackets{\int_{[0,T]\times\Gamma}\mathbf{1}_{\round{I\times B\times G}}\round{t,X,u}\Lambda\round{dt,du}}
\nonumber\end{eqnarray}
\noindent and disintegrate it as $\eta(dt,d\phi,du)=\tilde{\eta}(dt,d\phi)\lambda_{t,\phi}(du)$. Then:
\begin{eqnarray}
\eta\round{I\times B\times G}
&=&\int_{[0,T]\times\mathcal{X}\times}\int_{\Gamma}\mathbf{1}_{\round{I\times B\times G}}\round{t,\phi,u}\tilde{\eta}\round{dt,d\phi}\lambda_{t,\phi}\round{du}
\nonumber
\end{eqnarray}
\noindent for all $I\in\mathcal{B}([0,T])$, $B\in\mathcal{B}(\mathcal{X})$ and $G\in\mathcal{B}(\Gamma)$. By the disintegration Theorem, $(t,\phi)\mapsto \lambda_{t,\phi}(\cdot)\in\mathcal{P}(\Gamma)$ is Borel-measurable. Now set $\tilde{\mathcal{F}}^X_t\doteq \sigma(X_s,s\in[0,t])$ for each $t\in[0,T]$. We claim that:
\begin{eqnarray} 
\lambda_{t,X}\round{\cdot}&=&\mathbb{E}^{\Theta}\brackets{\Lambda_t\round{\cdot}\big| \tilde{\mathcal{F}}^X_t}\quad\Theta\text{-a.s. and for Lebesgue-a.e.}\,t\in[0,T]
\label{eq:lambdaProgmeas}
\end{eqnarray}
\noindent which is measurable and adapted, hence it has a progressively measurable modification $\lambda$. We show that for any bounded measurable functional $g:[0,T]\times\mathcal{X}\times\Gamma\rightarrow\mathbb{R}$ such that $g(t,\cdot,u)\in\tilde{\mathcal{F}}^X_t$ for all $t\in[0,T]$ and $u\in\Gamma$
\begin{eqnarray} 
\int_{\Gamma}g\round{t,X,u}\lambda_{t,X}\round{du}
&=&\int_{\Gamma}g\round{t,X,u}\mathbb{E}^{\Theta}\brackets{\Lambda_t\round{du}\big| \tilde{\mathcal{F}}^X_t}
\nonumber
\end{eqnarray}
\noindent $\Theta\text{-a.s. and for Lebesgue-a.e.}\,t\in[0,T]$. Indeed, for any other bounded measurable functional $h:[0,T]\times\mathcal{X}\rightarrow\mathbb{R}$ such that $h(t,\cdot)\in\tilde{\mathcal{F}}^X_t$ for all $t\in[0,T]$:
\begin{eqnarray} 
&&\frac{1}{T}\mathbb{E}^{\Theta}\brackets{\int_0^Th\round{t,X}\int_{\Gamma}g\round{t,X,u}\lambda_{t,X}\round{du}dt}\\
&=&\int_{[0,T]\times\mathcal{X}}h\round{t,\phi}\int_{\Gamma}g\round{t,\phi,u}\lambda_{t,\phi}\round{du}\tilde{\eta}\round{dt,d\phi}\nonumber\\
&=&\int_{[0,T]\times\mathcal{X}\times\Gamma}h\round{t,\phi}g\round{t,\phi,u}\eta\round{dt,d\phi,du}\nonumber\\
&=&\frac{1}{T}\mathbb{E}^{\Theta}\brackets{\int_0^Th\round{t,X}\int_{\Gamma}g\round{t,X,u}\Lambda_t\round{du}dt}
\nonumber
\end{eqnarray}
\noindent where the first equality comes from the definition of $\tilde{\eta}$, the second one is due to the disintegration of $\eta$ and the third one holds by definition of $\eta$.\\
\noindent Now, let $(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_t)_{t\in[0,T]},\mathbb{Q},W,X,\Lambda)$ be a weak solution of Eq.\eqref{eq:MFGRel3} with relaxed control $\Theta=\mathbb{Q}\circ(X,\Lambda)^{-1}$. By Corollary 3.11 in \cite{brunick} there exists a weak solution $(\tilde{\Omega}',\tilde{\mathcal{F}}',(\tilde{\mathcal{F}}'_t)_{t\in[0,T]},\mathbb{Q}',W',X')$ of
\begin{eqnarray} 
X'_t &=&\xi+\int_0^{t}\int_{\Gamma}b\round{s,X'_s,L(s;\mu),m(s;\mu),u}\lambda_{s,X'}(du)ds+\sigma W'_t,\quad t\in[0,T],
\nonumber
\end{eqnarray}
\noindent such that $\mathbb{Q}'\circ X'^{-1}=\mathbb{Q}\circ X^{-1}$. Define $\Theta'\doteq \mathbb{Q}'\circ (X',\Lambda')^{-1}$ where $\Lambda'(dt,du)\doteq dt\lambda_{t,X'}(du)$. Notice that if $\mu'$ is the flow of sub-probability measures associated to $\Theta'$ then $\mu'=\mu$. Finally, $\Theta'$ solves the same relaxed martingale problem as $\Theta$, and it has the same cost as $\Theta$ as required:
\begin{eqnarray} 
J^{\mu}\round{\Theta'}&=& 
\mathbb{E}^{\mathbb{Q}'}\brackets{\int_0^{\tau'}\int_{\Gamma}f\round{t,X'_t,L(t;\mu),m(t;\mu),u}\lambda_{t,X'}\round{du}dt+F\round{\tau',X'_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_0^{\tau}\int_{\Gamma}f\round{t,X_t,L(t;\mu),m(t;\mu),u}\lambda_{t,X}\round{du}dt+F\round{\tau,X_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_0^{\tau}\int_{\Gamma}f\round{t,X_t,L(t;\mu),m(t;\mu),u}\mathbb{E}^{\mathbb{Q}}\brackets{\Lambda_t\round{du}\big| \tilde{\mathcal{F}}^X_t}dt+F\round{\tau,X_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_0^{\tau}\int_{\Gamma}\mathbb{E}^{\mathbb{Q}}\brackets{f\round{t,X_t,L(t;\mu),m(t;\mu),u}\Lambda_t\round{du}\big| \tilde{\mathcal{F}}^X_t}dt+F\round{\tau,X_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_{[0,\tau]\times\Gamma}f\round{t,X_t,L(t;\mu),m(t;\mu),u}\Lambda\round{dt,du}+F\round{\tau,X_{\tau}}}\nonumber\\
&=&J^{\mu}\round{\Theta}.
\nonumber 
\end{eqnarray}
\end{proof}
\begin{rmk}
We observe that, due to the discontinuity induced by the exit time $\tau$, it is not possible in general to apply Theorem 3.6 of \citet{brunick} to $Z_t=(X_t,\mathbb{I}_{[0,\tau)}(t))$, $t\in[0,T]$, to obtain a control which is Markovian in $Z$. Moreover the few mimicking results available in the literature for discontinuous processes hold under very restrictive or hardly verifiable assumptions. Nonetheless, Theorem 3.6 in \cite{brunick} could still be applied in some particular cases when, for instance, $\mathcal{O}=(0,\infty)$ and $Z_t=(X_t,\inf_{s\in[0,t]} X_s)$.
\end{rmk}

\noindent\textit{Strict feedback MFG solutions}\quad Under additional convexity assumptions (\citet{filippov,haussmann}), we prove existence of feedback MFG solutions that can be written in strict form. Toward this aim, let $(\nu,\Theta,\mu)$ be a relaxed MFG solution according to Definition \ref{def:MFGsolRel2} and for all $(t,\varphi,u)\in[0,T]\times\mathcal{X}\times\Gamma$ set:
\begin{eqnarray} 
\tilde{b}(t,\varphi,\mu,u)&\doteq &b(t,\varphi(t),L(t;\mu),m(t;\mu),u),\nonumber \\
\tilde{f}(t,\varphi,\mu,u)&\doteq &f(t,\varphi(t),L(t;\mu),m(t;\mu),u).
\nonumber
\end{eqnarray}
\noindent For each $(t,\varphi)\in[0,T]\times\mathcal{X}$ define $K(t,\varphi,\mu)$ as:
\begin{eqnarray} 
K\round{t,\varphi,\mu}\doteq\cbr{\round{\tilde{b}\round{t,\varphi,\mu,u},z}\,:\,z\geq \tilde{f}\round{t,\varphi,\mu,u}\quad\text{and}\quad u\in\Gamma}.
\label{eq:setKConvex}
\nonumber
\end{eqnarray}
\noindent Existence of strict MFG solutions is established under the following additional convexity assumption:
\begin{itemize} 
\item[$\round{\mathrm{C2}}$] The running cost $f$ is convex in the control variable $u\in\Gamma$.
\end{itemize}
\begin{rmk}
\noindent Assumption (C2) is equivalent to requiring that the set $K(t,\varphi,\mu)$ is convex. This assumption is crucial to apply the measurable selection argument in \citet{haussmann,dufour}.
\end{rmk}
\begin{proposition}[\textit{Existence of strict feedback MFG solutions}]\label{prop:existenceStrictFeedlSolMFG} Grant assumptions (H1)-(H8), (C1) and assumption (C2). Let $(\nu,\Theta,\mu)$ be a relaxed MFG solution according to Definition \ref{def:MFGsolRel2}. Then there exists another relaxed MFG solution $(\nu,\Theta',\mu)$ and a progressively measurable functional $u\in\mathcal{U}_{fb}$ such that $\Theta'((\varphi,q)\in\mathcal{X}\times\mathcal{V}:q_t=\delta_{u(t,\varphi)})=1$ for Lebesgue-a.e. $t\in[0,T]$ and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$, i.e. $(\nu,u,\mu)$ is a strict and feedback solution of the MFG according to Definition \ref{def:MFGsol}.
\end{proposition}
\begin{proof}
\noindent Again, we follow the proof of Theorem 3.7 in \citet{lacker}, highlighting the main differences due to our different setting. The first part of the proof proceeds as in Proposition \ref{prop:existenceRelFeedlSolMFG}. Since for all $(t,\phi)\in[0,T]\times\mathcal{X}$ the pair $(\tilde{b}(t,\phi,\mu,u),\tilde{f}(t,\phi,\mu,u))$ is in $K(t,\phi,\mu)$ for all $u\in\Gamma$ and $K(t,\phi,\mu)$ is convex then also:
\begin{eqnarray} 
\int_{\Gamma}\round{\tilde{b}\round{t,\phi,\mu,u},\tilde{f}\round{t,\phi,\mu,u}}\lambda_{t,\phi}(du)\,\in K\round{t,\phi,\mu}.
\nonumber
\end{eqnarray}
\noindent By applying the measurable selection argument in \citet{haussmann,dufour} (with respect to the progressive $\sigma$-algebra, i.e. the $\sigma$-algebra generated by progressively measurable processes), we find a progressively measurable functional $u:[0,T]\times\mathcal{X}\rightarrow \Gamma$ such that:
\begin{eqnarray} 
\int_{\Gamma}\tilde{b}\round{t,\phi,\mu,u}\lambda_{t,\phi}(du)=\tilde{b}\round{t,\phi,\mu,u(t,\phi)}
\nonumber
\end{eqnarray}
\noindent and:
\begin{eqnarray} 
\int_{\Gamma}\tilde{f}\round{t,\phi,\mu,u}\lambda_{t,\phi}(du)\geq \tilde{f}\round{t,\phi,\mu,u(t,\phi)}
\nonumber
\end{eqnarray}
\noindent for all $(t,\phi)\in[0,T]\times\mathcal{X}$. Define $\Theta'\doteq \mathbb{Q}'\circ(X',\Lambda')^{-1}$ where $\mathbb{Q}'$ is as in the proof of Proposition \ref{prop:existenceRelFeedlSolMFG} and $\Lambda'(\phi,q)(dt,du)\doteq dt\delta_{u(t,\phi)}(du)$. $\Theta'$ solves the same relaxed martingale problem as $\Theta$. As for the costs
\begin{eqnarray} 
J^{\mu}\round{\Theta'}&=& 
\mathbb{E}^{\mathbb{Q}'}\brackets{\int_0^{\tau'}\int_{\Gamma}f\round{t,X'_t,L(t;\mu),m(t;\mu),u}\delta_{u(t,X')}(du)dt+F\round{\tau,X'_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}'}\brackets{\int_0^{\tau'}f\round{t,X'_t,L(t;\mu),m(t;\mu),u(t,X')}dt+F\round{\tau,X'_{\tau}}}\nonumber\\
&\leq &\mathbb{E}^{\mathbb{Q}'}\brackets{\int_0^{\tau'}\int_{\Gamma}f\round{t,X'_t,L(t;\mu),m(t;\mu),u}\lambda_{t,X'}(du)dt+F\round{\tau,X'_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_0^{\tau}\int_{\Gamma}f\round{t,X_t,L(t;\mu),m(t;\mu),u}\lambda_{t,X}\round{du}dt+F\round{\tau,X_{\tau}}}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_{[0,\tau]\times\Gamma}f\round{t,X_t,L(t;\mu),m(t;\mu),u}\Lambda\round{dt,du}+F\round{\tau,X_{\tau}}}\nonumber\\
&=&J^{\mu}\round{\Theta}.
\nonumber 
\end{eqnarray}
\noindent Given the optimality of $(\nu,\Theta,\mu)$ we already have the converse inequality, i.e. $J^{\mu}(\Theta)\leq J^{\mu}(\Theta')$. Hence $J^{\mu}(\Theta)=J^{\mu}(\Theta')$.
\end{proof}

\noindent We can finally give the proof of Theorem \ref{teo:existenceRelFeedSolMFG}.
\begin{proof}[Proof of Theorem \ref{teo:existenceRelFeedSolMFG}]
\noindent Grant assumptions (H1)-(H8) and (C1). Proposition \ref{prop:existenceRelSolMFG} guarantees existence of a relaxed MFG solution $(\nu,\Theta,\mu)$ according to Definition \ref{def:MFGsolRel2}. By Proposition \ref{prop:existenceRelFeedlSolMFG} there exists another relaxed MFG solution $(\nu,\Theta',\mu)$ and a progressively measurable functional $\lambda:[0,T]\times\mathcal{X}\rightarrow\mathcal{P}(\Gamma)$ such that $\Theta'((\varphi,q)\in\mathcal{X}\times\mathcal{V}:q_t=\lambda(t,\varphi))=1$ for Lebesgue-a.e. $t\in[0,T]$ and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$. Then $(\nu,\lambda,\mu)$ is a relaxed and feedback solution of the MFG according to Definition \ref{def:MFGsolRel}. 

Additionally grant assumption (C2). By Proposition \ref{prop:existenceStrictFeedlSolMFG} there exists another relaxed MFG solution $(\nu,\Theta',\mu)$ and a progressively measurable functional $u\in\mathcal{U}_{fb}$ such that $\Theta'((\varphi,q)\in\mathcal{X}\times\mathcal{V}:q_t=\delta_{u(t,\varphi)})=1$ for Lebesgue-a.e. $t\in[0,T]$ and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$. Then $(\nu,u,\mu)$ is a strict and feedback solution of the MFG according to Definition \ref{def:MFGsol}.
\end{proof}

\noindent\textit{Markovian MFG solutions}\quad We conclude this part with showing that  there exist relaxed and strict feedback solutions that are Markovian up to the exit time.
\begin{lemma}[\textit{Markovian MFG solutions}]\label{lem:existencemarkovSolMFG} Grant assumptions (H1)-(H8) and (C1). Let $(\nu,\Theta,\mu)$ be a relaxed MFG solution according to Definition \ref{def:MFGsolRel2}. Then there exist another relaxed MFG solution $(\nu,\Theta',\mu)$ and a function $\lambda:[0,T]\times\mathbb{R}^d\rightarrow\mathcal{P}(\Gamma)$ such that
\[ \mathcal L_T \otimes \Theta^\prime (\{(t,\varphi,q): q_t =\lambda(t,\varphi(t)), t \le \tau^X (\varphi)\})=1,\]
where $\mathcal L_T$ is the Lebesgue measure on $[0,T]$, and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$. Additionally, grant assumption (C2). Then there exists a function $u:[0,T]\times\mathbb{R}^d\rightarrow\Gamma$ such that
\[ \mathcal L_T \otimes \Theta^\prime (\{(t,\varphi,q): q_t =\delta_{u(t,\varphi(t))}, t \le \tau^X (\varphi)\})=1,\]
and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$.
\end{lemma}
\begin{proof}
\noindent We define the following processes:
\begin{eqnarray} 
Y_t\doteq (t,X_t),\quad
X^{\tau^X}_t\doteq X_{t\wedge\tau^X},\quad
Y^{\tau^X}_t\doteq  Y_{t\wedge\tau^X},
\nonumber
\end{eqnarray}
\noindent for $t\in[0,T]$. If $X$ satisfies Eq.\eqref{eq:MFGRel2} with flow of sub-probability measures $\mu$ and relaxed control $\Lambda$ then the SDE satisfied by $X^{\tau^X}$ is (on the same probability space)
\begin{eqnarray} 
X^{\tau^X}_t &=& \xi+\int_{[0,t]\times\Gamma}b\round{s,X^{\tau^X}_s,L(s;\mu),m(s;\mu),u}\mathbf{1}_{[0,\tau^X)}(s)\Lambda_s(du)ds\nonumber\\
&& +\sigma\int_0^t\mathbf{1}_{[0,\tau^X)}(s)dW_s,\quad t\in[0,T].
\label{eq:stoppedProcess}
\nonumber
\end{eqnarray}
\noindent Notice that until $t\leq \tau^X$ the stopped process $X^{\tau^X}$ coincides pathwise with the original process $X$. We now apply the mimicking result in  Corollary 3.7 of  \citet{brunick}, to the stopped process $Y^{\tau^X}$. To this end, we follow the proof of Theorem 3.7 in \citet{lacker} and the proofs of Propositions \ref{prop:existenceRelFeedlSolMFG} and \ref{prop:existenceStrictFeedlSolMFG} in the present paper.\\
First, we claim that there exists a measurable function $\lambda:[0,T]\times\mathbb{R}^{d+1}\rightarrow\mathcal{P}(\Gamma)$ such that
\begin{eqnarray} 
\lambda_{t,Y_t^{\tau^X}}(\cdot)&=&\mathbb{E}^{\Theta}\brackets{\Lambda_t(\cdot)\big| Y^{\tau^X}_t},\quad\Theta\text{-a.s. and for Lebesgue-a.e.}\,t\in[0,T],
\nonumber
\end{eqnarray}
\noindent and we construct it by disintegration. Let $\eta\in\mathcal{P}([0,T]\times\mathbb{R}^{d+1}\times\Gamma)$ as
\begin{eqnarray} 
\eta (B)&\doteq &\frac{1}{T}\mathbb{E}^{\Theta}\brackets{\int_{[0,T]\times\Gamma}\mathbf{1}_C\round{t,Y_t^{\tau^X},u}\Lambda(dt,du)}.
\nonumber
\end{eqnarray}
\noindent We define $\lambda$ through $\eta(dt,dy,du)\doteq\tilde{\eta}(dt,dy)\lambda_{t,y}(du)$. By Corollary 3.7 in \cite{brunick} applied to $\lambda_{t,Y_t^{\tau^X}}$ there exists a weak solution $(\tilde{\Omega}',\tilde{\mathcal{F}}',(\tilde{\mathcal{F}}'_t)_{t\in[0,T]},\mathbb{Q}',W',X')$ of
\begin{eqnarray} 
X'_t =\xi+\int_0^{t}\int_{\Gamma}b\round{s,X'_s,L(s;\mu),m(s;\mu),u}\mathbf{1}_{[0,\tau^{X'})}(s)\lambda_{s,Y_t^{\tau^{X'}}}(du)ds&&\nonumber\\
+\sigma\int_0^t\mathbf{1}_{[0,\tau^{X'})}(s) dW'_s,\quad t\in[0,T],&&
\nonumber
\end{eqnarray}
\noindent where $Y_t^{\tau^{X'}}\doteq(t\wedge\tau^{X'},X'_t)$ and $\mathbb{Q}'\circ (t\wedge\tau^{X'},X'_t)^{-1}=\mathbb{Q}\circ(t\wedge\tau^X,X^{\tau^X}_t)^{-1}$ for all $t\in[0,T]$, i.e. $Y^{\tau^{X'}}$ and $Y^{\tau^{X}}$ have the same time marginals. Now set $\tau'\doteq \tau^{X'}\wedge T$. Recall that $\Theta=\mathbb{Q}\circ (X,\Lambda)^{-1}$ and define $\Theta'\doteq \mathbb{Q}'\circ (X',\Lambda')^{-1}$ where $\Lambda'(dt,du)\doteq dt\lambda_{t,Y_t^{\tau^{X'}}}(du)$. Equality of the costs can be shown just as in the proof of Proposition \ref{prop:existenceRelFeedlSolMFG}:
\begin{eqnarray}
J^{\mu}\round{\Theta'}&=& 
\mathbb{E}^{\mathbb{Q}'}\brackets{\int_0^{\tau'}\int_{\Gamma}f(t,X'_t,L(t;\mu),m(t;\mu),u)\lambda_{t,t\wedge\tau^{X'},X'_t}(du)dt+F\round{\tau',X'_{\tau}}}\nonumber \\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_0^{\tau}\int_{\Gamma}f(t,X^{\tau^X}_t,L(t;\mu),m(t;\mu),u)\lambda_{t,t\wedge\tau^X,X^{\tau^X}_t}(u)dt+F\round{\tau,X^{\tau^X}_{\tau}}}\nonumber \\
&=&\mathbb{E}^{\mathbb{Q}}\brackets{\int_{[0,\tau]\times\Gamma}f(t,X^{\tau^X}_t,L(t;\mu),m(t;\mu),u)\Lambda(dt,du)+F\round{\tau,X^{\tau^X}_{\tau}}}\nonumber\\
&=& J^{\mu}\round{\Theta}.
\nonumber
\end{eqnarray}
Therefore, $\lambda:[0,T]\times[0,T]\times\mathbb{R}^d\rightarrow\mathcal{P}(\Gamma)$ is such that $\Theta'(q\in\mathcal{V}:q_t=\lambda(t,t\wedge\tau^{\hat{X}},\hat{X}^{\tau^{\hat{X}}}_t))=1$ for Lebesgue-a.e. $t\in[0,T]$ and $J^{\mu}(\Theta')=J^{\mu}(\Theta)=V^{\mu}$.\\
\noindent Consider now a weak solution $(\tilde{\Omega}'',\tilde{\mathcal{F}}'',(\tilde{\mathcal{F}}''_t)_{t\in[0,T]},\mathbb{Q}'',W'',X'')$ of
\begin{eqnarray} 
X''_t =\xi+\int_0^{t}\int_{\Gamma}b\round{s,X''_s,L(s;\mu),m(s;\mu),u}\lambda_{s,Y_t^{\tau^{X''}}}(du)ds+\sigma W''_t,\quad t\in[0,T],
\nonumber
\end{eqnarray}
\noindent where $Y_t^{\tau^{X''}}=(t\wedge\tau^{X''},X''_t)$. Set $\Theta''\doteq \mathbb{Q}''\circ (X'',\Lambda'')^{-1}$ where  $\Lambda''(dt,du)\doteq dt\lambda_{t,Y_t^{\tau^{X''}}}(du)$. In order not to make any confusion between specific solutions, here $(\hat{X},\hat{\Lambda})$ denotes the canonical process on $\mathcal{X}\times\mathcal{V}$. First, $\Theta'$ solves the martingale problem associated to
\begin{eqnarray}
\widehat{M}^{g}_t(\varphi,q)\doteq g(\varphi(t))-g(\varphi(0))
-\int_{[0,t]\times\Gamma}b(s,\varphi(s),L(s;\mu),m(s;\mu),u)\nabla g(\varphi(s))\mathbf{1}_{[0,\tau^{\hat{X}})}(s)q(ds,du)&&\nonumber\\
+\frac{1}{2}\int_0^t\sum_{i,j=1}^d\round{\sigma \sigma^T}_{i,j}\frac{\partial^2 g(\varphi(s))}{\partial x_i\partial x_j}\mathbf{1}_{[0,\tau^{\hat{X}})}(s)ds,\quad t\in[0,T].&&
\nonumber 
\end{eqnarray}
\noindent as well as the martingale problem associated to
\begin{eqnarray} 
M^{g}_t(\varphi,q)\doteq g(\varphi(t))-g(\varphi(0))-\int_{[0,t]\times\Gamma}b(s,\varphi(s),L(s;\mu),m(s;\mu),u)\nabla g(\varphi(s))q(ds,du)&&\nonumber\\
+\frac{1}{2}\int_0^t\sum_{i,j=1}^d\round{\sigma \sigma^T}_{i,j}\frac{\partial^2 g(\varphi(s))}{\partial x_i\partial x_j}ds,&&
\nonumber 
\end{eqnarray}
\noindent up to time $\tau^{\hat{X}}\wedge T$ in the sense that the martingale property is satisfied for all $s,t\in[0,T]$, $0\leq s<t< \tau^{\hat{X}}\wedge T$. Second, $\Theta''$ solves the latter martingale problem up to time $T$. Then $\Theta'$ and $\Theta''$ solve the same martingale problem up to time $\tau^{\hat{X}}\wedge T$. Moreover $\Theta''(q\in\mathcal{V}:q_t=\lambda(t,t\wedge\tau^{\hat{X}},\hat{X}_t))=1$ for Lebesgue-a.e. $t\in[0,T]$. If we set $\Theta_t\doteq\Theta\circ(\hat{X},\hat{\Lambda})_{\cdot\wedge t}^{-1}$ for all $\Theta\in\mathcal{P}(\mathcal{X}\times\mathcal{V})$ and $t\in[0,T]$, then by uniqueness of the solution of the martingale problem up to time $\tau^{\hat{X}}\wedge T$ \[ \Theta'_t (\cdot \cap \{t\le \tau^{\hat{X}} \wedge T\}) = \Theta''_t (\cdot \cap \{t \le \tau^{\hat{X}} \wedge T\}).\]
\noindent Hence $J^{\mu}(\Theta')=J^{\mu}(\Theta'')$. Now this $\Theta''$ satisfies item (iii) of Definition \ref{def:MFGsolRel2}.\\
To conclude notice that the process $Y^{\tau^{X''}}_t=(t\wedge\tau^{X''},X''_t)$ reduces to $(t,X''_t)$ before time $\tau^{X''}\wedge T$. Hence, also $\lambda_{t,Y_t^{\tau^{X''}}}$, with a slight abuse of notation, reduces to $\lambda_{t,X''_t}$. With the additional assumption (C2), the second part of this lemma follows from the proof of Proposition \ref{prop:existenceStrictFeedlSolMFG} applied to the stopped process $Y^{\tau^X}$.
\end{proof}
\section{Approximate Nash equilibria for the $N$-player game}\label{sec:Nplayer}
\noindent In this section, under conditions (H1)-(H8) with the additional assumptions (N1) and (N2) (see below), we show that if we have a feedback solution of the MFG, then we can construct a sequence of approximate Nash equilibria for the corresponding $N$-player game. The approximation is the content of Theorem \ref{teo:approxNashRel}  and Corollary \ref{cor:approxNash}. In order to prove this result, we interpret the $N$-player system as a system of $N$ interacting diffusions. The study of interacting particle systems goes back to \citet{mckean,sznitman,gartner}. While the usual mode of convergence of an $N$-particle system to its McKean-Vlasov limit is the convergence in law of the empirical measures, in the present paper we obtain a stronger form of propagation of chaos. Precisely, as in  \citet{lacker_chaos} but with possibly unbounded drift, we prove that the empirical measures converge in a stronger topology, the $\tau$-topology, which is widely used in the large deviations literature (see, for instance, Chapter 6.2 in \citet{dembo}); see Subsection \ref{subsec:chaos}.\medskip

\noindent In order to state the main results of this section, we need the following two additional assumptions, whose formulation requires some more terminology. We set $d^{TV}_t(\theta,\tilde{\theta})\doteq \sup_{B\in\mathcal{F}_t}|\theta(B)-\tilde{\theta}(B)|$ for all $\theta,\tilde{\theta}\in\mathcal{P}(\mathcal{X})$. For $t\in[0,T)$, $d_t$ is only a pseudo-metric on $\mathcal{P}(\mathcal{X})$. On the other hand, for $t=T$ it is a proper metric. In particular, $d^{TV}_T$ is called total variation distance. Wherever possible without confusion, we will refer to $d^{TV}_t$ as total variation distance for all $t \in \brackets{0,T}$.

\begin{itemize} 
\item[$\round{\mathrm{N1}}$] $b:[0,T]\times\mathcal{X}\times\mathcal{P}(\mathcal{X})\times\Gamma\rightarrow \mathbb{R}^d$ is Lipschitz continuous in $\theta\in\mathcal{P}(\mathcal{X})$, uniformly in the other variables in the total variation distance, i.e.:
\begin{eqnarray} 
\modl{b\round{t,\phi,\theta,u}-b\round{t,\phi,\theta',u}}\leq Ld^{TV}_t\round{\theta,\theta'}
\nonumber
\end{eqnarray}
\noindent for all $(t,\phi,u)\in[0,T]\times\mathcal{X}\times\Gamma$. 
\item[$\round{\mathrm{N2}}$] The running cost can be decomposed as:
\begin{eqnarray}
f(t,x,l,m,u)&=&f_0(t,x,u)+\bar{f}(t,x,l,m),\nonumber
\end{eqnarray}
where its components fulfill
\[
\modl{f_0(t,x,u)}\leq K\quad \text{and}\quad \modl{\bar{f}(t,x,l,m)}\leq  C\round{1+\modl{x}+\modl{m}},
\]
\noindent for all $(t,x,l,m,u)\in [0,T]\times\mathbb{R}^d\times[0,1]\times\mathbb{R}^{d_0}\times\Gamma$. Moreover, both $f_0:[0,T]\times\mathbb{R}^d\times\Gamma\rightarrow[0,\infty)$ and $\bar{f}:[0,T]\times \mathbb{R}^d\times [0,1]\times \mathbb{R}^{d_0}\rightarrow[0,\infty)$ are jointly continuous.
\end{itemize}
\begin{rmk} Assumption (N1) can be obtained, for instance, by choosing $w$ bounded, i.e. $\|w\|_{\infty}<\infty$. This leads to $L=C(1+\|w\|_{\infty})$.
\end{rmk}

\begin{theorem}[{Approximate Nash equilibria - relaxed}] Let $N \in \mathbb{N}$, $N>1$. Let $(\nu,\lambda,\mu)$ be a relaxed feedback MFG solution. Define $\boldsymbol{\lambda}^N\in\tilde{\mathcal{U}}^N_N$ as $\boldsymbol{\lambda}^N\doteq (\lambda^{N,i})_{i=1,\ldots,N}$ where $\lambda^{N,i}(t,\phi^N)\doteq \lambda(t,\phi^{N,i})$ for all $i=1,\ldots,N$, $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$.\\
\noindent Then under assumptions (H1)-(H8), (N1) and (N2), for every $\epsilon>0$ there exists a $N^{\epsilon}\in\mathbb{N}$ such that $\boldsymbol{\lambda}^N$ is an $\epsilon$-Nash equilibrium for the $N$-player game whenever $N\geq N^{\epsilon}$, i.e. for every $i=1,\ldots,N$ and for any deviation $\beta\in\tilde{\mathcal{U}}^N_1$:
\begin{eqnarray} 
J^{N,i}\round{\boldsymbol{\lambda}^{N}}\leq J^{N,i}\round{\brackets{\lambda^{N,-i},\beta}}+\epsilon
\nonumber
\label{eq:approxNashRel}
\end{eqnarray}
\noindent for all $N\geq N^{\epsilon}$.
\label{teo:approxNashRel}
\end{theorem}


\begin{corollary}[\textit{Approximate Nash equilibria - strict}] Let $N>1$. Let $(\nu,u,\mu)$ be a strict feedback MFG solution. Define $\textbf{\textit{u}}^N\in\mathcal{U}^N_N$ as $\textbf{\textit{u}}^N\doteq (u^{N,i})_{i=1,\ldots,N}$ where $u^{N,i}(t,\phi^N)\doteq u(t,\phi^{N,i})$ for all $i=1,\ldots,N$, $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$.\\
\noindent Then under assumptions (H1)-(H8), (N1) and (N2), for every $\epsilon>0$ there exists a $N^{\epsilon}\in\mathbb{N}$ such that $\textbf{\textit{u}}^N$ is an $\epsilon$-Nash equilibrium for the $N$-player game whenever $N\geq N^{\epsilon}$, i.e. for every $i=1,\ldots,N$ and for any deviation $v\in\mathcal{U}^N_1$:
\begin{eqnarray} 
J^{N,i}\round{\textbf{\textit{u}}^{N}}\leq J^{N,i}\round{\brackets{u^{N,-i},v}}+\epsilon
\nonumber
\label{eq:approxNash}
\end{eqnarray}
\noindent for all $N\geq N^{\epsilon}$.
\label{cor:approxNash}
\end{corollary}
\noindent 
Before proceeding, we define the empirical measure $\zeta^N$ of the $N$-player system (Eq.\eqref{eq:NgameRel}) as:
\begin{eqnarray} 
\zeta^N\round{\cdot}\doteq\frac{1}{N}\sum_{i=1}^N\delta_{X^{N,i}}\round{\cdot},
\label{eq:empiricalMeasure}
\end{eqnarray}
\noindent which is a $\mathcal{P}(\mathcal{X})$-valued random variable. Moreover, we fix a relaxed feedback MFG solution $(\nu,\lambda,\mu)$ and define (cfr. Theorem \ref{teo:approxNashRel} and Corollary \ref{cor:approxNash}) $\boldsymbol{\lambda}^N\in\tilde{\mathcal{U}}^N_N$ as $\boldsymbol{\lambda}^N\doteq (\lambda^{N,i})_{i=1,\ldots,N}$ where $\lambda^{N,i}(t,\phi^N)\doteq \lambda(t,\phi^{N,i})$ for all $i=1,\ldots,N$, $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$. In the next two subsections we consider the following $N$-particle system:
\begin{eqnarray} 
X^{N,1}_t &=& X^{N,1}_0 + \int_{[0,t]\times\Gamma}b\round{s, X^{N,1},\zeta^N,u}\beta\round{s,\textbf{X}^N}(du)ds+\sigma W^{N,1}_t\nonumber\\
X^{N,i}_t &=& X^{N,i}_0 + \int_{[0,t]\times\Gamma}b\round{s, X^{N,i},\mu_s^{N},u}\lambda\round{s,X^{N,i}}(du)ds+\sigma W^{N,i}_t,
\label{eq:Npart}
\end{eqnarray}
\noindent for $i = 2,\ldots,N$, $t\in[0,T]$ and where $\beta\in\tilde{\mathcal{U}}^N_1$ is a generic single-player control. Precisely, in Subsection \ref{subsec:chaos} we set $\beta(t,\phi^N)\doteq\lambda(t,\phi^{N,1})$ for $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$. In Subsection \ref{subsec:approxNash} instead, we let $\beta$ be generic, which means that we allow the first player to deviate from the MFG solution $\lambda$.

\subsection{Propagation of chaos results}\label{subsec:chaos}
\noindent In this subsection we regard Eq.\eqref{eq:Npart}, with $\beta(t,\phi^N)\doteq\lambda(t,\phi^{N,1})$ for $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$, as a system of $N$ interacting symmetric diffusions. We associate to this system a McKean-Vlasov equation and prove the particular instance of propagation of chaos needed to prove Theorem \ref{teo:approxNashRel} and Corollary \ref{cor:approxNash}.
\begin{definition}[\textit{McKean-Vlasov solution}] A law $\theta^*\in\mathcal{P}(\mathcal{X})$ is a McKean-Vlasov solution of equation:
\begin{eqnarray} 
X_t = X_0 + \int_{[0,t]\times\Gamma}b\round{s, X,\theta^*,u}\lambda\round{s,X}(du)ds + \sigma W_t,\quad t\in[0,T], \quad X_0 \overset{d}{\sim} \nu,
\label{eq:mcKean}
\end{eqnarray}
\noindent if there exists a weak solution $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,t]},\mathbb{P},X,W)$ with $\mathbb{P}\circ X^{-1}=\theta^*$ and $\mathbb{P}\circ X^{-1}_0=\nu$.
\label{def:mcKean}
\end{definition}
\noindent 
\noindent The following lemma ensures the well-posedness of Eq.\eqref{eq:mcKean}.
\begin{lemma}[\textit{Existence and uniqueness of McKean-Vlasov solutions}]\label{lem:mcKeanEU} Grant assumptions (H1)-(H8) and (N1)-(N2). Then existence and uniqueness holds for the McKean-Vlasov solution of Eq.\eqref{eq:mcKean}.
\end{lemma}
\begin{proof}
\noindent Let $\theta,\theta'\in\mathcal{Q}_{\nu,|\Gamma|,C}$ be two McKean-Vlasov solutions of Eq.\eqref{eq:mcKean}. By Pinsker's inequality there exists some constant $C_H>0$ such that $d^{TV}_T(\theta,\theta')\leq C_H\sqrt{\mathcal{H}(\theta\vert\theta')}$ where $\mathcal{H}(\theta\vert\theta')$ is the relative entropy between $\theta$ and $\theta'$. Hypothesis (N1) implies that:
\begin{eqnarray} 
\mathcal{H}(\theta_t\vert\theta'_t)&=&-\mathbb{E}^{\theta}\brackets{\log\round{\frac{d\theta'}{d\theta}\Bigr|_t}}
\leq  \frac{1}{2}|\sigma^{-1}|^2L^2\int_0^td_s(\theta,\theta')^2ds,\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent For $\alpha>0$, $\theta$ and $\theta' \in \mathcal{Q}_{\nu,|\Gamma|,C}$, let $d^{\alpha}(\theta,\theta')^2 \doteq \int_0^T\text{e}^{-\alpha t}d_t(\theta,\theta')^2dt$. In addition, let $\Psi:\mathcal{P}(\mathcal{X})\rightarrow\mathcal{P}(\mathcal{X})$ be the map that associates to each $\theta\in\mathcal{P}(\mathcal{X})$ the law of the solution of Eq.\eqref{eq:mcKean} with  $\theta$ in its drift. Assumption (N1) implies that 
\[ d^{\alpha}(\Psi(\theta),\Psi(\theta'))^2\leq \frac{1}{2}\frac{C_H}{\alpha}|\sigma^{-1}|^2L^2d^{\alpha}(\theta,\theta')^2.\] 
Therefore, $\Psi$ is a contraction for $\alpha$ large enough. The conclusion follows by Banach fixed point theorem.
\end{proof}

\noindent Now, we consider the sequence of empirical measures $(\zeta^N)_{N\in\mathbb{N}}$ in Eq.\eqref{eq:empiricalMeasure} associated to the $N$-particle systems in Eq.\eqref{eq:Npart}. We follow \citet{lacker_chaos} and we prove the convergence, both in law and in probability in the $\tau$-topology, of $(\zeta^N)_{N\in\mathbb{N}}$ to the McKean-Vlasov solution $\theta^*\in\mathcal{P}(\mathcal{X})$ of Eq.\eqref{eq:mcKean}. We remind that the $\tau$-topology on $\mathcal{X}$, denoted with $\tau(\mathcal{P}(\mathcal{X}))$, is the topology generated by the following sets:
\begin{eqnarray}
B_{f,x,\delta} \doteq\cbr{\pi\in\mathcal{P}(\mathcal{X}):\modl{\int_{\mathcal{X}}f(y)\pi(dy)-x}<\delta},
\nonumber
\end{eqnarray}
\noindent where $f:\mathcal{X}\rightarrow\mathbb{R}$ is a measurable and bounded function, $x\in\mathbb{R}$ and $\delta$ is a strictly positive constant. In particular, the $\tau$-topology is the coarsest topology that makes the maps $\pi\mapsto \int_{\mathcal{X}}f(y)\pi(dy)$ continuous for all functions $f:\mathcal{X}\rightarrow\mathbb{R}$ measurable and bounded (see, for instance, Chapter 6.2 in \citet{dembo}). 

Moreover, we denote by $w(\mathcal{P}(\mathcal{X}))$ the weak topology on $\mathcal{P}(\mathcal{X})$ and with $\mathcal{B}(\mathcal{P}(\mathcal{X}))$ the Borel $\sigma$-algebra on $\mathcal{X}$ generated by the open sets of the weak topology. The following lemma adapts Theorem 2.5.1-2 in \citet{lacker_chaos} to our working framework, in particular to the case of diffusions with possibly unbounded drift.
\begin{lemma}[\textit{Propagation of chaos}]\label{lem:propagationChaos} Grant assumptions (H1)-(H8) and (N1)-(N2). Let $\theta^* \in\mathcal{P}(\mathcal{X})$ be the unique McKean-Vlasov solution of Eq.\eqref{eq:mcKean}. Then $\zeta^N\overset{\mathcal{L}}{\longrightarrow} \theta^*$ as $N\rightarrow \infty$. Moreover
\begin{eqnarray} 
\lim_{N\rightarrow\infty}\mathbb{P}^N\round{\zeta^N \not\in B}=0
\nonumber
\end{eqnarray}
\noindent for all sets $B$ which are (open) neighbourhoods of $\theta^*$ in the $\tau$-topology and are measurable with respect to $\mathcal{B}(\mathcal{P}(\mathcal{X}))$.
\end{lemma}
\begin{proof}
\noindent Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space that supports an i.i.d. sequence of $\mathcal{X}$-valued random variables with law $\theta^*$. For each $N\in\mathbb{N}$ set $(\mathcal{F}^N_t)_{t\in[0,T]}$ to be the filtration generated by $X^1,\ldots,X^N$. Define:
\begin{eqnarray} 
W^i_t\doteq \sigma^{-1}\round{X^i_t-\xi-\int_{[0,t]\times\Gamma}b(s,X^i,\theta^*,u)\lambda(s,X^i)(du)ds},\,\, t\in[0,T],\,\, i\in\{1,\ldots,N\}.
\nonumber
\end{eqnarray}
\noindent In particular, $W^1,\ldots,W^N$ are independent Wiener processes on $(\Omega,\mathcal{F},(\mathcal{F}^N_t)_{t\in[0,T]},\mathbb{P})$. Fix $N\in\mathbb{N}$, and consider the tuple $(\Omega,\mathcal{F},(\mathcal{F}^N_t)_{t\in[0,T]},\mathbb{P},(X^{N,1},\ldots,X^{N,N}),(W^1,\ldots,W^N))$, with $X^{N,i}\doteq X^i$, for all $i=1,\ldots,N$. This is a weak solution of:
\begin{eqnarray} 
X^{N,i}=\xi+\int_{[0,t]\times\Gamma}b(s,X^{N,i},\theta^*,u)\lambda(s,X^{N,i})(du)ds+\sigma W^i_t,\quad t\in[0,T],\quad i\in\{1,\ldots,N\}.
\nonumber
\end{eqnarray}
\noindent Now, define $\mathbb{P}^N$ as the probability measure under which:
\begin{eqnarray} 
X^{N,i}_t=\xi+\int_{[0,t]\times\Gamma}b(s,X^{N,i},\zeta^N,u)\lambda(s,X^{N,i})(du)ds+\sigma W^{N,i}_t,\quad t\in[0,T],\,\,\, i\in\{1,\ldots,N\}
\nonumber
\end{eqnarray}
\noindent for some $\mathbb{P}^N$-Wiener process $W^N$. In particular, $\frac{d\mathbb{P}^N}{d\mathbb{P}}\doteq Z^N_T$ where, for all $t\in [0,T]$,
\begin{eqnarray} 
Z^N_t\doteq\mathcal{E}_t\round{\int_0^{\cdot}\sum_{i=1}^N\int_{\Gamma}\sigma^{-1}\round{b(s,X^{N,i},\zeta^N,u)-b(s,X^{N,i},\theta^*,u)}\lambda(s,X^{N,i})(du)dW^i_s}.
\nonumber
\end{eqnarray}
\noindent  Notice that $(\Omega,\mathcal{F},(\mathcal{F}^N_t)_{t\in[0,T]},\mathbb{P}^N,X^N,W^N)$ is a weak solution of the $N$-particle system in Eq.\eqref{eq:Npart}, with $\beta(t,\phi^N)\doteq\lambda(t,\phi^{N,1})$ for $t\in[0,T]$ and $\phi^N\in\mathcal{X}^N$. 

At this point, the rest of the proof can be performed as in \citet{lacker_chaos}, Theorem 2.5.1-2, along the following steps:
\begin{itemize} 
\item[$(i)$] $F_{t_1,t_2}:\mathcal{P}(\mathcal{X})\rightarrow\mathbb{R}$ defined as:
\begin{eqnarray} 
F_{t_1,t_2}(\theta)\doteq \int_{\mathcal{X}}\int_{t_1}^{t_2}\modl{\int_{\Gamma}\sigma^{-1}\round{b(s,\phi,\theta,u)-b(s,\phi,\theta^*,u)}\lambda(s,\phi)(du)}^2ds\theta^*(d\phi)
\nonumber
\end{eqnarray}
is $\tau$-continuous for all $t_1,t_2\in[0,T]$, $t_1<t_2$ and $\mathcal{B}(\mathcal{P}(\mathcal{X}))$-measurable. Moreover $F_{t_1,t_2}(\theta)\leq \tilde{L}(t_2-t_1)\mathcal{H}(\theta\vert\theta^*)$ for all $t_1,t_2\in[0,T]$, $t_1<t_2$ and for all $\theta\in\mathcal{P}(\mathcal{X})$.
\item[$(ii)$] $X^{N,1},X^{N,2},\ldots X^{N,N}$ are i.i.d. under $\mathbb{P}$, and therefore we can apply  Sanov's Theorem (e.g. Theorem 6.2.10 in \citet{dembo})
to $\mathbb{P}\circ(\zeta^N)^{-1}$.
\item[$(iii)$] Derive a large deviation principle for $\mathbb{P}^N\circ(\zeta^N)^{-1}$, precisely:
\begin{eqnarray} 
\underset{N\rightarrow\infty}{\lim\sup}\frac{1}{N}\log\mathbb{P}^N\round{\zeta^N\not\in B}\leq -\text{e}^{-\tilde{L}T}\inf_{\theta\not\in B}\mathcal{H}\round{\theta\vert\theta^*}
\nonumber
\end{eqnarray}
\noindent for all open neighbourhoods $B$ of $\theta$ in the $\tau$-topology that are $\mathcal{B}(\mathcal{P}(\mathcal{X}))$, $\tilde{L}>0$. Conclude by showing that $\inf_{\theta\not\in B}\mathcal{H}(\theta\vert\theta^*)\geq 0$.
\end{itemize}
\end{proof}
\subsection{Approximate Nash equilibria}\label{subsec:approxNash}
This section is devoted to the proof of one of the main results of the present paper, namely the construction of approximate Nash equilibria for the $N$-player game from a solution of the limit problem, whose existence has been proved in Section \ref{sec:mfg}. In particular, results of previous Subsection \eqref{subsec:chaos} enable us to perform the passage to the many player limit even if feedback MFG strategies are in general only progressively measurable functionals, possibly discontinuous in the state variable. Indeed, as we have observed in the introduction, in \citet{campi} the proof of the existence of approximated Nash equilibria for the $N$-player games was crucially based on the continuity of the MFG solution with respect to the state variable. In our setting, such a regularity property is no longer feasible due to the possibly unboundedness of the coefficients. Therefore, in order to circumvent this difficulty, we will make use of the strong form of propagation of chaos in Lemma \ref{lem:propagationChaos}, which allows to pass to the limit even without any continuity property for the MFG solution, at the price of stronger continuity for the drift.

We start with some preliminary estimates ensuring that the costs remain bounded in the mean-field limit despite the sub-linear growth.
\begin{lemma}[\textit{A-priori estimates}]\label{lem:NgameEstimates} Grant assumptions (H1)-(H8). Consider the dynamics in Eq.\eqref{eq:Npart}. Then for any $\alpha\geq 1$,
\begin{eqnarray} 
\sup_{N\in\mathbb{N}}\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorm{X^{N,i}}^{\alpha}}&\leq & K(\alpha)
\nonumber
\end{eqnarray}
\noindent for $i\in\{1,\ldots,N\}$ and where $K(\alpha)<\infty$ is a positive constant independent of $N$.
\end{lemma}
\begin{proof}
\noindent We show how to prove the a-priori estimates for $\alpha=1$ both in case the first player chooses the MFG control $\lambda$ and in the case of a generic deviation $\beta$. For $\alpha>1$ similar estimates to those in Lemma \ref{lem:alphaEstimate} apply here as well. First set $\lambda^{N,1}(t,\phi^N)\doteq\lambda(t,\phi^{N,1})$ for $(t,\phi^N)\in[0,T]\times\mathcal{X}^N$. Then by symmetry and Gr\"onwall's lemma:
\begin{eqnarray} 
\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnormt{X^{N,1}}}&\leq &
C(t)+C\round{1+C}\int_0^t\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorms{X^{N,1}}}ds \nonumber\\
&\leq &C(t)\text{e}^{C\round{1+C}t},\quad t\in[0,T]
\nonumber
\end{eqnarray}
\noindent where $C(t)\doteq \mathbb{E}{|\xi|}+T(|\Gamma|+2C+C^2)+|\sigma|E^{\mathbb{P}^{N}}[\infnormt{W^{N,1}}]$ and $\xi\overset{d}{\sim} \nu$.\\
\noindent Now set $\lambda^{N,1}(t,\phi^N)\doteq\beta(t,\phi^N)$ for $(t,\phi^N)\in[0,T]\times\mathcal{X}^N$ and for a generic $\beta\in\tilde{\mathcal{U}}^N_1$. In this case, we lose the symmetry among the players and so we have to deal with a system of inequalities:
\begin{eqnarray} 
\begin{cases}
\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnormt{X^{N,1}}}\leq C(t)+C\int_0^t\round{\round{1+\frac{C}{N}}\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorms{X^{N,1}}}+C\frac{N-1}{N}\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorms{X^{N,2}}}}ds,\\
\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnormt{X^{N,2}}}\leq C(t)+C\int_0^t \round{\frac{C}{N}\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorms{X^{N,1}}}+\round{1+C\frac{N-1}{N}}\mathbb{E}^{\mathbb{P}^{N}}\brackets{\infnorms{X^{N,2}}}}ds.
\end{cases}
\nonumber
\end{eqnarray}
The conclusion now follows by applying Gr\"onwall's lemma to both inequalities, inserting the second inequality into the first one and then applying Gr\"onwall's lemma again.
\end{proof}
\noindent We now prove the tightness of the sequence of laws $(\mathbb{P}^N\circ (\zeta^N)^{-1})_{N\in\mathbb{N}}$. Then, thanks to Lemma \ref{lem:propagationChaos}, we characterize the limit points of $(\mathbb{P}^N\circ (\zeta^N)^{-1})_{N\in\mathbb{N}}$ as McKean-Vlasov solutions of Eq.\eqref{eq:mcKean}; see Lemma \ref{lem:nplayerLimitPoints}.
\begin{lemma}[\textit{Tightness}]\label{lem:nplayerTight} Grant assumptions (H1)-(H8) and (N1)-(N2). Then the sequence $(\mathbb{P}^N\circ (\zeta^N)^{-1})_{N\in\mathbb{N}}$ is tight in $\mathcal{P}(\mathcal{P}(\mathcal{X}))$.
\end{lemma}
\begin{proof}
\noindent The tightness of such a sequence is a consequence of \citet{sznitman}, Proposition 2.2, and of the Kolmogorov-Chentsov criterion (see, for instance, Corollary 14.9 in \citet{kallenberg}).
\end{proof}
\begin{lemma}[\textit{Characterization of limit points}]\label{lem:nplayerLimitPoints} Grant assumptions (H1)-(H8) and (N1)-(N2). Let $(\mathbb{P}^n\circ (\zeta^n)^{-1})_{n\in\mathbb{N}}$ be a convergent subsequence of $(\mathbb{P}^N\circ (\zeta^N)^{-1})_{N\in\mathbb{N}}$. Let $\zeta$ be a random variable defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ with values in $\mathcal{P}(\mathcal{X})$ such that $\zeta^n\overset{\mathcal{L}}{\longrightarrow}\zeta$. Then
\begin{itemize} 
\item[(i)] $\zeta$ coincides $\mathbb{P}$-a.s. with the unique McKean-Vlasov solution $\theta^*$ of Eq.\eqref{eq:mcKean}.
\item[(ii)] The sequence $(\zeta^N)_{N\in\mathbb{N}}$ converges in probability in the $\tau$-topology (hence also in law) to $\theta^*$.
\end{itemize}
\end{lemma}
\begin{proof}
\noindent By Lemma \ref{lem:nplayerTight} there exists a convergent subsequence $(\mathbb{P}^n\circ (\zeta^n)^{-1})_{n\in\mathbb{N}}\subset \mathcal{P}(\mathcal{P}(\mathcal{X}))$ and a limit $\mathbb{P}\circ\zeta^{-1}\in\mathcal{P}(\mathcal{P}(\mathcal{X}))$ such that $\zeta^n\overset{\mathcal{L}}{\longrightarrow}\zeta$. Lemma \ref{lem:propagationChaos} guarantees the convergence in law of the entire sequence to the deterministic limit $\theta^*\in\mathcal{Q}_{\nu,|\Gamma|,C}$, which is the unique McKean-Vlasov solution of Eq.\eqref{eq:mcKean}. By uniqueness in law of the weak limit we have $\mathbb{P}\circ\zeta^{-1}=\delta_{\theta^*}$. This means $\zeta=\theta^*$ $\mathbb{P}$-a.s.. Lemma \ref{lem:propagationChaos} also gives convergence in probability in the $\tau$-topology of $(\zeta^N)_{N\in\mathbb{N}}$ to $\theta^*$.
\end{proof}
\begin{corollary}[\textit{Characterization of the convergence}]\label{cor:nplayerLimitPoints} Under the assumptions of Lemma \ref{lem:nplayerLimitPoints}:
\begin{itemize} 
\item[$(i)$] For all $f:\mathcal{X}\rightarrow\mathbb{R}$ Borel-measurable and bounded  and such that $\theta\mapsto\int_{\mathcal{X}}f(\phi)\theta(d\phi)$ is $\tau(\mathcal{P}(\mathcal{X}))$-continuous and $\mathcal{B}(\mathcal{P}(\mathcal{X}))$-measurable:
\begin{eqnarray} 
\mathbb{E}^{\mathbb{P}^N}\brackets{\int_{\mathcal{X}}f(\phi)\zeta^N(d\phi)}\underset{N\rightarrow\infty}{\longrightarrow}\mathbb{E}^{\mathbb{P}}\brackets{\int_{\mathcal{X}}f(\phi)\zeta(d\phi)}.
\nonumber
\end{eqnarray}
\item[$(ii)$] $\mathbb{P}^N\circ(X^{N,1},\zeta^N)^{-1}\overset{w}{\rightharpoonup} \theta^*\otimes\delta_{\theta^*}$. Moreover $\mathbb{P}^N\circ(X^{N,1})^{-1}\overset{w}{\rightharpoonup} \theta^*$ and $\mathbb{P}^N\circ(\zeta^N)^{-1}\overset{w}{\rightharpoonup} \delta_{\theta^*}$.
\end{itemize}
\end{corollary}
\begin{proof}
\noindent (i)\, This is a consequence of Lemma \ref{lem:propagationChaos}, Lemma \ref{lem:nplayerLimitPoints} and of the almost sure equality $\zeta=\theta^*$.\\
\indent (ii)\, We already know that $\mathbb{P}^N\circ(\zeta^N)^{-1}\overset{w}{\rightharpoonup} \delta_{\theta^*}$ from Lemma \ref{lem:nplayerLimitPoints}. Convergence of $\mathbb{P}^N\circ(X^{N,1})^{-1}$ to $\theta^*$ then follows from \citet{sznitman}, Proposition 2.2, and the symmetry of the system in Eq.\eqref{eq:NgameRel}.
\end{proof}
\begin{rmk}  Let $D\doteq\{\phi\in\mathcal{X}:\tau(\phi)\text{ is discontinuous at }\phi\}$. Since $\zeta\overset{a.s.}{=}\theta^*\in\mathcal{Q}_{\nu,|\Gamma|,C}$, by Lemma \ref{lem:regularity} $\theta^*(D)=0$ and the statement of Corollary \ref{cor:nplayerLimitPoints} holds for $f=\mathbf{1}_D$ as well.
\end{rmk}
\noindent Finally, we conclude this section with the proof of Theorem \ref{teo:approxNashRel} and of Corollary \ref{cor:approxNash}.
\begin{proof}[Proof of Theorem \ref{teo:approxNashRel}]
\noindent We prove Theorem \ref{teo:approxNashRel} through the following steps:
\begin{itemize} 
\item[$(j)$]\quad $\lim_{N\rightarrow\infty}J^{N,1}(\boldsymbol{\lambda}^N)=J^{\mu}(\lambda)$.
\item[$(jj)$]\quad Let $\beta^{N,1}\in\mathcal{U}^N_1$ be such that
\begin{eqnarray}
J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta^{N,1}])\leq\inf_{\beta\in\mathcal{U}^N_1}J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta])+\frac{\epsilon}{2}.
\nonumber
\end{eqnarray}
\noindent Then
\begin{eqnarray} 
\underset{N\rightarrow\infty}{\lim\inf}J^{N,1}\round{\brackets{\boldsymbol{\lambda}^{N,-1},\beta^{N,1}}}\geq  J^{\mu}(\lambda).
\nonumber
\end{eqnarray}
\item[$(jjj)$]\quad $J^{N,1}(\boldsymbol{\lambda}^N)\leq\inf_{\beta\in\mathcal{U}^N_1}J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta])+\epsilon$.
\end{itemize}
\noindent We consider the dynamics in Eq.\eqref{eq:Npart}. In $(j)$ we set $\lambda^{N,1}(t,\phi^N)=\lambda(t,\phi^{N,i})$ for all $(t,\phi)\in[0,T]\times\mathcal{X}$ and prove convergence of the first-player cost functional to the cost functional of the MFG. In $(jj)$ instead we allow the first player to deviate and choose $\lambda^{N,1}(t,\phi^N)=\beta^{N,1}(t,\phi^{N})$ for all $(t,\phi)\in[0,T]\times\mathcal{X}$ where $\beta^{N,1}\in\tilde{\mathcal{U}}^N_1$ generic single-player relaxed control. We conclude the proof in $(jjj)$ by combining the results in $(j)$ and $(jj)$.\\
\indent\textit{Proof of (j).}\, To prove that $J^{N,1}(\boldsymbol{\lambda}^N)\rightarrow J^{\mu}(\lambda)$ as $N\rightarrow \infty$ we split each cost functional in the sum of two terms:
\begin{eqnarray} 
J^{N,1}(\boldsymbol{\lambda}^N)&=&\mathbb{E}^{\mathbb{P}^N}\brackets{\int_{[0,T]\times\Gamma}\int_{\mathcal{X}}\mathbf{1}_{[0,\tau(\phi))}(t)f_0(t,\phi,u)\lambda(t,\phi)(du)\zeta^N(d\phi)dt}\nonumber\\
&&+\,\mathbb{E}^{\mathbb{P}^N}\brackets{\int_{0}^{T}\mathbf{1}_{[0,\tau^{N,1})}(t)\bar{f}(t,X^{N,1},\zeta^N)dt+F(\tau^{N,1},X^{N,1}_{\tau^{N,1}})}
\nonumber
\end{eqnarray}
\noindent and
\begin{eqnarray}
J^{\mu}(\lambda)&=&\mathbb{E}^{\mathbb{P}}\brackets{\int_{[0,T]\times\Gamma}\int_{\mathcal{X}}\mathbf{1}_{[0,\tau(\phi))}(t)f_0(t,\phi,u)\lambda(t,\phi)(du)\zeta(d\phi)dt}\nonumber\\
&& +\,\mathbb{E}^{\mathbb{P}}\brackets{\int_{0}^{T}\mathbf{1}_{[0,\tau)}(t)\bar{f}(t,X,\zeta)dt+F(\tau,X_{\tau})}.
\nonumber
\end{eqnarray}
\noindent Convergence of the first terms is a consequence of Corollary \ref{cor:nplayerLimitPoints} and of the boundedness of $f_0$. Convergence of the second terms is again a consequence of Corollary \ref{cor:nplayerLimitPoints}, of Lemma \ref{lem:nplayerLimitPoints}, sub-linearity of $\bar{f}$ and $F$, the fact that $\theta^*\in\mathcal{Q}_{\nu,|\Gamma|,C}$ together with Lemma \ref{lem:regularityConvergence}.\\
\indent\textit{Proof of (jj).}\, We follow the proof of Theorem 3.10 in \citet{lacker2018} with suitable modifications due to the possibly unbounded drift and history-dependence induced by the presence of the first exit time from $\mathcal{O}$.\\
\noindent Let $(\Omega^N,\mathcal{F}^N,(\mathcal{F}^N_t)_{t\in[0,T]},\mathbb{Q}^N,Y^N,W^N)_{N\in\mathbb{N}}$ be a weak solutions of the $N$-player system in Eq.\eqref{eq:Npart}. Let $(\zeta^N)_{N\in\mathbb{N}}$ be the associated empirical measures. Under $\mathbb{Q}^N$ the first player's dynamics is:
\begin{eqnarray} 
Y^{N,1}_t &=& Y^{N,1}_0 + \int_{[0,t]\times\Gamma}b(s,Y^{N,1},\zeta_Y^N,u)\beta^{N,1}(s,\textbf{Y}^{N})(du)ds + \sigma W^{N,1}_t,\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent Now let $\mathbb{P}^N$ the probability measure under which the first player's dynamics is instead:
\begin{eqnarray} 
Y^{N,1}_t &=& Y^{N,1}_0 + \int_{[0,t]\times\Gamma}b(s, Y^{N,1},\zeta_Y^N,u)\lambda(s,Y^{N,1})(du)ds + \sigma \tilde{W}^{N,1}_t,\quad t\in[0,T].
\nonumber
\end{eqnarray}
\noindent Precisely $\frac{d\mathbb{Q}^N}{d\mathbb{P}^N}= Z^N_T$ where, for all $t\in[0,T]$,
\begin{eqnarray} 
Z^N_t=\mathcal{E}_{t}\round{\int_0^{\cdot}\int_{\Gamma}b(s,Y^{N,1},\zeta_Y^N,u)(\beta^{N,1}(s,\textbf{Y}^{N})-\lambda(s,Y^{N,1}))(du)d\tilde{W}_s}.
\nonumber
\end{eqnarray}
\noindent Corollary \ref{cor:expmartMoments} gives the uniform integrability of the sequence of  exponential martingales $(Z^N)_{N\in\mathbb{N}}$. Therefore convergence of the empirical measures to $\theta^*$ in probability in the $\tau$-topology under $\mathbb{P}^N$ implies convergence of the empirical measures to the same limit in probability in the $\tau$-topology under $\mathbb{Q}^N$. Hence $\zeta_Y^N\overset{\mathcal{L}}{\longrightarrow}\theta^*$ under $\mathbb{Q}^N$ and:
\begin{eqnarray} 
\lim_{N\rightarrow\infty}\mathbb{Q}^N\round{\zeta_Y^N\not\in B}=0
\nonumber
\end{eqnarray}
\noindent for all sets $B$ which are neighbourhoods of $\theta$ in the $\tau$-topology and belong to $\mathcal{B}(\mathcal{P}(\mathcal{X}))$. The tightness of $(Y^{N,1})_{N\in\mathbb{N}}$ under $\mathbb{Q}^N$ still follows from their tightness under $\mathbb{P}^N$. Consider $(\beta^{N,1}(t,\textbf{Y}^N))_{t\in[0,T]}$ as a single-player relaxed stochastic open-loop control and denote it simply by $(\beta^{N,1}_t)_{t\in[0,T]}$. 
Regard $(Y^{N,1},\beta^{N,1},\zeta^N_Y)_{N\in\mathbb{N}}$ as a sequence of random variables with values in $\mathcal{X}\times\mathcal{V}\times\mathcal{P}(\mathcal{X})$. 
Compactness of $\mathcal{V}$ and tightness of $(Y^{N,1},\zeta^N_Y)_{N\in\mathbb{N}}$ imply the tightness of $(Y^{N,1},\beta^{N,1},\zeta^N_Y)_{N\in\mathbb{N}}$ under $\mathbb{Q}^N$. 

Let $(Y,\beta,\theta^*)$ be a limit point defined on some probability space with probability measure $\mathbb{Q}$. Then by a standard martingale argument it can be shown to satisfy:
\begin{eqnarray} 
Y_t=\xi+\int_{[0,t]\times\Gamma}b(s,Y,\theta^*,d)\beta_t(du)ds+\sigma W_t,\quad t\in[0,T],
\label{eq:representDeviation}
\end{eqnarray}
\noindent where $W$ is a $\mathbb{Q}$-Wiener process. As in $(j)$ we split $J^{N,1}([\lambda^{N,-1},\beta^{N,1}])$ in two terms:
\begin{eqnarray} 
J^{N,1}([\lambda^{N,-1},\beta^{N,1}])
&=&\mathbb{E}^{\mathbb{Q}^N}\brackets{\int_{[0,T]\times\Gamma}\mathbf{1}_{[0,\tau^{N,1})}(t)f_0(t,Y^{N,1},u)\beta^{N,1}_t(du)dt}\nonumber\\
&&+ \, \mathbb{E}^{\mathbb{Q}^N}\brackets{\int_{0}^T\mathbf{1}_{[0,\tau^{N,1})}(t)\bar{f}(t,Y^{N,1},\zeta^N_Y)dt+F(\tau^{N,1},Y^{N,1}_{\tau^{N,1}})}.
\nonumber
\end{eqnarray}
\noindent We move along a weakly converging subsequence of $(Y^{N,1},\beta^{N,1},W^{N,1})_{N\in\mathbb{N}}$ under $\mathbb{Q}^N$ to the limit point $(Y,\beta,W)$ in Eq.\eqref{eq:representDeviation}. Convergence of the first and second summands above now works as in the proof of $(j)$. Considering again the whole sequence, we obtain
\begin{eqnarray} 
\underset{N\rightarrow\infty}{\lim\inf}J^{N,1}([\lambda^{N,-1},\beta^{N,1}])
&\geq &\inf_{\beta}E^{\mathbb{Q}^N}\brackets{\int_{[0,T]\times\Gamma}\mathbf{1}_{[0,\tau)}(t)f(t,Y,\theta^*,u)\beta_t(du)dt+F(\tau,Y_{\tau})}\nonumber\\
&= &V^{\mu}
\nonumber
\end{eqnarray}
\noindent where the last infimum is taken over all relaxed stochastic open-loop controls and the last equality follows from the embedding of the set of strict controls into the set of relaxed controls and from the chattering lemma (\citet{karoui,fleming,bahlali}).\\
\indent\textit{Proof of (jjj).}\, This is a consequence of steps $(j)$ and $(jj)$. Indeed 
\begin{eqnarray} 
J^{N,1}(\boldsymbol{\lambda}^N)-\inf_{\beta\in\mathcal{U}^N_1}J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta])\leq
J^{N,1}(\boldsymbol{\lambda}^N)-J^{\mu}(\lambda)+J^{\mu}(\lambda)-J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta^{N,1}])+\frac{\epsilon}{2}.
\nonumber
\end{eqnarray}
\noindent Now by steps $(j)$ and $(jj)$ there exists $N^{\epsilon}\in\mathbb{N}$ such that for all $N\geq N^{\epsilon}$
\begin{eqnarray} 
J^{N,1}(\boldsymbol{\lambda}^N)-J^{\mu}(\lambda)\leq\frac{\epsilon}{4}\quad\text{and}\quad J^{\mu}(\lambda)-J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta^{N,1}])\leq\frac{\epsilon}{4}.
\nonumber
\end{eqnarray}
Hence we conclude that $J^{N,1}(\boldsymbol{\lambda}^N)\leq\inf_{\beta\in\mathcal{U}^N_1}J^{N,1}([\boldsymbol{\lambda}^{N,-1},\beta])+\epsilon$ for all $N\geq N^{\epsilon}$, which establishes the statement of Theorem \ref{teo:approxNashRel}.
\end{proof}
