# Mean-Field Liquidation Games with Market Drop-out

Guanxing Fu Paul P. Hager Ulrich Horst The Hong Kong Polytechnic University,
Department of Applied Mathematics, Hung Hom, Kowloon, Hong Kong;
<EMAIL_ADDRESS>G. Fu’s research is supported by Hong Kong RGC (ECS)
Grant No. 25215122 and NSFC Grant No. 12101523.Humboldt University Berlin,
Department of Mathematics, Unter den Linden 6,10099 Berlin; paul.hager@hu-
berlin.de.Humboldt University Berlin, Department of Mathematics and School of
Business and Economics, Unter den Linden 6, 10099 Berlin<EMAIL_ADDRESS>berlin.de.

###### Abstract

We consider a novel class of portfolio liquidation games with market drop-out
(“absorption”). More precisely, we consider mean-field and finite player
liquidation games where a player drops out of the market when her position
hits zero. In particular round-trips are not admissible. This can be viewed as
a no statistical arbitrage condition. In a model with only sellers we prove
that the absorption condition is equivalent to a short selling constraint. We
prove that equilibria (both in the mean-field and the finite player game) are
given as solutions to a non-linear higher-order integral equation with
endogenous terminal condition. We prove the existence of a unique solution to
the integral equation from which we obtain the existence of a unique
equilibrium in the MFG and the existence of a unique equilibrium in the
$N$-player game. We establish the convergence of the equilibria in the finite
player games to the obtained mean-field equilibrium and illustrate the impact
of the drop-out constraint on equilibrium trading rates.

AMS Subject Classification: 93E20, 91B70, 60H30

Keywords: portfolio liquidation, mean-field game, Nash equilibrium,
absorption, non-linear integral equations

##  1 Introduction

Models of optimal portfolio liquidation under market impact have received
substantial consideration in the financial mathematics literature in recent
years. Starting with the work of Almgren and Chriss [2] existence and
uniqueness of solutions in different settings have been established by a
variety of authors including [3], [4], [15], [19], [23], [24], [25], [27],
[29], [30], [31], and [32]. In this paper we consider a game-theoretic model
of optimal portfolio liquidation where a player drops out of the market as
soon as her position hits zero; both finite player games and the corresponding
mean-field game are considered.

Mean-field games (MFGs) of optimal liquidation without a strict liquidation
constraint and without market drop-out have been analyzed in the literature
before. Cardaliaguet and Lehalle [9] considered an MFG where each player has a
different risk aversion. Casgrain and Jaimungal [10, 11] considered games with
partial information and different beliefs, respectively. Huang et al. [28]
considered a game between a major agent who is liquidating a large number of
shares and many minor agents that trade against the major player. Mean-field
(type) games with strict liquidation constraint have so far only been analyzed
by Fu and co-workers [16, 17, 18] and Evangelista and Thamsten [14].

Our model is different, due to the market drop-out (“absorption”) constraint.
In particular, we do not allow round-trips where players with zero (initial)
position trade the asset to benefit from favorable future market dynamics. It
has been shown in many papers including [16, 17] that round-trips arise
naturally in equilibrium when players with different initial holdings interact
in the same market. At the same time beneficial round-trips are usually
considered statistical arbitrage. Although the players do not generate
positive profits in any state of the world, they generate profits on average
from trading an asset that they do not hold. Our absorption constraint may
hence be viewed as a “no statistical arbitrage condition.”

Requiring a player to drop out of the market as soon as her position hits zero
also avoids “hot potato effects” as they occur in [33, 34] where different
players repeatedly take long and short positions in the same asset to benefit
from their own positive impact on market dynamics. It has been argued by many
authors including [1, 20] that cyclic fluctuations in the players’ asset
holdings should be viewed as model irregularities or a form of arbitrage and
should hence be avoided.

In a benchmark model with only sellers we show that our drop-out constraint is
equivalent to a short-selling constraint where the players are not allowed to
change the direction of trading. This is in accordance with legal requirements
that ban market participants that liquidate portfolios on a client’s behalf to
take the opposite side of the market. In models with both buyers and sellers
we prove that if sellers initially dominate, then the drop-out constraint is
still equivalent to a no short-selling constraint on sellers, while buyers
with sufficiently small initial conditions may still benefit from first
selling the asset and then buying it back when the market environment is more
favorable. We illustrate numerically that the drop-out condition leads to
initially slower and eventually faster aggregate equilibrium trading rates
compared to models without drop-out constraint. In the absence of a drop-out
constraint some traders will initially “oversell” and buy the stock back
later.

The assumption of absorption leads to endogenously controlled trading
horizons. In stochastic settings games with endogenous trading horizons are
challenging to analyze; the literature on such games is hence sparse. Specific
stochastic MFGs with absorption have been considered by, e.g. Campi and
coauthors in [6, 7, 8]. Their arguments heavily rely on the non-degenerate
assumption on the state dynamics, thus cannot be applied to our model because
liquidation problems are degenerate. Graber and Bensoussan [21] studied
Bertrand and Cournot MFGs with absorption, where the state was not allowed to
be degenerate and thus it excludes the situation we are interested in. Graber
and Sircar [22] studied a similar model by solving a master equation with
Dirichlet boundary condition. Their arguments, too, rely on the non-degeneracy
assumption on the state dynamics. Dumitrescu et al. [13] developed a linear
programming approach to study MFGs with stopping and absorption. They use
relaxed solutions which allows them to prove existence results under weak
assumptions and which easily lends itself to numerical implementation. Cesari
and Zheng [12] established a necessary stochastic maximum principle for a
class of stochastic control problems with absorption under strong assumptions
that are difficult to verify in general.

The situation is simpler for deterministic MFGs. The work of Bonnans et al [5]
establishes an abstract existence of solutions result for a class of finite-
time MFGs of controls with mixed state-control and terminal state constraints.
Their analysis is based on a sophisticated, yet abstract fixed point argument
which makes it difficult to solve MFGs in closed form. A very different
approach that applies to a different class of deterministic MFGs has recently
been taken by Graewe et al. [26]. They considered a deterministic linear-
quadratic MFG of control in which players extract an exhaustible resource and
drop out of the game as soon as their resources hit zero. In their model the
representative player problem can be rephrased as a standard convex control
problem with state constraints and unknwon terminal value of the adjoint
process. The authors consider the Hamiltonian of the representative player
model only up a candidate optimal exploitation time in terms of which a
candidate terminal condition for the adjoint process can be obtained. The key
observation is that the dynamics of the amount of resource extracted by an
individual player up to any given time by using a strategy that would be
optimal if the optimal exhaustion time was equal to that given time is
independent of the equilibrium mean production rate.

Our approach is inspired by the work of Graewe et al [26]. However, unlike
their model, our control problem cannot be rewritten as a control problem with
pointwise constraints on the controls and/or the state process. This makes it
much more difficult to identify the Hamiltonian associated with the
representative player’s control problem. In the mean-field model we establish
a verification result that states that given the optimal liquidation time, the
“usual” Hamiltonian - and the forward-backward system derived from it by an
application of Pontryagin’s principle - where the absorption constraint is
ignored is indeed the “correct” Hamiltonian - and hence giving us the correct
forward-backward system on an endogenously determined time interval - of the
representative player’s problem if the exogenous mean trading rate does not
change sign. This reduces the problem of solving the MFG to finding the
optimal liquidation time for the representative agent problem.

While the sign condition on the mean trading rate is highly non-trivial and
calls for a novel approach to solving the MFG, it does provide a clear path
towards solving the game. Identifying Nash equilibria in the $N$-player game
is much more difficult. Due to the endogenously determined trading horizons it
seems impossible to derive a Nash equilibrium by solving a forward-backward
system of Pontryagin-type.

Our key idea is to replace the mean-trading rate in the co-state equations
(and only there) by an exogenous trading rate. This provides a unified
mathematical framework within which to solve both the $N$-player game and the
MFG. If $N=\infty$, then the system reduces to the forward-backward system for
the representative player’s problem in the MFG. If $N<\infty$, the system has
no direct economic meaning but turns out to be very useful for establishing
the existence of Nash equilibria in the $N$-player game.

The solutions to the modified forward-backward system allow us to define a
family of admissible trading strategies as functions of exogenous mean trading
rates in terms of which we derive a general fixed-point equation for the
candidate Nash equilibria in both games. The fixed-point equation can be
rewritten in terms of a complex integral equation that seems hard to solve at
first sight. However, motivated by the previously established verification
result for the MFG we expect any solution to the said equation to be of
constant sign. This motivates an a priori estimate from which we do indeed
deduce this property. The “constant sign property” substantially simplifies
the integral equation and eventually allows us to prove an existence and
uniqueness of solutions result.

Subsequently, we prove a necessary and sufficient maximum principle that
states that the solution to our integral equations yields the unique mean-
field equilibrium in the class of equilibria with continuous mean trading
rates. The analysis of the $N$-player game is more subtle as we do not have a
general verification result. Instead, we prove that a verification result
applies “in equilibrium” from which we deduce the existence of a unique Nash
equilibrium with continuous aggregate trading rate in the finite player game.
In a final step, we prove the convergence of the unique equilibria in the
$N$-player game to the mean-field equilibrium established before when the
number of players tends to infinity.

The reminder of this paper is organized as follows. In Section 2 we introduce
our liquidation models along with a family of benchmark trading strategies in
terms of which we can formulate our fixed-point condition for equilibrium
aggregate trading rates. The fixed-point equation is analyzed in Section 3.
Existence, uniqueness of convergence of equilibrium results are established in
Section 4. Section 5 illustrates the impact of drop-out constraints on
equilibrium trading. Our numerical simulations suggest that the MFG
equilibrium is a very good approximation of finite-player equilibria even for
relatively small numbers of players. Section 6 concludes.

Notation. For $y\in\mathbb{R}^{d_{1}\times d_{2}}$, $\|y\|$ denotes its
$2$-norm. For $y\in L^{\infty}([0,T]),C^{0}([0,T])$ we denote by
$\|y\|_{\infty}$ the supremum norm. For an almost everywhere differentiable
function $y$, we denote by $\dot{y}$ its derivative. For space $\mathcal{S}$
of real-valued functions we denote by $\mathcal{S}_{\pm}$ the subspace of
$\mathcal{S}$ whose elements take nonnegative/nonpositive values.

##  2 The Model

In this section we introduce a game-theoretic liquidation model with permanent
price impact where a player drops out of the game as soon as her portfolio
process hits zero.

###  2.1 Single player model with market drop-out

We consider the problem of a large investor that needs to liquidate a large
number $x\in\mathbb{R}$ of shares over the time interval $[0,T]$. Following
the majority of the optimal liquidation literature we assume that only
absolutely continuous trading strategies are allowed. In particular, the
portfolio process satisfies

$X_{t}=x-\int_{0}^{t}\xi_{s}\,ds,\quad t\in[0,T]$

where $\xi_{t}$ denotes the trading rate at time $t\in[0,T]$; positive rates
mean that the trader is selling the asset while negative rates mean she is
buying it. We assume that the player drops out of the market as soon as her
position hits zero. As a result, the set of admissible trading strategies is
given by the set

$\mathcal{A}_{x}:=\left\\{\xi\in
L^{2}([0,T])\;\bigg{|}\;\exists\tau\in(0,T]\text{ s.t. }\lvert
X_{t}\rvert>0\mbox{ for }t\in[0,\tau)\mbox{ and }X_{t}=0\mbox{ for
}t\in[\tau,T]\right\\}$

of all square integrable processes $\xi$ that satisfy almost surely the
liquidation constraint $X_{T}=0$ and whose associated portfolio process is
absorbed at zero.

We also assume that the unaffected price process against which the trading
costs are benchmarked follows some Brownian martingale $S$, and that the
trader’s transaction price process is given by

$\tilde{S}_{t}=S_{t}-\int_{0}^{t}\kappa_{s}\xi_{s}ds-\eta_{t}\xi_{t},\quad
t\in[0,T]$

for deterministic positive impact processes $\kappa$ and $\eta$. The integral
term accounts for permanent price impact while the term $\eta_{t}\xi_{t}$
accounts for the instantaneous price impact that does not affect future
transactions. Introducing an additional risk term $\lambda_{t}X_{t}^{2}$ for
some deterministic non-negative process $\lambda$ that penalizes slow
liquidation, the trader’s objective is to minimize the expectation of the cost
functional

$J(\xi):=\int_{0}^{T}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\xi_{t}X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt.$

Since all cost coefficients are deterministic it is well known that the
stochastic control problem reduces to the deterministic problem

$\min_{\xi\in{\mathcal{A}}_{x}}J(\xi)\quad\mbox{ s.t. }\quad
dX_{t}=-\xi_{t}dt.$ (2.1)

###  2.2 Game-theoretic liquidation models with market drop-out

In $N$-player games of optimal liquidation it is usually assumed that
transaction prices are driven by the average trading rate

$\displaystyle\overline{\xi}^{N}_{t}~{}:=~{}\frac{1}{N}\sum_{i=1}^{N}\xi^{k}_{t}$

and that the transaction price process for some player $i=1,...,N$ is of the
form

$\tilde{S}^{i}_{t}=S_{t}-\int_{0}^{t}\kappa_{s}\overline{\xi}^{N}_{s}ds-\eta_{t}\xi^{i}_{t}.$

Assuming that all players are homogeneous, the cost functional for a generic
player $i$ equals

$J(\xi^{i};\xi^{-i}):=\int_{0}^{T}\left(\frac{1}{2}\eta_{t}(\xi^{i}_{t})^{2}+\frac{\kappa_{t}X^{i}_{t}}{N}\sum_{j=1}^{N}\xi^{j}_{t}+\frac{1}{2}\lambda_{t}(X^{i}_{t})^{2}\right)\,dt$

and his optimization problem reads

$\min_{\xi\in{\mathcal{A}}_{x_{i}}}J(\xi^{i},\xi^{-i})\quad\mbox{ s.t. }\quad
dX^{i}_{t}=-\xi^{i}_{t}dt,\quad X_{0}^{i}=x_{i}.$ (2.2)

In the corresponding MFG the average trading rate is replaced by an exogenous
trading rate $\mu$, the representative player’s cost functional is given by

$J(\xi;\mu):=\int_{0}^{T}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\mu_{t}X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt$

and her control problem reads

$\min_{\xi\in\mathcal{A}_{x}}J(\xi;\mu)\quad\mbox{ s.t. }\quad
dX_{t}=-\xi_{t}dt,\quad X_{0}=x.$ (2.3)

Given an optimal trading rate $\xi^{*}(\mu)$ as a function of the exogenous
mean trading rate $\mu$ the equilibrium condition reads $\mu$ equals the
integral of $\xi^{*}(\mu)$ w.r.t. the initial distribution of the position. We
impose the following standing assumption on the cost coefficients.

######  Assumption 2.1.

The cost coefficients satisfy111The differentiability of $\kappa$ is only
needed to solve the $N$-player game. It is not needed to solve the MFG.

$\lambda\in L^{\infty}_{+}([0,T]),~{}\kappa\in C^{1}_{+}([0,T]),\quad\mbox{
and }\quad 1/\eta,\eta\in C^{1}([0,T];(0,\infty)).$

###  2.3 Heuristics

In this section we heuristically derive a family of strategies in terms of
which we can formulate a fixed-point problem for equilibrium mean trading
rates. Following Graewe et al [26] the main idea in solving our liquidation
games is to reduce the problem of finding the optimal liquidation strategies
to the problem of finding the optimal liquidation time for which we derive an
explicit representation.

One of the main difficulties when solving our liquidation games with market
drop-out is that the underlying control problems are control problems of
absorption that cannot be rewritten as control problems with pointwise
constraints on the controls and/or state process. As a result, it is not
obvious how to identify the Hamiltonians associated with a single/the
representative player’s optimization problem.

Our educated guess is that - in equilibrium, and hence for the optimal
liquidation time $\tau^{i}$ \- the “usual” Hamiltonian

$H(t,\xi^{i},X^{i},Y^{i};\xi^{-i})=-\xi^{i}Y^{i}+\frac{1}{2}\eta_{t}(\xi^{i})^{2}+\kappa_{t}\overline{\xi}^{N}_{t}X^{i}_{t}+\frac{1}{2}\lambda_{t}(X^{i}_{t})^{2}$

is in fact the “correct” Hamiltonian for player $i$’s optimization problem in
the $N$-player game. Assuming that this is in fact true, and taking partial
derivatives with respect to $X^{i}$ and $\xi^{i}$, we obtain the candidate
best response strategy

$\xi^{i}_{t}=\frac{Y^{i}_{t}-\frac{\kappa_{t}}{N}X^{i}_{t}}{\eta_{t}}$ (2.4)

in terms of the solution to a system of fully coupled systems of forward-
backward differential equations

$\left\\{\begin{split}\bigg{.}\dot{X}^{i}_{t}=&~{}-\frac{Y^{i}_{t}-\frac{\kappa_{t}}{N}X^{i}_{t}}{\eta_{t}}1_{\\{t\leq\tau^{i}\\}}\\\
\bigg{.}-\dot{Y}^{i}_{t}=&~{}\left(\lambda_{t}X^{i}_{t}+\kappa_{t}\overline{\xi}^{N}_{t}\right)1_{\\{t\leq\tau^{i}\\}}\\\
X_{0}^{i}=&~{}x_{i},\quad X_{T}^{i}=0\end{split}\right.,\qquad\text{for a.e.
}t\in[0,T].$ (2.5)

Solving these systems simultaneously for all players is challenging, due to
the dependence on the endogenous absorption times. Our idea is to consider
instead - for any $\delta\in[0,1]$, any initial position $x\in\mathbb{R}$,
aggregate trading rate $\mu$ and candidate liquidation time $\tau_{\mu}(x)$ \-
the auxiliary forward-backward system

$\left\\{\begin{split}\bigg{.}\dot{X}_{t}=&~{}-\frac{Y_{t}-\delta\kappa_{t}X_{t}}{\eta_{t}}1_{\\{t\leq\tau_{\mu}(x)\\}}\\\
\bigg{.}-\dot{Y}_{t}=&~{}\left(\lambda_{t}X_{t}+\kappa_{t}\mu_{t}\right)1_{\\{t\leq\tau_{\mu}(x)\\}}\\\
X_{0}=&~{}x,\quad X_{T}=0\end{split}\right.,\qquad\text{for a.e. }t\in[0,T].$
(2.6)

For $\delta=\frac{1}{N}$ this system corresponds to the system (2.5) where the
aggregate trading rate $\overline{\xi}^{N}$ in the co-state equation is
replaced by a generic trading rate $\mu$. For $\delta=0$ the system reduces to
the corresponding system for the MFG where $N=\infty$ and the term
$\frac{\kappa_{t}}{N}X_{t}$ drops out of the equation.

The above system provides a unified mathematical framework within which to
analyze both the $N$-player game and the MFG. Specifically, we prove that the
above system has a unique solution $(X^{\delta,x,\mu},Y^{\delta,x,\mu})$, that
the strategy

$\xi^{\delta,x,\mu}:=\frac{Y^{\delta,x,\mu}+\delta\kappa
X^{\delta,x,\mu}}{\eta}$ (2.7)

is admissible and that $\mu^{*}$ is an equilibrium aggregate trading rate in
the MFG, respectively the $N$-player game if and only if

$\mu^{*}=\int_{\mathbb{R}}\xi^{\delta,x,\mu^{*}}\nu_{0}(dx)$ (2.8)

for $\delta=0$, respectively $\delta=\frac{1}{N}$. Here, $\nu_{0}$ is a
probability measure on $\mathbb{R}$ that specifies the distribution of the
players’ initial positions.

####  2.3.1 Candidate strategies

The forward-backward system (2.6) can be solved by making the usual linear
ansatz $Y=AX+B$. The ansatz yields the following dynamics for the coefficient
processes:

$\left\\{\begin{split}-\dot{A}^{\delta}_{t}=&~{}-\frac{(A^{\delta})^{2}_{t}}{\eta_{t}}+\delta\frac{\kappa_{t}}{\eta_{t}}A^{\delta}_{t}+\lambda_{t}\\\
-\dot{B}^{\delta}_{t}=&~{}\left(-\frac{A^{\delta}_{t}B^{\delta}_{t}}{\eta_{t}}+\kappa_{t}\mu_{t}\right)1_{\\{t\leq\tau_{\mu}(x)\\}}\\\
\lim_{t\nearrow T}A^{\delta}_{t}=&~{}\infty,\quad
B^{\delta}_{\tau_{\mu}(x)}=0\end{split}\right..$ (2.9)

######  Remark 2.2.

The singular terminal condition on the process $A^{\delta}$ reflects the
liquidation constraint at the terminal time. We notice that the dynamics of
the process $A^{\delta}$ is independent of $\tau_{\mu}(x)$.

The existence of a unique process $A^{\delta}$ that satisfies the singular
terminal value problem follows from Lemma A.3. The second equality in (2.9)
then yields that

$B^{\delta,x,\mu}_{t}=1_{\\{t\leq\tau_{\mu}(x)\\}}\int_{t}^{\tau_{\mu}(x)}e^{-\int_{t}^{s}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\kappa_{s}\mu_{s}\,ds,\qquad
t\in[0,T].$ (2.10)

In particular, the strategy $\xi^{\delta,x,\mu}$ is well-defined. In what
follows we prove that the strategy is admissible if $\tau_{\mu}(x)$ is a
liquidation time.

If $\tau_{\mu}(x)$ is a liquidation time, then the forward-backward systems
(2.6) and (2.9) are equivalent, and the corresponding portfolio process can be
represented as

$\displaystyle
X_{t}=\left\\{xe^{-\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}-\int_{0}^{t}\frac{1}{\eta_{s}}e^{-\int_{s}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\int_{s}^{\tau_{\mu}(x)}\kappa_{u}\mu_{u}e^{-\int_{s}^{u}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\,du\,ds\right\\}1_{\\{t\leq\tau_{\mu}(x)\\}}.$
(2.11)

The representation allows us to identify the liquidation time. We start with
the following lemma that shows that liquidation always takes place at or
before the terminal time. The proof is standard and follows from (2.11)
together with Lemma A.1 and Lemma A.3.

######  Proposition 2.3.

If $\tau_{\mu}(x)=T$, then $X_{\tau_{\mu}(x)}=0$.

Let us now turn to the case where early liquidation takes place. In this case
the liquidation time satisfies $\tau_{\mu}(x)<T$ and hence the equation

$\int_{0}^{\tau_{\mu}(x)}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\int_{s}^{\tau_{\mu}(x)}\kappa_{u}\mu_{u}e^{-\int_{s}^{u}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\,du\,ds=x.$
(2.12)

To identify those initial positions for which early liquidation takes place we
introduce the function

$h^{\delta}_{t}:=e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{t}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{2A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds.$

As we will see, the function $h^{\delta}$ features in the characterization of
the absorption time. The proof of the following lemma is given in the
appendix.

######  Lemma 2.4.

The function $h^{\delta}$ is strictly increasing, differentiable, and both
$h^{\delta},$ $\dot{h}^{\delta}$ and $1/\dot{h}^{\delta}$ are bounded
uniformly in $\delta\in[0,1]$. Moreover,

$\alpha^{\delta}_{T}:=\left(\lim_{t\to T}h^{\delta}_{t}\right)^{-1}=\lim_{t\to
T}A^{\delta}_{t}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\in(0,\infty),$
(2.13)

and for almost every $t\in[0,T]$ it holds that $h_{t}^{\delta}\rightarrow
h_{t}^{0}$ and $\dot{h}_{t}^{\delta}\rightarrow\dot{h}_{t}^{0}$, as
$\delta\rightarrow 0$.

We can apply Fubini’s theorem to rewrite the left hand side of the equation
(2.12) as a function of time

$\begin{split}&\int_{0}^{t}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\int_{s}^{t}\kappa_{u}\mu_{u}e^{-\int_{s}^{u}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\,du\,ds\\\
=&~{}\int_{0}^{t}\kappa_{u}\mu_{u}e^{-\int_{0}^{u}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{u}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{2A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds\,du\\\
=&~{}\int_{0}^{t}\kappa_{u}\mu_{u}h^{\delta}_{u}\,du\\\
=:&~{}f_{\mu}(t).\end{split}$ (2.14)

In view of the preceding lemma the function $f_{\mu}$, along with its running
maximum and minimum

$\overline{f}_{\mu}(t):=\max_{0\leq s\leq
t}f_{\mu}(s),\qquad\underline{f}_{\mu}(t):=\min_{0\leq s\leq t}f_{\mu}(s)$
(2.15)

is well defined. Using the convention $\inf\emptyset=T$ the candidate
liquidation time is then given by

$\tau_{\mu}(x)=\inf\left\\{t\in[0,T]:f_{\mu}(t)=(x\wedge\overline{f}_{\mu}(T))\vee\underline{f}_{\mu}(T)\right\\}.$
(2.16)

In particular, we expect that early liquidation takes place if the player’s
initial position $x$ satisfies

$x\in\Big{(}\underline{f}_{\mu}(T),\overline{f}_{\mu}(T)\Big{)}.$

######  Remark 2.5.

The above analysis also suggests that $\tau_{\mu}(x)=T$ if the sign of $\mu$
is always positive while the sign of $x$ is negative, or vice versa. That is,
if the aggregate trading rate is always positive (negative), then we expect
early liquidation for a buyer (seller) not to be beneficial in equilibrium as
the price process is driven in a favorable direction.

The following lemma shows that with our choice of the liquidation time the
strategy $\xi^{\delta,x,\mu}$ is admissible and establishes uniform estimates
that will be useful for our subsequent equilibrium analysis.

######  Lemma 2.6.

  * i)

Let $\mu\in L^{1}([0,T])$, $x\in\mathbb{R}$ and $\tau_{\mu}(x)$ be given by
(2.16). Then there exists a unique solution

$(Y^{\delta,x,\mu},X^{\delta,x,\mu})\in C^{0}([0,T])\times C^{0}([0,T])$

to the forward-backward ODE (2.6). Moreover, the candidate strategy

$\xi^{\delta,x,\mu}:=\frac{Y^{\delta,x,\mu}-\delta\kappa
X^{\delta,x,\mu}}{\eta}\in\mathcal{A}_{x}$ (2.17)

is admissible and absolutely continuous on $[0,T]$, for all $x\in\mathbb{R}$
and $\delta\in[0,1]$.

  * ii)

There exists a constant $C>0$ that depends only on $\mu,\eta,\lambda,\kappa$
and $T$ such that

$\displaystyle\|\xi^{\delta,x,\mu}\|_{\infty}+\|\dot{\xi}^{\delta,x,\mu}\|_{\infty}\leq
C(1+\lvert x\rvert),\qquad x\in\mathbb{R}.$ (2.18)

###### Proof.

  * i)

Since $X^{\delta,x,\mu}_{\tau_{\mu}(x)}=0$ the ODE systems (2.6) and (2.9) are
equivalent and we only need to show that $Y^{\delta,x,\mu}$ is absolutely
continuous, which then also implies that the strategy is admissible.

Being the solution of (2.6), the process $Y^{\delta,x,\mu}$ is absolutely
continuous on $[0,T)$. If $\tau_{\mu}(x)<T$, then

$Y^{\delta,x,\mu}_{\tau_{\mu}(x)}=A^{\delta}_{\tau_{\mu}(x)}X^{\delta,x,\mu}_{\tau_{\mu}(x)}+B_{\tau_{\mu}(x)}^{\delta,x,\mu}=0$

and so $Y^{\delta,x,\mu}=0$ on $[\tau_{\mu}(x),T]$, which implies absolute
continuity on the closed interval $[0,T]$. If $\tau_{\mu}(x)=T$, recalling the
definition of $\alpha_{T}^{\delta}$ in (2.13), absolute continuity on $[0,T]$
follows from

$\lim_{t\to T}Y^{\delta,x,\mu}_{t}=\lim_{t\nearrow
T}(A^{\delta}_{t}X^{\delta,x,\mu}_{t}+B_{t}^{\delta,x,\mu})=\alpha^{\delta}_{T}(x-f_{\mu}(T))<\infty.$
(2.19)

  * ii)

For any $t\in[0,T]$, using Lemma A.4 in the second and Lemma 2.4 in the third
step, there exists a constant $c>0$ such that

$\displaystyle\lvert\xi^{\delta,x,\mu}_{t}\rvert\leq$
$\displaystyle~{}\frac{1}{\eta_{t}}\left(A^{\delta}_{t}-\delta\kappa_{t}\right)e^{-\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\left\\{\lvert
x\rvert+\int_{0}^{T}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\int_{s}^{T}\kappa_{u}\lvert\mu_{u}\rvert
e^{-\int_{s}^{u}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\,du\,ds\right\\}$
$\displaystyle+\frac{1}{\eta_{t}}\int_{0}^{T}e^{-\int_{t}^{s}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\kappa_{s}\lvert\mu_{s}\rvert\,ds$
$\displaystyle\leq$
$\displaystyle~{}\left(A_{0}^{\delta}+\|\kappa\|_{\infty}\right)\left\|1/\eta\right\|_{\infty}\left\\{\lvert
x\rvert+\int_{0}^{T}h^{\delta}_{t}\kappa_{t}\lvert\mu_{t}\rvert\,dt\right\\}+\left\|1/\eta\right\|_{\infty}\int_{0}^{T}\kappa_{s}\lvert\mu_{s}\rvert\,ds$
$\displaystyle\leq$
$\displaystyle~{}\left(A_{0}^{\delta}+\|\kappa\|_{\infty}\right)\left\|1/\eta\right\|_{\infty}\left\\{\lvert
x\rvert+\|h^{\delta}\|_{\infty}\|\kappa\|_{\infty}\|\mu\|_{L^{1}([0,T])}\right\\}+\left\|1/\eta\right\|_{\infty}\|\kappa\|_{\infty}\|\mu\|_{L^{1}([0,T])}$
$\displaystyle\leq$ $\displaystyle~{}c(1+\lvert x\rvert).$

Furthermore, for any $t\in[0,\tau_{\mu}(x))$ where $\xi^{\delta,x,\mu}$ is
differentiable we have from (2.6) that

$\displaystyle\lvert\dot{\xi}^{\delta,x,\mu}_{t}\rvert$
$\displaystyle\leq~{}\left|{\frac{\dot{\eta}_{t}}{\eta_{t}}}\right||\xi^{\delta,x,\mu}_{t}|+\frac{1}{\eta_{t}}\lvert\dot{Y}^{\delta,x,\mu}_{t}-\delta\dot{\kappa}_{t}X_{t}^{*,x,\mu}-\delta\kappa_{t}\dot{X}_{t}^{*,x,\mu}\rvert$
$\displaystyle\leq~{}{\|\dot{\eta}\|_{\infty}}\|1/\eta\|_{\infty}\|\xi^{\delta,x,\mu}\|_{\infty}$
$\displaystyle~{}\quad+\|1/\eta\|_{\infty}\Big{(}\|\kappa\|_{\infty}\|\mu\|_{\infty}+(\|\lambda\|_{\infty}+\|\dot{\kappa}\|_{\infty})\|X^{\delta,x,\mu}\|_{\infty}+\|\kappa\|_{\infty}\|\xi^{\delta,x,\mu}\|_{\infty}\Big{)}.$

Since $\|X^{\delta,x,\mu}\|_{\infty}\leq\lvert
x\rvert+T\|\xi^{\delta,x,\mu}\|_{\infty}\leq(cT+1)\lvert x\rvert+cT$, we
conclude that

$\displaystyle\lvert\dot{\xi}^{\delta,x,\mu}_{t}\rvert\leq C(1+\lvert
x\rvert),$

for some constant $C>c$. Finally, since $\xi^{\delta,x,\mu}$ is almost
everywhere differentiable and constant on the interval $(\tau_{\mu}(x),T]$ the
stated estimate follows.

∎

####  2.3.2 Verification in the mean-field model

For the MFG ($\delta=0$) the system (2.6) reduces to the forward-backward
system that results from an application of Pontryagin’s maximum principle to
the candidate Hamiltonian. The following proposition verifies that in this
case the admissible strategy $\xi^{0,x,\mu}$ solves the representative player
problem if the aggregate trading rate does not change sign.

The verification result is key to our equilibrium analysis for at least two
reasons. First, it motivates the fixed-point equation (2.8). Second, we prove
in Section 3 that the fixed-point equation reduces to a complex integral
equation that seems difficult to solve. From the verification result we expect
the solutions to the equation to not change sign, which motivates an important
a priori estimate that substantially simplifies the analysis that follows.

######  Proposition 2.7.

If $\mu\in L^{1}_{-}([0,T])\cup L^{1}_{+}([0,T])$ then the strategy defined in
(2.17) with $\delta=0$ is the a.s. unique optimal strategy that solves the
representative player’s optimization problem (2.3).

###### Proof.

Most of the arguments given below are a blend of arguments given in [16] and
[26]. We present a detailed proof to keep the paper self-contained.

Let us now assume that $\mu\in L^{1}_{-}([0,T])\cup L^{1}_{+}([0,T])$ and let
$\xi$ be any other admissible strategy with corresponding state process $X$
and liquidation time $\tau$. We need to distinguish two cases, depending on
which strategy liquidates first.

  * •

$\tau\leq\tau_{\mu}(x)$. Note that $\xi_{t}=X_{t}=0$ on $(\tau,T]$ and thus we
have

$\begin{split}&~{}J(\xi;\mu)-J(\xi^{0,x,\mu};\mu)\\\
=&~{}\int_{0}^{\tau_{\mu}(x)}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\mu_{t}X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt\\\
&\quad-\int_{0}^{\tau_{\mu}(x)}\left(\frac{1}{2}\eta_{t}(\xi^{0,x,\mu}_{t})^{2}+\kappa_{t}\mu_{t}X^{0,x,\mu}_{t}+\frac{1}{2}\lambda_{t}(X^{0,x,\mu}_{t})^{2}\right)\,dt\\\
\geq&~{}\int_{0}^{\tau_{\mu}(x)}\eta_{t}\xi^{0,x,\mu}_{t}(\xi_{t}-\xi^{0,x,\mu}_{t})+(\kappa_{t}\mu_{t}+\lambda_{t}X^{0,x,\mu}_{t})(X_{t}-X^{0,x,\mu}_{t})\,dt.\end{split}$

Since $Y^{0,x,\mu}$ is continuous, integration by parts along with equation
(2.6) yields that

$\begin{split}0=&~{}Y^{0,x,\mu}_{\tau_{\mu}(x)}(X_{\tau_{\mu}(x)}-X^{0,x,\mu}_{\tau_{\mu}(x)})\\\
=&~{}-\int_{0}^{\tau_{\mu}(x)}Y^{0,x,\mu}_{t}(\xi_{t}-\xi^{0,x,\mu}_{t})\,dt-\int_{0}^{\tau_{\mu}(x)}(X_{t}-X^{0,x,\mu}_{t})(\kappa_{t}\mu_{t}+\lambda_{t}X^{0,x,\mu}_{t})\,dt.\end{split}$
(2.20)

Plugging this into the difference of cost functionals we see that

$J(\xi;\mu)-J(\xi^{0,x,\mu};\mu)\geq\int_{0}^{\tau_{\mu}(x)}(\eta_{t}\xi^{0,x,\mu}_{t}-Y^{0,x,\mu}_{t})(X_{t}-X^{0,x,\mu}_{t})\,dt=0,$

where the last equality follows from the definition of $\xi^{0,x,\mu}$.

  * •

$\tau>\tau_{\mu}(x)$. In this case $\tau_{\mu}(x)<T$ and since $\mu$ does not
change sign we must have that
$\operatorname{sign}(\mu_{0})=\operatorname{sign}(x)$. Since $\tau$ is the
absorption time of $X$ at zero this shows that

$\mu_{t}X_{t}\geq 0\quad\mbox{for all}\quad t\in[0,T].$

As a result, the same arguments as before show that

$\begin{split}&~{}J(\xi;\mu)-J(\xi^{0,x,\mu};\mu)\\\
=&~{}\int_{0}^{\tau_{\mu}(x)}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\mu_{t}X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt\\\
&\quad-\int_{0}^{\tau_{\mu}(x)}\left(\frac{1}{2}\eta_{t}(\xi^{0,x,\mu}_{t})^{2}+\kappa_{t}\mu_{t}X^{0,x,\mu}_{t}+\frac{1}{2}\lambda_{t}(X^{0,x,\mu}_{t})^{2}\right)\,dt\\\
&\quad\quad+\int_{\tau_{\mu}(x)}^{\tau}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\mu_{t}X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt\\\
\geq&~{}0.\end{split}$

Since the above inequalities are strict if
$\|\xi^{0,x,\mu}-\xi\|_{L^{2}([0,T])}\neq 0$, it follows that $\xi^{0,x,\mu}$
is the unique optimizer. From this we conclude that (2.3) admits a unique
solution. ∎

######  Remark 2.8 (Monotonicity of trading).

The explicit representation of the optimal trading strategy allows us to
deduce an important implication for trading. If $\mu$ does not change sign,
then a representative seller always sells if $\mu>0$ and a representative
buyer always buys if $\mu<0$. At the same time, if $\mu>0$, then for any buyer
the strictly positive process $B^{0,\mu,\tau_{\mu}(x)}=B^{0,\mu,T}$ is
independent of $x$. For small enough negative initial position this means that

$\xi^{0,x,\mu}_{0}=\frac{A^{0}_{0}x+B^{0,\mu,T}_{0}}{\eta}>0.$

Hence, if sellers dominate, then for a buyer with small enough initial
position, it is beneficial to initially sell the asset to benefit from the
positive price drift when buying it back.

##  3 The equilibrium equation

In what follows we denote by $\xi^{\delta,\mu}$ the vector of strategies
introduced in (2.17) and introduce the mapping

$F:L^{1}([0,T])\to\mathbb{R}^{[0,T]},\qquad\mu\mapsto\int_{\mathbb{R}}\xi^{\delta,x,\mu}\nu_{0}(dx).$

The verification result given in Section 2 suggests that any fixed-point of
the mapping $F$ for $\delta=0$ that does not change sign yields an MFG
equilibrium. A verification result for a generic aggregate trading rate $\mu$
cannot be expected in the $N$-player game. However, we except a verification
result to hold for equilibrium mean trading rates, that is, for fixed-points
of the map $F$. This suggests that both games can be solved within a common
mathematical framework by following four non-standard steps:222For MFGs the
fourth step reduces to verifying that $\mu^{*}$ does not change sign.

$\displaystyle\left\\{\begin{array}[]{l}1.\text{ Fix }\mu\in L^{1}([0,T]).\\\
2.\text{ Consider the candidate strategy profile $\xi^{\delta,\mu}$ obtained
in \eqref{xi*} for $\delta=0$, resp. $\delta=\frac{1}{N}$}.\\\ 3.\text{ Find
the fixed-points $\mu^{*}$ of the mapping $\mu\mapsto F(\mu)$ in
$L^{1}([0,T])$}.\\\ 4.\text{ Verify that $\xi^{\delta,\mu^{*}}$ is a Nash
equilibrium.}\end{array}\right.$ (3.5)

In what follows, we prove that for any fixed $\delta\in[0,1]$ there exists a
fixed point

$\displaystyle\mu^{*}_{t}$ $\displaystyle=~{}F(\mu^{*})_{t},\qquad t\in[0,T]$
(3.6)

that does not change sign and thereby we address the third step of (3.5) for
both the MFG and the $N$-player. To guarantee that the fixed-point mapping is
always well defined we impose the following assumption on the initial
distribution $\nu_{0}$ of the players’ portfolios.

######  Assumption 3.1.

The distribution of initial position $\nu_{0}$ has a finite first absolute
moment and

$~{}\lambda+\delta\dot{\kappa}\geq 0.$

###  3.1 A non-linear integral equation

We begin our fixed-point analysis by deriving a more explicit form of the
fixed-point map that will later allow us to rewrite the fixed point property
in terms of a non-linear integral equation.

######  Lemma 3.2.

For any $\mu\in L^{1}([0,T])$ it holds for all $t\in[0,T]$ that

$\displaystyle
F(\mu)_{t}~{}=F(\mu)_{T}+\int_{t}^{T}\frac{1}{\eta_{s}}\int_{I_{\mu}(s)}\Big{(}\kappa_{s}\mu_{s}+(\lambda_{s}+\delta\dot{\kappa}_{s})X^{\delta,x,\mu}_{s}+(\dot{\eta}_{s}-\delta\kappa_{s})\xi^{\delta,x,\mu}_{s}\Big{)}\nu_{0}(dx)\,ds,$
(3.7)

where
$I_{\mu}(t):=(-\infty,\underline{f}_{\mu}(t)]\cup[\overline{f}_{\mu}(t),\infty)$.
In particular, $F$ maps $L^{1}([0,T])$ into the space of absolutely continuous
functions on $[0,T]$.

###### Proof.

From the definition of $\xi^{\delta,x,\mu}$ and $\tau_{\mu}(x)$ in Section 2
it holds that

$\xi^{\delta,x,\mu}\big{|}_{(\tau_{\mu}(x),T]}\equiv 0,\quad\textrm{ for all
}x\textrm{ such that }\tau_{\mu}(x)<T.$

Furthermore, by Lemma 2.6 it follows that $\xi^{\delta,x,\mu}$ is almost
everywhere differentiable and the derivative has at most linear growth in $x$
uniformly in $t$. Therefore, by Assumption 3.1 on $\nu_{0}$ we can apply
Fubini’s theorem to the right-hand side of equation (3.6) to obtain by the
definition of $\xi^{\delta,x,\mu}$ and $\tau_{\mu}(x)$

$\displaystyle\int_{\mathbb{R}}\xi^{\delta,x,\mu}_{t}\nu_{0}(dx)$
$\displaystyle=~{}\int_{\mathbb{R}}\xi^{\delta,x,\mu}_{T}\nu_{0}(dx)-\int_{t}^{T}\int_{I_{\mu}(s)}\dot{\xi}^{\delta,x,\mu}_{s}\nu_{0}(dx)\;ds$
$\displaystyle=~{}F(\mu)_{T}-\int_{t}^{T}\int_{I_{\mu}(s)}\frac{d}{ds}\left(\frac{Y^{\delta,x,\mu}_{s}-\delta\kappa_{s}X_{s}^{\delta,x,\mu}}{\eta_{s}}\right)\nu_{0}(dx)\;ds,\qquad
t\in[0,T].$

The assertion now follows by using that $(X^{\delta,x,\mu},Y^{\delta,x,\mu})$
solves the forward-backward equation (2.6). ∎

In what follows we denote the left and right tail distribution function of
$\nu_{0}$ by

$\displaystyle p_{0}:\mathbb{R}\to[0,1],$ $\displaystyle\quad
x\mapsto\nu_{0}((-\infty,x]),$ $\displaystyle q_{0}:\mathbb{R}\to[0,1],$
$\displaystyle\quad x\mapsto\nu_{0}([x,\infty)).$

In other words, $q_{0}$ represents the tail distribution function for sellers
and $p_{0}$ the tail distribution function for buyers. Furthermore, we
introduce the mean position among all players

$\mathbb{E}[\nu_{0}]:=\int_{-\infty}^{\infty}x\nu_{0}(dx).$

We now derive a representation of a solution $\mu$ to our fixed-point equation
in terms of an integral equation. Similar to the proof of Lemma 3.2 and using
that $X^{\delta,x,\mu}_{T}=0$ we have that

$\displaystyle\int_{I_{\mu}(t)}X^{\delta,x,\mu}_{t}\nu_{0}(dx)=\int_{I_{\mu}(t)}\int_{t}^{T}\xi^{\delta,x,\mu}_{s}\,ds\,\nu_{0}(dx)=\int_{t}^{T}\int_{\mathbb{R}}\xi^{\delta,x,\mu}_{s}\nu_{0}(dx)\,ds=\int_{t}^{T}F(\mu)_{s}ds.$

Using the definition of the tail probabilities $q_{0}$ and $p_{0}$ equation
(3.7) can be rewritten as

$\begin{split}F(\mu)_{t}=&~{}F(\mu)_{T}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}\left(q_{0}\big{(}\overline{f}_{\mu}(s)\big{)}+p_{0}\big{(}\underline{f}_{\mu}(s)\big{)}\right)\mu_{s}\,ds+\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}F(\mu)_{u}du\right)ds\\\
&~{}+\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}F(\mu)_{s}\;ds,\qquad
t\in[0,T].\end{split}$ (3.8)

Recalling the definition of $\overline{f}_{\mu}$ and $\underline{f}_{\mu}$ in
(2.15), we see that any fixed point $\mu=F(\mu)$ solves the non-linear higher
order integral equation (3.8). The following proposition shows that converse
is also true. Any solution to this equation yields a fixed point $\mu=F(\mu)$.

######  Proposition 3.3.

A process $\mu\in L^{1}([0,T])$ solves the fixed-point equation (3.6) if and
only if $\mu_{T}=F(\mu)_{T}$ and $\mu$ solves the equation

$\begin{split}\mu_{t}=&~{}\mu_{T}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}\left(q_{0}\big{(}\overline{f}_{\mu}(s)\big{)}+p_{0}\big{(}\underline{f}_{\mu}(s)\big{)}\right)\mu_{s}\,ds+\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}du\right)ds\\\
&~{}+\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\mu_{s}\;ds,\qquad
t\in[0,T].\end{split}$ (3.9)

###### Proof.

We have already proven that any solution to the fixed point equation (3.6)
satisfies the integral equation (3.9). For the converse direction, define the
linear operator $G$ on $L^{1}([0,T])$ by

$\displaystyle
G(\mu)_{t}=\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}du\right)ds+\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\mu_{s}\;ds,\qquad
t\in[0,T].$

Now let $\mu\in L^{1}([0,T])$ be a solution to (3.9). Comparing the equations
(3.8) and (3.9) we then obtain

$\displaystyle\mu-F(\mu)$ $\displaystyle=~{}G(\mu)-G(F(\mu))$
$\displaystyle=~{}G(\mu-F(\mu)).$

An application of Grönwall’s inequality [35, Theorem 2.7] shows that any
solution to the linear integral equation $\varphi=G(\varphi)$ is necessarily
trivial and hence $\mu=F(\mu)$. ∎

The integral equation (3.9) is rather intricate as it depends on the running
maximum and minimum of the function $f_{\mu}$ and hence depends in a non-
linear way on the entire process $\mu$. Furthermore, the terminal condition
$\mu_{T}=F(\mu)_{T}$ is endogenous and needs to be characterized more
explicitly.

The following lemma provides an a priori estimate that allows us to
substantially simplify the equation.

######  Lemma 3.4.

Let $\mu\in L^{1}([0,T])$ be a solution to the equation (3.9). Then,

$e^{-K_{1}(T-t)}\frac{\eta_{T}}{\eta_{t}}|\mu_{T}|\leq\lvert\mu_{t}\rvert\leq\lvert\mu_{T}\rvert
e^{K_{2}(T-t)},\qquad t\in[0,T],$ (3.10)

where $K_{1}:=\delta\|\kappa\|_{\infty}\|1/\eta\|_{\infty}$ and
$K_{2}:=\|1/\eta\|_{\infty}((1+\delta)\|\kappa\|_{\infty}+\delta
T\|\dot{\kappa}\|_{\infty}+T\|\lambda\|_{\infty}+\|\dot{\eta}\|_{\infty})$. In
particular,

$\operatorname{sign}(\mu_{t})=\operatorname{sign}(\mu_{T}),\qquad t\in[0,T].$

###### Proof.

It follows from the equation (3.9) that for all $t\in[0,T]$,

$\displaystyle\lvert\mu_{t}\rvert\leq$
$\displaystyle~{}\lvert\mu_{T}\rvert+\|1/\eta\|_{\infty}\|\kappa\|_{\infty}\int_{t}^{T}\lvert\mu_{s}\rvert\,ds+\|1/\eta\|_{\infty}(\|\lambda\|_{\infty}+\delta\|\dot{\kappa}\|_{\infty})\int_{t}^{T}\int_{s}^{T}\lvert\mu_{r}\rvert\,dr\,ds$
$\displaystyle+\|1/\eta\|_{\infty}(\|\dot{\eta}\|_{\infty}+\delta\|\kappa\|_{\infty})\int_{t}^{T}\lvert\mu_{s}\rvert\,ds$
$\displaystyle\leq$
$\displaystyle~{}\lvert\mu_{T}\rvert+K_{2}\int_{t}^{T}\lvert\mu_{s}\rvert\,ds.$

Hence, the upper estimate in (3.10) follows from Grönwall’s inequality. In
particular $\mu\equiv 0$ for $\mu_{T}=0$. Let us thus suppose that $\mu_{T}>0$
and put

$t_{0}:=\inf\\{t\in[0,T)\;|\;\mu_{s}>0,\;t\leq s\leq T\\}.$

By Assumption 3.1 it holds that $(\lambda+\delta\dot{\kappa})$ is non-
negative.333For the MFG this assumption is always satisfied. Since $\mu$ is
almost everywhere differentiable we conclude from (3.9) that

$\displaystyle\dot{\mu}_{t}~{}\leq~{}-\frac{\dot{\eta}_{t}-\delta\kappa_{t}}{\eta_{t}}\mu_{t},\qquad\text{
for a.e. }t\in[t_{0},T].$

Hence, it follows from the modified Grönwall’s inequality in Lemma A.5

$\displaystyle\mu_{t}~{}\geq~{}\mu_{T}\exp\left(\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\,ds\right)~{}\geq~{}\mu_{T}\frac{\eta_{T}}{\eta_{t}}e^{-K_{1}(T-t)}\quad
t\in[t_{0},T].$

In particular, $\mu_{t_{0}}>0$. From the definition of $t_{0}$ and the
continuity of $\mu$ it thus follows that $t_{0}=0$. An analogous argument
applies to the case $\mu_{T}<0$. ∎

The fact that a fixed-point $\mu$ does not change sign implies that either
$\overline{f}_{\mu}\equiv f_{\mu}$ and $\underline{f}_{\mu}\equiv 0$ or
$\overline{f}_{\mu}\equiv 0$ and $\underline{f}_{\mu}\equiv f_{\mu}$, which
considerably simplifies the equation (3.9). The following proposition shows
that the sign of $\mu$ depends only on the sign of the average initial
position $\mathbb{E}[\nu_{0}]$.

######  Proposition 3.5.

Let $\mu\in L^{1}([0,T];\mathbb{R})$ be a solution to the fixed-point equation
(3.6). Then,

$\operatorname{sign}(\mu_{t})=\operatorname{sign}(\mathbb{E}[\nu_{0}]),\quad
0\leq t\leq T.$

In case $\mathbb{E}[\nu_{0}]>0$, then $\mu$ is a solution to the fixed-point
equation (3.6) if and only if it satisfies the equation444Due to symmetry, if
$\mathbb{E}[\nu_{0}]<0$ we equivalently can solve for the fixed-point problem
with reflected initial measure
$\widetilde{\nu_{0}}((-\infty,x])=\nu_{0}((-x,\infty])$ and
$\mathbb{E}[\widetilde{\nu_{0}}]>0$.

$\begin{split}\mu_{t}=&~{}\mu_{T}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}\left(q_{0}\big{(}f_{\mu}(s)\big{)}+p_{0}(0)\right)\mu_{s}\,ds+\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}du\right)ds\\\
&~{}+\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\mu_{s}\;ds,\qquad
t\in[0,T],\end{split}$ (3.11)

subject to the the terminal condition

$\displaystyle\mu_{T}=\frac{\alpha_{T}^{\delta}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-(1-q_{0}(0))x_{\mu}-\int_{0}^{x_{\mu}}q_{0}(x)dx\right),$
(3.12)

where $x_{\mu}:=f_{\mu}(T)$.

###### Proof.

By Proposition 3.3 $\mu$ is a fixed-point of (3.6) if and only if it solves
the integral equation (3.9) and satisfies $F(\mu)_{T}=\mu_{T}$. According to
Lemma 3.4 the sign of $\mu$ is then determined by the sign of the terminal
value $\mu_{T}$. In order to analyze the terminal value $\mu_{T}$ we will take
the limit for $t\nearrow T$ in the equation (3.6). To this end, recall the
limiting behavior of $Y^{\delta,x,\mu}_{t}$ from (2.19) and note that

$\displaystyle\lim_{t\nearrow T}\xi^{\delta,x,\mu}_{t}=\lim_{t\nearrow
T}\frac{Y^{\delta,x,\mu}_{t}-\delta\kappa_{t}X^{\delta,x,\mu}_{t}}{\eta_{t}}=\frac{\alpha^{\delta}_{T}}{\eta_{T}}(x-x_{\mu})\mathbf{1}_{\\{x\in
I_{\mu}(T)\\}},\qquad x\in\mathbb{R}.$

Further, recalling the a priori estimate on $\xi^{\delta,x,\mu}$ from Lemma
2.6 we obtain from dominated convergence

$\begin{split}\mu_{T}=F(\mu)_{T}=&~{}\lim_{t\nearrow
T}\int_{\mathbb{R}}\xi^{\delta,x,\mu}_{t}\nu_{0}(dx)\\\
=&~{}\int_{I_{\mu}(T)}\frac{\alpha^{\delta}_{T}(x-x_{\mu})}{\eta_{T}}\nu_{0}(dx)\\\
=&~{}\frac{\alpha^{\delta}_{T}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-x_{\mu}-\int_{\underline{f}_{\mu}(T)}^{\overline{f}_{\mu}(T)}(x-x_{\mu})\nu_{0}(dx)\right).\end{split}$

Now assume that $\mu_{T}>0$. Then it follows from Lemma 3.4 that $\mu_{t}>0$
for all $t\in[0,T]$. Hence $\overline{f}_{\mu}\equiv f_{\mu}$ and
$\underline{f}_{\mu}\equiv 0$; in particular $x_{\mu}>0$. Therefore, the above
expression for the terminal value simplifies to

$\displaystyle\mu_{T}$
$\displaystyle=~{}\frac{\alpha^{\delta}_{T}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-x_{\mu}-\int_{0}^{x_{\mu}}(x-x_{\mu})\nu_{0}(dx)\right)$
$\displaystyle=~{}\frac{\alpha^{\delta}_{T}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-(1-q_{0}(0))x_{\mu}-\int_{0}^{x_{\mu}}q_{0}(x)dx\right).$

From this we conclude that $\mathbb{E}[\nu_{0}]>0$ since otherwise all the
terms in the bracket would be non-positive. Hence, $\mu$ satisfies the
equation (3.11). Analogously, we see that $\mu_{T}<0$ implies
$\mathbb{E}[\nu_{0}]<0$. Finally, if $\mu_{T}=0$, then by Lemma 3.4 it holds
$\mu\equiv 0$ and therefore
$x_{\mu}=\underline{f}_{\mu}(T)=\overline{f}_{\mu}(T)=0$, hence
$\mathbb{E}[\nu_{0}]=0$. ∎

###  3.2 Existence and uniqueness of solutions

In this section we prove that the integral equation (3.11) with terminal
condition (3.12) has a unique solution; by Proposition 3.5 this implies that
the fixed-point problem (3.6) has a unique solution.

Apart from the non-linearity induced by the function $q_{0}$ the main
difficulty in solving the equation is that it involves forward and backward
integrals of $\mu$. More precisely,

$\displaystyle
f_{\mu}(t)=\int_{0}^{t}\kappa_{s}h^{\delta}_{s}\mu_{s}\,ds,\qquad t\in[0,T]$

is defined as a forward integral, while all other integrals in (3.11) are
backward integrals. Furthermore, the terminal value of $\mu$ is endogenous as
it depends on $x_{\mu}=f_{\mu}(T)$. To overcome these problems we introduce
the following backward integral equations

$\begin{split}\mu_{t}=&~{}\mu^{c}_{T}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}\left(q_{0}\\!\left(c-\int_{s}^{T}\kappa_{r}h^{\delta}_{r}\mu_{r}\,dr\right)+p_{0}(0)\right)\mu_{s}\,ds+\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}du\right)ds\\\
&~{}+\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\mu_{s}\;ds,\qquad
t\in[0,T].\end{split}$ (3.13)

for any $c\in\mathbb{R}$ with exogenous terminal condition

$\displaystyle\mu^{c}_{T}:=\frac{\alpha^{\delta}_{T}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-(1-q_{0}(0))c-\int_{0}^{c}q_{0}(x)dx\right).$

A solution $\mu^{c}$ to the equation (3.13) yields a solution to the original
equation (3.11) if the additional condition $f_{\mu^{c}}(T)=c$ is satisfied.
The following result shows that such a $c$ always exists and is unique.

######  Theorem 3.6.

  * i)

For each $c\in\mathbb{R}$ there exists a unique solution $\mu^{c}$ in
$C^{0}([0,T])$ to the integral equation (3.13).

  * ii)

There exists a unique $\widehat{x}\in\mathbb{R}$ such that
$f_{\mu^{\hat{x}}}(T)=\hat{x}$. Moreover, $\mu^{\hat{x}}$ is the unique
solution to (3.11) and it holds that
$0<\widehat{x}<\sup(\mathrm{supp}(\nu_{0}))$.555The quantity $\hat{x}$ can be
understood as the largest initial position for which early liquidation is
beneficial.

###### Proof.

  * i)

By Assumption 2.1 the coefficients $\eta,\kappa,\lambda$ and $1/\eta$ are non-
negative and bounded, and the derivatives $\dot{\kappa}$ and $\dot{\eta}$ are
bounded. By Lemma 2.4 the functions $h^{\delta}$ and $\dot{h}^{\delta}$ are
bounded uniformly in $\delta\in[0,1]$ and $1/h^{\delta}$ is bounded on compact
subsets of $(0,T]$ uniformly for $\delta\in[0,1]$.

In what follows $K$ denotes a positive constant that may change from line to
line, but only depends on the aforementioned bounds.

We will first consider the linear and non-linear parts of equation (3.13)
separately. Similarly to the proof of Proposition 3.3, we define the linear
operator $G$ mapping $C^{0}([0,T])$ into the space of absolutely continuous
functions on $[0,T]$ by

$\displaystyle G(\mu)_{t}=$
$\displaystyle~{}\int_{t}^{T}\frac{\lambda_{s}+\delta\dot{\kappa}_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}du\right)ds+\int_{t}^{T}\frac{\dot{\eta}_{s}+(p(0)-\delta)\kappa_{s}}{\eta_{s}}\mu_{s}\;ds,\qquad
t\in[0,T].$

We readily see that the following estimate holds for all $\mu\in
C^{0}([0,T])$:

$\displaystyle|G(\mu)_{t}|\leq K\int_{t}^{T}|\mu_{s}|\,ds,\qquad t\in[0,T].$
(3.14)

The remaining parts of (3.13) are described by the non-linear operator $H_{c}$
mapping $C^{0}([0,T])$ into the space of absolutely continuous functions on
$[0,T]$ defined by

$\displaystyle H_{c}(\mu)_{t}=$
$\displaystyle~{}\mu^{c}_{T}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}q_{0}\\!\left(c-\int_{s}^{T}\kappa_{u}h^{\delta}_{u}\mu_{u}\,du\right)\mu_{s}\,ds,\qquad
t\in[0,T],\quad c\geq 0.$

Since $0\leq q_{0}\leq 1$ we immediately obtain the following estimate:

$\displaystyle|H_{c}(\mu)_{t}|\leq$
$\displaystyle~{}|\mu^{c}_{T}|+K\int_{t}^{T}|\mu_{s}|\,ds,\qquad
t\in[0,T],\quad c\geq 0.$ (3.15)

Since $H_{c}$ is non-linear, this estimate is not sufficient to proof the
existence result. Introducing the integrated tail probability
$Q_{0}(x):=\int_{0}^{x}q_{0}(y)\,dy$, integration by parts yields the
following representation for all $\mu\in C^{0}([0,T])$

$\displaystyle H_{c}(\mu)_{t}=~{}$
$\displaystyle\mu^{c}_{T}+Q_{0}(c)\frac{1}{h^{\delta}_{T}\eta_{T}}-Q_{0}\left(c-\int_{t}^{T}\kappa_{s}h^{\delta}_{s}\mu_{s}ds\right)\frac{1}{h^{\delta}_{t}\eta_{t}}$
$\displaystyle+\int_{t}^{T}Q_{0}\left(c-\int_{s}^{T}\kappa_{u}h^{\delta}_{u}\mu_{u}du\right)\frac{\dot{h}^{\delta}_{s}\eta_{s}+h^{\delta}_{s}\dot{\eta}_{s}}{(h^{\delta}_{s})^{2}\eta_{s}^{2}}ds,\qquad
t\in(0,T].$

Note that $Q_{0}$ is globally Lipschitz continuous with coefficient
$\|Q_{0}\|_{\mathrm{Lip}}=\|q_{0}\|_{\infty}=1$ and similarly also the mapping
$c\mapsto\mu^{c}_{T}$ is globally Lipschitz continuous.

Since $1/h^{\delta}$ is only bounded on compact sets of $(0,T]$ we introduce
the time $t_{k}:=1/k$ for some $k\in\mathbb{N}$. For any
$\mu,\widetilde{\mu}\in C^{0}([t_{k},T])$ and $c,\widetilde{c}\geq 0$ we
obtain the following estimate:

$\begin{split}|H_{c}(\mu)_{t}-H_{\widetilde{c}}(\widetilde{\mu})_{t}|\leq&~{}|\mu^{c}_{T}-\mu^{\widetilde{c}}_{T}|+\frac{1}{h^{\delta}_{T}\eta_{T}}|c-\widetilde{c}|+\frac{1}{h^{\delta}_{t}\eta_{t}}\left(\int_{t}^{T}\kappa_{s}h^{\delta}_{s}|\mu_{s}-\widetilde{\mu}_{s}|\,ds+|c-\widetilde{c}|\right)\\\
&~{}+\int_{t}^{T}\left|\frac{\dot{h}^{\delta}_{s}\eta_{s}+h^{\delta}_{s}\dot{\eta}_{s}}{(h^{\delta}_{s})^{2}\eta_{s}^{2}}\right|\left(\int_{s}^{T}\kappa_{u}h^{\delta}_{u}|\mu_{u}-\widetilde{\mu}_{u}|\,du+|c-\widetilde{c}|\right)ds\\\
\leq&~{}K|c-\widetilde{c}|+K\int_{t}^{T}|\mu_{s}-\widetilde{\mu}_{s}|ds,\qquad
t\in[t_{k},T].\end{split}$ (3.16)

Now let $c\geq 0$ be arbitrary but fixed. Iterating the estimates (3.14) and
(3.16), we see that for $n\in\mathbb{N}$ it holds that

$\displaystyle\left|[H_{c}+G]^{n}(\mu)_{t}-[H_{c}+G]^{n}(\widetilde{\mu})_{t}\right|$
$\displaystyle\leq~{}K\int_{t}^{T}\left|[H_{c}+G]^{n-1}(\mu)_{t}-[H_{c}+G]^{n-1}(\widetilde{\mu})_{t}\right|\,ds$
$\displaystyle\leq~{}\frac{K^{n}T^{n}}{n!}\|\mu-\widetilde{\mu}\|_{\infty},\qquad
t\in[t_{k},T].$

From [35, Theorem 2.4] it then follows that $[H_{c}+G]$ has a unique fixed-
point $\mu^{c,k}$ in $C^{0}({[t_{k},T]})$.

From the uniqueness it follows that the pointwise limit
$\mu^{c}_{t}:=\lim_{k\to\infty}\mu_{t}^{c,k}$ is well defined and satisfies
$[H_{c}+G](\mu^{c})_{t}=\mu^{c}_{t}$ for all $t\in(0,T]$. From the estimates
(3.14), (3.15) and Grönwall’s inequality it then follows that

$\displaystyle|\mu^{c}_{t}|~{}\leq~{}|\mu^{c}_{T}|+K\int_{t}^{T}|\mu^{c}_{s}|\,ds~{}\leq~{}|\mu^{c}_{T}|e^{K(T-t)}\qquad
t\in(0,T].$ (3.17)

Hence $\mu^{c}$ is bounded which allows us to extend $\mu^{c}$ to a continuous
function on $[0,T]$ using dominated convergence in the representation
$\mu^{c}_{t}=[H_{c}+G](\mu^{c})_{t}$. By construction it follows that
$\mu^{c}$ is the unique fixed-point of $[H_{c}+G]$ in $C^{0}([0,T])$, hence
the unique solution to the equation (3.13).

  * ii)

We are first going to prove the existence of $\hat{x}$ and afterwards we will
prove that $\mu^{\hat{x}}$ is the unique solution to equation (3.11). For all
$c\geq 0$ we denote by $\mu^{c}$ the unique solution to (3.13). Our aim is to
show that the map

$\displaystyle\psi:[0,\infty)\to\mathbb{R},\quad c\mapsto
c-f_{\mu^{c}}(T)=c-\int_{0}^{T}h^{\delta}_{s}\kappa_{s}\mu^{c}_{s}\,ds,$

has a unique root. Note that from the above estimates (3.14) and (3.16) we
obtain that for any $t_{0}\in(0,T]$ it holds

$\displaystyle|\mu^{c_{1}}_{t}-\mu^{c_{2}}_{t}|$
$\displaystyle=~{}\|[H_{c_{1}}+G](\mu^{c_{1}})-[H_{c_{2}}+G](\mu^{c_{2}})\|_{\infty}$
$\displaystyle\leq~{}K|c_{1}-c_{2}|+K\int_{t}^{T}|\mu^{c_{1}}_{s}-\mu^{c_{2}}_{s}|\,ds$
$\displaystyle\leq~{}K|c_{1}-c_{2}|e^{K(T-t)},\qquad t\in[t_{0},T],$

where the last estimate follows from Grönwall’s inequality. Since additionally
the estimate (3.17) holds we can use dominated convergence to conclude that
the map $\psi$ is continuous.

To prove the existence of a root it suffices to prove that $\psi$ changes
sign. To this end we note that the bounds of (3.10) also hold for solutions to
the equation (3.13); in particular,
$\operatorname{sign}(\mu^{c})=\operatorname{sign}(\mu^{c}_{T})$ for all $c\geq
0$. Indeed, an inspection of the proof of Lemma 3.10 shows that only the
estimate $0\leq q_{0}\leq 1$ is used to estimate the factor involving the
$q_{0}$ term. We now need to distinguish two cases.

    * –

$q_{0}(0)<1$. By the continuity and the monotonicity of the mapping
$c\mapsto\mu^{c}_{T}$ there exists a unique $c_{0}>0$ such that
$\mu^{c_{0}}_{T}=0$ and we see that the unique solution to the equation (3.13)
is trivial in this case, i.e., $\mu^{c_{0}}\equiv 0$. Therefore it holds
$\psi(c_{0})=c_{0}>0$.

On the other hand, since $\mu^{0}_{T}>0$ it follows that $\mu^{0}>0$ and as a
result,

$\psi(0)=-\int_{0}^{T}h^{\delta}_{s}\kappa_{s}\mu^{0}_{s}ds<0.$

    * –

$q_{0}(0)=1$. In this case, there are only sellers in the market and

$\mu^{c}_{T}=\frac{\alpha^{\delta}_{T}}{\eta_{T}}\left(\mathbb{E}[\nu_{0}]-\int_{0}^{c}q_{0}(x)\,dx\right)\to
0\quad\mbox{ for }\quad c\to\infty;$

In particular, it follows again from (3.10) that

$\|\mu^{c}\|_{\infty}\to 0\quad\mbox{ for }\quad c\to\infty.$

Thus, by dominated convergence there exists $c_{0}>0$ such that

$\psi(c_{0})=c_{0}-\int_{0}^{T}h^{\delta}_{s}\kappa_{s}\mu^{c_{0}}_{s}ds>0.$

At the same time we obtain analogously to the first case that $\psi(0)<0$.

Hence, in both cases $\psi$ changes it sign and the existence of at least one
root $\hat{x}$ follows. To prove the uniqueness of $\hat{x}$ we show that the
mapping $\psi$ is strictly increasing. To this end, we fix $c_{1}>c_{2}$ and
define

$\displaystyle
f^{c_{i}}_{t}:=c_{i}-\int_{t}^{T}h^{\delta}_{s}\kappa_{s}\mu^{c_{i}}_{s}\,ds,\qquad
t\in[0,T],\quad i=1,2.$

Note that by definition it holds $\mu^{c_{1}}_{T}<\mu^{c_{2}}_{T}$ and
$f^{c_{1}}_{T}>f^{c_{2}}_{T}$. Now let us denote by $t^{\mu}_{0}$ and
$t_{0}^{f}$ the last times when $\mu^{c_{2}}-\mu^{c_{1}}$, respectively
$f^{c_{1}}-f^{c_{2}}$ changes its sign, i.e.,

$\displaystyle t_{0}^{\mu}$
$\displaystyle=\inf\big{\\{}t\in[0,T]\;\big{|}\;\mu^{c_{2}}_{s}-\mu^{c_{1}}_{s}>0,\;s\in[t,T]\big{\\}},$
$\displaystyle t_{0}^{f}$
$\displaystyle=\inf\big{\\{}t\in[0,T]\;\big{|}\;f^{c_{1}}_{s}-f^{c_{2}}_{s}>0,\;s\in[t,T]\big{\\}}.$

For any point $t\in[t_{0}^{\mu}\vee t^{f}_{0},T]$ where $\mu^{c_{1}}$ and
$\mu^{c_{2}}$ are differentiable it holds that

$\begin{split}\dot{\mu}^{c_{2}}_{t}-\dot{\mu}^{c_{1}}_{t}\leq&~{}\frac{\kappa_{t}}{\eta_{t}}\left\\{q_{0}(f^{c_{1}}_{t})\mu^{c_{1}}_{t}-q_{0}(f^{c_{2}}_{t})\mu^{c_{1}}_{t}\right\\}+\frac{\kappa_{t}}{\eta_{t}}\left\\{q_{0}(f^{c_{2}}_{t})\mu^{c_{1}}_{t}-q_{0}(f^{c_{2}}_{t})\mu^{c_{2}}_{t}\right\\}-\frac{\dot{\eta}_{t}-\delta\kappa_{t}}{\eta_{t}}(\mu^{c_{2}}_{t}-\mu^{c_{1}}_{t})\\\
\leq&~{}-\frac{\dot{\eta}_{t}-\delta\kappa_{t}}{\eta_{t}}(\mu^{c_{2}}_{t}-\mu^{c_{1}}_{t}).\end{split}$

Since $\mu^{c_{1}}$ and $\mu^{c_{2}}$ are almost everywhere differentiable it
follows by the version of Grönwall’s inequality in Lemma A.5 that

$\mu^{c_{2}}_{t}-\mu^{c_{1}}_{t}\geq(\mu^{c_{2}}_{T}-\mu^{c_{1}}_{T})\exp\left(\int_{t}^{T}\frac{\dot{\eta}_{s}-\delta\kappa_{s}}{\eta_{s}}\,ds\right)>0,\qquad
t\in[t_{0}^{\mu}\vee t^{f}_{0},T].$ (3.18)

From this it also follows

$\dot{f}^{c_{1}}_{t}-\dot{f}^{c_{2}}_{t}=h_{t}\kappa_{t}(\mu^{c_{1}}_{t}-\mu^{c_{2}}_{t})<0,\qquad
t\in[t_{0}^{\mu}\vee t^{f}_{0},T].$ (3.19)

If $t_{0}^{\mu}\leq t^{f}_{0}$, then the inequality (3.19) implies that
$t\mapsto f^{c_{1}}_{t}-f^{c_{2}}_{t}$ is strictly decreasing on the interval
$[t^{f}_{0},T]$. By the definition of $t_{0}^{f}$ it then holds that
$t_{0}^{f}=0$, which implies that $t_{0}^{\mu}=0$ and thus $t\mapsto
f^{c_{1}}_{t}-f^{c_{2}}_{t}$ is strictly decreasing on $[0,T]$. In particular
it holds that

$\psi(c_{1})-\psi(c_{2})=f_{0}^{c_{1}}-f^{c_{2}}_{0}>f_{T}^{c_{1}}-f^{c_{2}}_{T}>0$

and so the map $\psi$ is strictly increasing.

If $t_{0}^{\mu}\geq t^{f}_{0}$, then the inequality (3.18) implies that
$\mu^{c_{2}}_{t}-\mu^{c_{1}}_{t}>0$ on the interval $[t_{0}^{\mu},T]$. Thus,
by the definition of $t_{0}^{\mu}$ it must hold $t_{0}^{\mu}=0$, which also
implies that $t^{f}_{0}=0$. Using the inequality (3.19) again, we conclude
that $c\mapsto f^{c}_{0}$ is also strictly increasing in this case.

We are now proving the second part of the assertion in ii). From the
definition of $\hat{x}$ it follows that

$\displaystyle\hat{x}-\int_{t}^{T}h^{\delta}_{s}\kappa_{s}\mu^{\hat{x}}_{s}\,ds=\int_{0}^{t}h^{\delta}_{s}\kappa_{s}\mu^{\hat{x}}_{s}\,ds=f_{\mu}^{\hat{x}}(t),\qquad
t\in[0,T].$

Inserting this relation into (3.13) for $\mu^{\hat{x}}$ we immediately see
that $\mu^{\hat{x}}$ also solves (3.11). Conversely, if $\mu$ is a solution to
(3.11), then defining $c:=f_{\mu}(T)$ we immediately see that $\mu$ also
solves (3.13) and by uniqueness it must holds $\mu=\mu^{c}$. Since then
$f_{\mu^{c}}(T)=f_{\mu}(T)=c$ it follows from uniqueness of $\hat{x}$ that
$\mu=\mu^{\hat{x}}$.

∎

##  4 Equilibrium analysis

For MFGs the verification argument given in Section 2.3.2 along with the
solvability of the fixed-point problem readily yields the existence of a
unique equilibrium for our liquidation game.

######  Theorem 4.1.

Under Assumption 2.1 and Assumption 3.1, the mean-field liquidation game
admits an equilibrium. The equilibrium is unique in the class of equilibria
with continuous aggregate trading rates.

###### Proof.

W.l.o.g. we assume that $\mathbb{E}[\nu_{0}]>0$. According to Theorem 3.6
there exists a unique solution $\mu\in C^{0}([0,T])$ to the equation (3.11).
By Proposition 3.3 it follows that $\mu$ is the unique solution to the fixed-
point problem (3.6) and that
$\operatorname{sign}(\mu)\equiv\operatorname{sign}(\mathbb{E}[\nu_{0}])$.
Therefore, it follows from Proposition 2.7 that $\xi^{\delta,x,\mu}$ is the
unique optimal liquidation strategy for the representative player with initial
position $x\in\mathbb{R}$. Since $\mu$ solves the fixed-point problem (3.6),
we conclude that $(\xi^{\delta,x,\mu})_{x\in\mathbb{R}}$ is an equilibrium of
the MFG.

Now let $(\widetilde{\xi}^{x})_{x\in\mathbb{R}}$ be another equilibrium of the
MFG, such that the aggregated trading rate $\widetilde{\mu}$ is continuous.
Then there exists a minimal $t_{0}\in[0,T]$ such that
$\widetilde{\mu}|_{[t_{0},T]}\in L^{1}_{-}([0,T])\cup L^{1}_{+}([0,T])$ and
$\widetilde{\mu}|_{[t_{0},T]}$ is a solution to the MFG restricted on
$[t_{0},T]$ with initial measure
$\nu_{t_{0}}=Law((\widetilde{X}^{x}_{t_{0}})_{x\in\mathbb{R}})$. By the
definition of $t_{0}$, the aggregate rate $\widetilde{\mu}|_{[t_{0},T]}$ does
not change sign on $[t_{0},T]$. Hence, it follows from Proposition 2.7 that
$\widetilde{\xi}^{x}|_{[t_{0},T]}=\xi^{\delta,\widetilde{\mu},x}|_{[t_{0},T]}$.
As a result, $\widetilde{\mu}|_{[t_{0},T]}$ solves the fixed-point problem
(3.6) with initial measure $\nu_{t_{0}}$ and we eventually see that
$\operatorname{sign}(\widetilde{\mu}_{t})=\operatorname{sign}(\mathbb{E}[\nu_{t_{0}}])$
for all $t\in[t_{0},T]$. From the minimality of $t_{0}$ it follows that
$t_{0}=0$, and we conclude from the uniqueness of solutions to (3.6) that
$\widetilde{\mu}=\mu$. ∎

###  4.1 Equilibria in the $N$-player game

Establishing the existence of equilibria in the $N$-player liquidation game is
more complex, due to the lack of a general verification result. To obtain a
convex optimization problem in the $N$-player game we need to lightly
strengthen our assumptions on the cost parameters.

######  Assumption 4.2.

In addition to Assumption 2.1, we assume that the cost parameters satisfy
$\eta-\delta\kappa>0$ and $\lambda-\delta\kappa\geq 0$, with
$\delta=\frac{1}{N}$.

In the $N$-player game, we denote the initial distribution of the players’
positions by $\nu_{0}=\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{j}}$, which
trivially satisfies Assumption 3.1. The following theorem establishes the
existence of an equilibrium in the $N$-player game. While some estimates in
the verification argument seem similar to the MFG, the situation is different.
In the MFG we verified that the candidate strategy is a best response to any
given mean trading rate $\mu$ that does not change sign. In our current
setting this argument only applies in equilibrium, which complicates the
uniqueness argument.

In what follows we put

$\xi^{\mu,i}:=\xi^{\frac{1}{N},x_{i},\mu},\quad\xi^{\mu,-i}:=\big{(}\xi^{\frac{1}{N},x_{j},\mu}\big{)}_{j\neq
i},\quad X^{\mu,i}=X^{\frac{1}{N},x_{i},\mu},\quad
Y^{\mu,i}=Y^{\frac{1}{N},x_{i},\mu}$

######  Theorem 4.3.

Let $\mu\in L^{1}([0,T])$ be the unique solution to (3.6). Then, for every
$i=1,\dots,N$ the admissible control $\xi^{\mu,i}$ is the unique solution to
the optimal control problem

$\displaystyle\inf_{\xi\in\mathcal{A}_{x_{i}}}J(\xi;\xi^{\mu,-i}),\qquad
X_{t}=x_{i}-\int_{0}^{t}\xi_{s}\,ds,\qquad t\in[0,T].$ (4.1)

As a result, $(\xi^{\mu,1},\dots,\xi^{\mu,N})$ is a Nash equilibrium of the
$N$-player game.

###### Proof.

For $i=1,\dots,N$ let $\xi\in\mathcal{A}_{x_{i}}$ be an admissible control
with corresponding state process $X$ and absorption time $\tau$. Let
$\tau^{i}:=\tau_{\mu}(x^{i})$. We distinguish again two cases.

  1. (i)

$\tau\leq\tau^{i}$. Using the fact that $\mu$ satisfies the equation (3.6)
shows that

$\begin{split}J(\xi;\xi^{\mu,-i})&-J(\xi^{\mu,i};\xi^{\mu,-i})\\\
=&~{}\int_{0}^{\tau}\left(\frac{1}{2}\eta_{t}\xi^{2}_{t}+\kappa_{t}\left(\frac{1}{N}\xi_{t}+\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)X_{t}+\frac{1}{2}\lambda_{t}X^{2}_{t}\right)\,dt\\\
&\quad-\int_{0}^{\tau^{i}}\left(\frac{1}{2}\eta_{t}(\xi^{\mu,i}_{t})^{2}+\kappa_{t}\left(\frac{1}{N}\xi^{\mu,i}_{t}+\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)X^{\mu,i}_{t}+\frac{1}{2}\lambda_{t}(X^{\mu,i}_{t})^{2}\right)\,dt\\\
=&~{}\int_{0}^{\tau^{i}}\left(\frac{1}{2}\left(\eta_{t}-\frac{\kappa_{t}}{N}\right)\xi^{2}_{t}+\frac{1}{2}\frac{\kappa_{t}}{N}\left(\xi_{t}+X_{t}\right)^{2}+\frac{1}{2}\left(\lambda_{t}-\frac{\kappa_{t}}{N}\right)X^{2}_{t}\right)\,dt\\\
&\quad-\int_{0}^{\tau^{i}}\left(\frac{1}{2}\left(\eta_{t}-\frac{\kappa_{t}}{N}\right)(\xi^{\mu,i}_{t})^{2}+\frac{1}{2}\frac{\kappa_{t}}{N}\left(\xi^{\mu,i}_{t}+X^{\mu,i}_{t}\right)^{2}+\frac{1}{2}\left(\lambda_{t}-\frac{\kappa_{t}}{N}\right)(X^{\mu,i}_{t})^{2}\right)\,dt\\\
&\quad+\int_{0}^{\tau^{i}}\kappa_{t}\left(\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)(X_{t}-X^{\mu,i}_{t})dt\\\
\geq&~{}\int_{0}^{\tau^{i}}\left(\left(\eta_{t}-\frac{\kappa_{t}}{N}\right)\xi^{\mu,i}_{t}(\xi_{t}-\xi^{\mu,i}_{t})+\frac{\kappa_{t}}{N}(\xi^{\mu,i}_{t}+X^{\mu,i}_{t})\left(\xi_{t}-\xi^{\mu,i}_{t}+X_{t}-X^{\mu,i}_{t}\right)\right)\,dt\\\
&~{}+\int_{0}^{\tau^{i}}\left(\lambda_{t}-\frac{\kappa_{t}}{N}\right)X^{\mu,i}_{t}(X_{t}-X^{\mu,i}_{t})\,dt+\int_{0}^{\tau^{i}}\kappa_{t}\left(\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)(X_{t}-X^{\mu,i}_{t})dt\\\
\geq&~{}\int_{0}^{\tau^{i}}\left(\left(\eta_{t}\xi^{\mu,i}_{t}+\frac{\kappa_{t}}{N}X^{\mu,i}_{t}\right)(\xi_{t}-\xi^{\mu,i}_{t})+\left(\lambda_{t}X^{\mu,i}_{t}+\kappa_{t}\mu_{t}\right)(X_{t}-X^{\mu,i}_{t})\right)\,dt.\end{split}$

Since $\|Y^{\mu,i}\|_{\infty}<\infty$ and
$X^{\mu,i}_{\tau^{i}}=X_{\tau^{i}}=0$, the same integration by parts argument
given in the proof of Proposition 2.7 yields that

$\displaystyle
0=Y^{\mu,i}_{\tau^{i}}(X_{\tau^{i}}-X^{\mu,i}_{\tau^{i}})=-\int_{0}^{\tau^{i}}Y^{\mu,i}_{t}(\xi_{t}-\xi^{\mu,i}_{t})\,dt-\int_{0}^{\tau^{i}}(\kappa_{t}\mu_{t}+\lambda_{t}X^{\mu,i}_{t})(X_{t}-X^{\mu,i}_{t})\,dt.$
(4.2)

Using that $(X^{\mu,i},Y^{\mu,i})$ solves the equation (2.6) with
$\delta=\frac{1}{N}$ this shows that

$J(\xi;\xi^{\mu,-i})-J(\xi^{\mu,i};\xi^{\mu,-i})\geq\int_{0}^{\tau^{i}}\left(\eta_{t}\xi^{\mu,i}_{t}+\frac{\kappa_{t}}{N}X^{\mu,i}_{t}-Y^{\mu,i}_{t}\right)(\xi_{t}-\xi^{\mu,i}_{t})\,dt=0.$

  2. (ii)

$\tau>\tau^{i}$. In this case, $\tau^{i}<T$ and
$\xi^{\mu,i}_{t}=Y^{\mu,i}_{t}=0$ for all $t\in[\tau,T]$. Since $\tau^{i}=T$
if $x^{i}$ and $\mu_{T}$ have different signs, $\tau>\tau^{i}$ holds only if
$x^{i}$ and $\mu_{T}$ have the same sign in which case

$\mu_{t}X_{t}\geq 0\quad\mbox{ for all }t\in[0,T].$

Moreover, the integration by parts argument still applies since
$Y^{\mu,i}_{\tau^{i}}=0$. Hence,

$\displaystyle J(\xi;\xi^{\mu,-i})-J(\xi^{\mu,i};\xi^{\mu,-i})\geq$
$\displaystyle~{}\int_{\tau^{i}}^{\tau}\left(\frac{1}{2}\left(\eta_{t}-\frac{\kappa_{t}}{N}\right)\xi^{2}_{t}+\frac{1}{2}\frac{\kappa_{t}}{N}\left(\xi_{t}+X_{t}\right)^{2}+\frac{1}{2}\left(\lambda_{t}-\frac{\kappa_{t}}{N}\right)X^{2}_{t}\right)\,dt$
$\displaystyle\quad+\int_{\tau^{i}}^{\tau}\kappa_{t}\left(\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)X_{t}dt$ $\displaystyle\geq$
$\displaystyle~{}\int_{\tau^{i}}^{\tau}\kappa_{t}\left(\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}\right)X_{t}dt.$

The key observation is now that $\xi^{\mu,i}_{t}=0$ for $t\in[\tau^{i},T]$
yields

$\displaystyle\frac{1}{N}\sum_{j\neq
i}\xi^{\mu,j}_{t}=\frac{1}{N}\sum_{j=1}^{N}\xi^{\mu,j}_{t}=\mu_{t},\qquad
t\in[\tau^{i},T].$

Hence,

$\displaystyle J(\xi;\xi^{\mu,-i})-J(\xi^{\mu,i};\xi^{\mu,-i})\geq$
$\displaystyle~{}\int_{\tau^{i}}^{\tau}\mu_{t}X_{t}dt\geq 0.$

∎

We are now going to show that there exists a unique Nash equilibrium with
continuous aggregate trading rate. To do so, we first establish a uniqueness
of equilibrium result when the aggregate trading rate does not change sign.

######  Theorem 4.4.

  1. 1.

Let
$(\xi^{\ast,1},\dots,\xi^{\ast,N})\in\mathcal{A}_{x_{1}}\times\dots,\mathcal{A}_{x_{N}}$
be a Nash equilibrium of the $N$-player game, such that the average trading
rate does not change sign:

$\mu:=\frac{1}{N}\sum_{i=1}^{N}\xi^{\ast,i}\in L^{1}_{-}([0,T])\cup
L^{1}_{+}([0,T]).$

Then $\xi^{\ast,i}=\xi^{\mu,i}$ for all $i=1,\dots,N$ and $\mu$ is the unique
solution to the fixed-point equation (3.6).

  2. 2.

There exists a unique equilibrium with continuous aggregate trading rate.

###### Proof.

  1. 1.

For every $i=1,\dots,N$ the strategy $\xi^{\ast,i}\in\mathcal{A}_{x_{i}}$ is a
maximizer of $J(\cdot;\xi^{\ast,-i})$. Let $X^{\ast,i}$ be the corresponding
state process and $\tau^{\ast,i}$ the corresponding absorption time, and
$\tau^{i}:=\tau_{\mu}(x^{i})$. Assume to the contrary that

$\|\xi^{\ast,i}-\xi^{\mu,i}\|_{L^{2}}\neq 0.$

    1. (i)

$\tau^{i}\geq\tau^{\ast,i}$: The argument is different from the one given in
the proof Theorem 4.3 as we now working with the equilibrium strategy
$\xi^{\ast}$ and not with the admisible strategy $\xi^{\mu}$. Nevertheless, we
use again the integration by parts argument based on the dynamics of
$\xi^{\mu,i}$. Therefore, we need to separate cross terms from squared terms
in the convexity estimate:

$\begin{split}J(\xi^{\ast,i};\xi^{\ast,-i})&-J(\xi^{\mu,i};\xi^{\ast,-i})\\\
=&~{}\int_{0}^{\tau^{i}}\left(\frac{1}{2}\eta_{t}(\xi^{\ast,i}_{t})^{2}+\frac{1}{2}\lambda_{t}(X^{\ast,i}_{t})^{2}-\frac{1}{2}\eta_{t}(\xi^{\mu,i}_{t})^{2}-\frac{1}{2}\lambda_{t}(X^{\mu,i}_{t})^{2}\right)\,dt\\\
&+~{}\int_{0}^{\tau^{i}}\frac{\kappa_{t}}{N}\left(\xi^{\ast,i}_{t}X^{\ast,i}_{t}-\xi^{\mu,i}_{t}X^{\mu,i}_{t}\right)dt+\int_{0}^{\tau^{i}}\kappa_{t}\left(\frac{1}{N}\sum_{j\neq
i}\xi^{\ast,j}_{t}\right)(X^{\ast,i}_{t}-X^{\mu,i}_{t})dt\\\
=&~{}\int_{0}^{\tau^{i}}\left(\frac{1}{2}\eta_{t}(\xi^{\ast,i}_{t})^{2}+\frac{1}{2}\lambda_{t}(X^{\ast,i}_{t})^{2}-\frac{1}{2}\eta_{t}(\xi^{\mu,i}_{t})^{2}-\frac{1}{2}\lambda_{t}(X^{\mu,i}_{t})^{2}\right)\,dt\\\
&+~{}\int_{0}^{\tau^{i}}\frac{\kappa_{t}}{N}\left(\xi^{\ast,i}_{t}-\xi^{\mu,i}_{t}\right)X^{\mu,i}_{t}dt+\int_{0}^{\tau^{i}}\kappa_{t}\mu_{t}(X^{\ast,i}_{t}-X^{\mu,i}_{t})dt\\\
>&~{}\int_{0}^{\tau^{i}}\left(\eta_{t}\xi^{\mu,i}_{t}(\xi^{\ast,i}_{t}-\xi^{\mu,i}_{t})+\lambda_{t}X^{\mu,i}_{t}(X^{*,i}_{t}-X^{\mu,i}_{t})\right)\,dt\\\
&+~{}\int_{0}^{\tau^{i}}\frac{\kappa_{t}}{N}\left(\xi^{\ast,i}_{t}-\xi^{\mu,i}_{t}\right)X^{\mu,i}_{t}dt+\int_{0}^{\tau^{i}}\kappa_{t}\mu_{t}(X^{\ast,i}_{t}-X^{\mu,i}_{t})dt\\\
=&~{}\int_{0}^{\tau^{i}}\left(\left(\eta_{t}\xi^{\mu,i}_{t}+\frac{\kappa_{t}}{N}X^{\mu,i}_{t}\right)(\xi^{\ast,i}_{t}-\xi^{\mu,i}_{t})+\left(\lambda_{t}X^{\mu,i}_{t}+\kappa_{t}\mu_{t}\right)(X^{\ast,i}_{t}-X^{\mu,i}_{t})\right)\,dt,\\\
=&~{}0\end{split}$

The last argument follows by the same integration by parts argument as in the
proof of Theorem 4.3. The above estimate is a contradiction to the optimality
of $\xi^{\ast,i}$, hence it must hold $\xi^{\ast,i}=\xi^{\mu,i}$.

    2. (ii)

$\tau^{i}<\tau^{\ast,i}$: Since $\mu$ is either strictly non-negative or
strictly non-positive we again only have to consider the case $x_{i}\mu\geq
0$. We then have immediately

$\displaystyle J(\xi^{\ast,i};\xi^{\ast,-i})-$ $\displaystyle
J(\xi^{\mu,i};\xi^{\ast,-i})$ $\displaystyle\geq$
$\displaystyle~{}\int_{\tau^{i}}^{\tau^{\ast,i}}\left(\frac{1}{2}\eta_{t}(\xi^{\ast,i}_{t})^{2}+\frac{1}{N}\left(\xi^{\ast,i}_{t}+\sum_{j\neq
i}\xi^{\ast,j}_{t}\right)\kappa_{t}X^{\ast,i}_{t}+\frac{1}{2}\lambda_{t}(X^{\ast,i}_{t})^{2}\right)\,dt$
$\displaystyle=$
$\displaystyle~{}\int_{\tau^{i}}^{\tau^{\ast,i}}\left(\frac{1}{2}\eta_{t}(\xi^{\ast,i}_{t})^{2}+\mu_{t}\kappa_{t}X^{\ast,i}_{t}+\frac{1}{2}\lambda_{t}(X^{\ast,i}_{t})^{2}\right)\,dt$
$\displaystyle>$ $\displaystyle~{}0,$

where the last inequality follows from the fact that
$\operatorname{sign}(X^{\ast,i}_{t})=\operatorname{sign}(x_{i})=\operatorname{sign}(\mu_{t})$
for all $t\in[0,\tau^{\ast,i})$. The above estimate contradicts the optimality
of $\xi^{\ast,i}$ and therefore $\xi^{\ast,i}=\xi^{\mu,i}$.

  2. 2.

The uniqueness of equilibria with continuous mean trading rate now follows
from the first part of this theorem and Theorem 4.3 by using the same
arguments as in the proof of Theorem 4.1.

∎

###  4.2 Convergence to MFG equilibrium

In this section we prove the convergence of the $N$-player equilibrium to the
mean field equilibrium if the number of players tends to infinity. For each
integer $N$ we denote by $\mu^{N}$ the mean equilibrium trading rate in the
$N$-player game with initial distribution
$\nu^{N}_{0}=\frac{1}{N}\sum_{j=0}^{N}\delta_{x_{j}}$ and tail probability
function $q_{0}^{N}$.

######  Proposition 4.5.

Assume that tail distribution $q^{N}_{0}$ converges toward $q_{0}$ in the
following sense:

$\lim_{N\rightarrow\infty}\sup_{x\in[0,\infty)}|q^{N}_{0}(x)-q_{0}(x)|=0.$
(4.3)

Then

$\lim_{N\to\infty}\|\mu^{N}-\mu\|_{\infty}=0,$ (4.4)

where $\mu\in C^{0}([0,T];\mathbb{R})$ satisfies the mean field equilibrium
equation (3.6).

###### Proof.

We assume w.l.o.g. that $\mathbb{E}[\nu_{0}]>0$ and consider $N$ large enough
such that $\mathbb{E}[\nu^{N}_{0}]>0$ and $\lambda-\frac{\kappa}{N}\geq 0$ and
$\eta-\frac{\kappa}{N}>0$. The equilibrium equation (3.11) applied to the
empirical measure $\nu_{0}^{N}$ yields

$\displaystyle 0<\mu^{N}_{t}$
$\displaystyle\leq\frac{\alpha^{N}_{T}}{\eta_{T}}\left\\{\mathbb{E}[\nu_{0}^{N}]-(1-q_{0}^{N}(0))x_{\mu^{N}}-\int_{0}^{x_{\mu^{N}}}q_{0}^{N}(y)\,dy\right\\}+C\int_{t}^{T}\left(\mu^{N}_{s}+\int_{s}^{T}\mu^{N}_{u}\,du\right)\,ds$
$\displaystyle\leq
C\left\\{\mathbb{E}[\nu_{0}^{N}]-(1-q_{0}^{N}(0))x_{\mu^{N}}-\int_{0}^{x_{\mu^{N}}}q_{0}^{N}(y)\,dy\right\\}+C\int_{t}^{T}\mu^{N}_{s}\,ds,$

where $C>0$ is independent of $N$. In view of (4.3) the first integral term in
last line above is bounded uniformly in $N$. Hence it follows from Grönwall’s
inequality that $(\mu^{N})_{N\in\mathbb{N}}$ is uniformly bounded in
$C^{0}([0,T];\mathbb{R})$. Similarly, from (3.11) applied to $\nu^{N}_{0}$ we
have for any $0\leq s\leq t\leq T$ that

$\displaystyle 0<|\mu^{N}_{t}-\mu^{N}_{s}|$ $\displaystyle\leq
C(1+T)\left(\sup_{N\geq 1}\|\mu^{N}\|_{\infty}\right)|t-s|.$

Thus, the sequence $(\mu^{N})_{N\in\mathbb{N}}$ is uniformly Lipschitz
continuous. By the Arzelà–Ascoli theorem there exists $\mu\in
C^{0}([0,T];\mathbb{R})$ and a subsequence $(N_{k})_{k\geq 1}$ that converges
to $\mu$.

In what follows we prove that $\mu$ is the unique solution to our integral
equation (3.11) corresponding to $\nu_{0}$. In particular, the sequence
$(\mu^{N})_{N\in\mathbb{N}}$ has a unique accumulation point and hence
converges to $\mu$. To this end, we consider the $N$-player equilibrium
equation (3.11):

$\displaystyle\begin{split}\mu^{N_{k}}_{t}=&~{}\frac{\alpha^{1/N_{k}}_{T}}{\eta_{T}}\left\\{\mathbb{E}[\nu_{0}^{N}]-(1-q_{0}^{N}(0))x_{\mu^{N_{k}}}-\int_{0}^{x_{\mu^{N_{k}}}}q_{0}^{N_{k}}(y)\,dy\right\\}\\\
&~{}+\int_{t}^{T}\frac{\lambda_{s}+\frac{\dot{\kappa}_{s}}{N_{k}}}{\eta_{s}}\left(\int_{s}^{T}\mu^{N_{k}}_{u}\,du\right)\,ds+\int_{t}^{T}\left(\frac{\dot{\eta}_{s}+\kappa_{s}(1-q_{0}^{N_{k}}(0))}{\eta_{s}}-\frac{\kappa_{s}}{N_{k}\eta_{s}}\right)\mu^{N_{k}}_{s}\,ds\\\
&~{}+\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}q_{0}^{N_{k}}\left(\int_{0}^{s}h^{1/N_{k}}_{u}\kappa_{u}\mu^{N_{k}}_{u}\,du\right)\mu^{N_{k}}_{s}\,ds,\qquad
t\in[0,T].\end{split}$ (4.5)

Our goal is to take the limit for $k\to\infty$ in the above equation. The
convergence $\alpha^{1/N}_{T}\to\alpha^{0}_{T}$ is proven in Lemma A.4. By
Lemma 2.4 the family $(h^{1/N})_{N\in\mathbb{N}}$ is uniformly bounded. By
dominated convergence theorem we have that

$\lim_{k\to\infty}x_{\mu^{N_{k}}}=\lim_{k\to\infty}\int_{0}^{T}h^{{1/N_{k}}}_{s}\kappa_{s}\mu^{N_{k}}_{s}\,ds=\int_{0}^{T}h^{0}_{s}\kappa_{s}\mu_{s}\,ds=x_{\mu}.$

Similarly for the terms in the second line of (4.5), it holds by the dominated
convergence theorem

$\lim_{k\to\infty}\left(\int_{t}^{T}\frac{\lambda_{s}+\frac{\dot{\kappa}_{s}}{N_{k}}}{\eta_{s}}\left(\int_{s}^{T}\mu^{N_{k}}_{u}\,du\right)\,ds+\int_{t}^{T}\left(\frac{\dot{\eta}_{s}+\kappa_{s}(1-q^{N_{k}}_{0}(0))}{\eta_{s}}-\frac{\kappa_{s}}{N_{k}\eta_{s}}\right)\mu^{N_{k}}_{s}\,ds\right)\\\
=\int_{t}^{T}\frac{\lambda_{s}}{\eta_{s}}\left(\int_{s}^{T}\mu_{u}\,du\right)\,ds+\int_{t}^{T}\frac{\dot{\eta}_{s}+\kappa_{s}(1-q_{0}(0))}{\eta_{s}}\mu_{s}\,ds,\quad
t\in[0,T].$

Regarding the last term in (4.5), we use a similar estimate as in the proof
Theorem 3.6. More precisely using integration by parts and the Lipschitz
continuity of $Q_{0}(x):=\int_{0}^{x}q_{0}(y)\,dy$ and
$Q^{N_{k}}_{0}(x):=\int_{0}^{x}q^{N_{k}}_{0}(y)\,dy$, it follows that for any
$t\in(0,T]$ there exists a constant $K>0$ that only depends on the bounds of
the coefficients and the functions $h^{\delta}$, $1/h^{\delta}$ and
$\dot{h}^{\delta}$ on the interval $[t,T]$, such that

$\displaystyle~{}\left|\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}q_{0}^{N_{k}}\left(x_{\mu^{N_{k}}}-\int_{s}^{T}h^{1/N_{k}}_{u}\kappa_{u}\mu^{N_{k}}_{u}\,du\right)\mu^{N_{k}}_{s}\,ds-\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}q_{0}\left(x_{\mu}-\int_{s}^{T}h^{0}_{u}\kappa_{u}\mu_{u}\,du\right)\mu_{s}\,ds\right|$
$\displaystyle\leq$
$\displaystyle~{}\left|\frac{1}{\eta_{T}h_{T}^{1/N_{k}}}Q_{0}^{N_{k}}(x_{\mu^{N_{k}}})-\frac{1}{\eta_{T}h_{T}^{0}}Q_{0}(x_{\mu})\right|$
$\displaystyle~{}+\left|\frac{1}{\eta_{t}h_{t}^{1/N_{k}}}Q_{0}^{N_{k}}\left(x_{\mu^{N_{k}}}-\int_{t}^{T}h^{1/N_{k}}_{u}\kappa_{u}\mu^{1/N_{k}}_{u}\,du\right)-\frac{1}{\eta_{t}h_{t}^{0}}Q_{0}\left(x_{\mu}-\int_{t}^{T}h^{0}_{u}\kappa_{u}\mu_{u}\,du\right)\right|$
$\displaystyle~{}+\int_{t}^{T}\left|Q_{0}^{N_{k}}\left(x_{\mu^{N_{k}}}-\int_{s}^{T}h^{1/N_{k}}_{u}\kappa_{u}\mu^{1/N_{k}}_{u}\,du\right)\frac{\dot{\eta}_{s}h^{1/N_{k}}_{s}-\eta_{s}\dot{h}^{1/N_{k}}_{s}}{(\eta_{s}h_{s}^{1/N_{k}})^{2}}\right.$
$\displaystyle\left.-~{}Q_{0}\left(x_{\mu}-\int_{s}^{T}h^{0}_{u}\kappa_{u}\mu_{u}\,du\right)\frac{\dot{\eta}_{s}h^{0}_{s}-\eta_{s}\dot{h}^{0}_{s}}{(\eta_{s}h^{0}_{s})^{2}}\right|\,ds$
$\displaystyle\leq$
$\displaystyle~{}K\left(\sup_{x\in[0,\infty)}|q^{N_{k}}_{0}(x)-q_{0}(x)|\right)\|\mu^{N_{k}}-\mu\|_{\infty}+K|x_{\mu^{N_{k}}}-x_{\mu}|+K\|\mu^{N_{k}}-\mu\|_{\infty}.$

Taking $k\to\infty$ in (4.5), it follows from (4.3) that $\mu$ satisfies the
mean field equilibrium equation (3.11). ∎

##  5 Examples

In this section we illustrate the impact of market drop-out on equilibrium
trading rates in the mean field and $N$-player game on a few numerical
examples. In particular, we compare the equilibria to the mean field game of
portfolio liquidation without market drop-out as presented in [16]. For
simplicity we will consider constant cost parameters; more precisely we choose

$\displaystyle\eta\equiv 5,\quad\kappa\equiv 10,\quad\lambda\equiv 5.$

Figure 1: The evolution of the state processes in equilibrium of the
liquidation game _with market drop-out_ (left) and _without market drop-out_
(right) for several representative players in the presence of only sellers. We
have highlighted the moments of drop-out and $\hat{x}$ (which represents the
smallest initial position for which no early exit takes place).

A numerical approximation of the mean-field equilibrium is obtained by solving
the backward integral equation (3.13) using a standard numerical solver for
several values c777Technically one rather defines a function that outputs for
any given parameter $c$ the evaluated solution $f_{\mu^{c}}(T)-c$ and then
applies a standard numerical root finding method to that function. and then
finding the root of888Note that in the two sided situation we use the root of
$c\mapsto\mu_{T}^{c}$ as an upper bound in the root finding procedure. In the
one sided situation we instead use the upper bound of support of $\nu_{0}$ or
the upper bound from the a priori estimate (3.10). $c\mapsto f^{c}_{0}$.

We first consider a one-sided situation with only sellers. Specifically we
choose $\nu_{0}$ to be an exponential distribution with mean $1.5$, i.e.
$q_{0}(x)=e^{-\frac{2}{3}x}$ and $p_{0}\equiv 0$. Figure 1 presents the
evolution of the equilibrium mean-field state process with and without market
drop-out for several representative players. In a model without market drop-
out constraint, players with smaller initial portfolio use short selling and
round trip strategies, whereas in the presence of the market drop-out
constraint all players stick to strict selling strategies (as it is generally
the case for the dominating side of the market, see Section 2.3.2).

Figure 2: Comparison of the mean trading rate in the equilibrium of the
liquidation game _with market drop-out_ (solid line) and _without market drop-
out_ (dashed line). left: one-sided situation, right: two sided situation.

Figure 2 (left) compares the mean equilibrium trading rate for both
situations. For our choice of parameters the rates only deviate slightly. As
expected, the drop-out constraint initially leads to slower liquidation and to
an increasingly faster liquidation halfway through the trading period.

Secondly, we consider a two-sided situation with $q_{0}(x)=0.8\cdot
e^{-\frac{2}{3}x}$ and $p_{0}(x)=0.2\cdot e^{x}$. This leads to a mean initial
position of $\mathbb{E}[\nu_{0}]=1$, hence, to a situation in which sellers
dominate the market. Figure 3 (left) shows the evolution of the representative
state processes while Figure 3 (right) displays the equilibrium mean trading
rate. We observe that the equilibrium term indeed does not change its positive
sign, which again leads to the fact that sellers do not use short selling
strategies in the situation with market drop-out.

Figure 4 shows the solution to the $N$-player games. In this game, the two
smallest players on the selling side drop out early, while all other players
liquidate at the terminal time. Our simulations suggest that the convergence
to the MFG equilibrium is rather fast. We emphasize that mean trading rate in
the $N$-player game is not smooth; the slope changes discontinuously when a
player drops out of the market.

Figure 3: The evolution of the state processes in equilibrium of the
liquidation game in the two sided case: _with market drop-out_ (left) and
_without market drop-out_ (right).

Figure 4: left: Evolution of states of $N=7$ players in the Nash equilibrium.
right: Comparison of the mean trading rate in the equilibrium of the _mean
field game_ (solid line) and the $N$-player game (dashed/dotted line) for
$N=7,15$ and $100$.

##  6 Conclusion

In this paper we studied novel mean field liquidation games, where each player
drops out of the market once her position hits zero. We characterized the
unique Nash equilibrium by a nonlinear integral equation. We established
existence and uniqueness of equilibrium result for both the $N$-player game
and the MFG, and proved the convergence of equilibria to the MFG equilibrium.
Our result suggests that in a market dominated by sellers, the sellers’
optimization problem with market drop-out is the same as the optimization
problem with short selling constraint. However, the buyers with small initial
positions would trade in an opposite direction initially, thus, the buyers’
optimization problem may be different from the one with short selling
constraint, which is left as an independent ongoing work.

##  Appendix A Appendix

Throughout this section we assume that Assumption 2.1 holds.

######  Lemma A.1 (cf. Lemma A.1 in [16]).

There exists a unique solution to the terminal value problem

$\displaystyle-\dot{A}_{t}=$
$\displaystyle~{}-\frac{A^{2}_{t}}{\eta_{t}}+\lambda_{t},\qquad\lim_{t\nearrow
T}A_{t}=\infty.$ (A.1)

Moreover, the following estimates hold for all $0\leq s\leq t<T$:

$\displaystyle\frac{1}{\|\eta^{-1}\|_{\infty}(T-t)}\;\;\leq\;\;\frac{1}{\int_{t}^{T}\frac{1}{\eta_{s}}ds}\;\;\leq\;\;A_{t}\;\;\leq\;\;\frac{1}{(T-t)^{2}}\int_{t}^{T}\left(\eta_{s}+(T-s)^{2}\lambda_{s}\right)\,ds$

and

$\displaystyle\exp\left(-\int_{s}^{t}\frac{A_{r}}{\eta_{r}}dr\right)\leq{\|\eta\|_{\infty}}{\|\eta^{-1}\|_{\infty}}\frac{T-t}{T-s}.$

######  Lemma A.2.

Let $A$ be the solution to the equation (A.1). Then the map

$\alpha^{0}:[0,T)\to[0,\infty),\quad t\mapsto
A_{t}e^{-\int_{0}^{t}\frac{A_{r}}{\eta_{r}}\,dr},$

is non-increasing and

$\alpha^{0}_{T}=\lim_{t\to
T}A_{t}e^{-\int_{0}^{t}\frac{A_{r}}{\eta_{r}}dr}\in(0,\infty).$

###### Proof.

Differentiation yields

$\displaystyle\frac{d}{dt}\left(A_{t}e^{-\int_{0}^{t}\frac{A_{r}}{\eta_{r}}\,dr}\right)=\left(\dot{A}_{t}-\frac{A^{2}_{t}}{\eta_{t}}\right)e^{-\int_{0}^{t}\frac{A_{r}}{\eta_{r}}\,dr}=-\lambda_{t}e^{-\int_{0}^{t}\frac{A_{r}}{\eta_{r}}\,dr}\leq
0,\qquad t\in[0,T).$

which proves the first statement. Since $\alpha^{0}$ is positive it remains to
show that $\alpha^{0}_{T}$ is strictly positive. The differentiability of
$\eta$ at $T$ and along with the fact that $\eta_{T}>0$ yields $L>0$ and
$\varepsilon\in(0,\frac{\eta_{T}}{L})$ such that

$|\eta_{T}-\eta_{t}|\leq L(T-t),$

for all $t\in(T-\varepsilon,T]$. For all such $t$ it then follows from Lemma
A.1 that

$\displaystyle\int_{T-\epsilon}^{t}\frac{A_{r}}{\eta_{r}}\,dr$
$\displaystyle\leq~{}\int_{T-\varepsilon}^{t}\frac{\frac{1}{(T-s)^{2}}\int_{s}^{T}\eta_{u}\,du+T\|\lambda\|_{\infty}}{\eta_{s}}\,ds$
$\displaystyle\leq~{}\int_{T-\varepsilon}^{t}\frac{\eta_{T}\frac{1}{(T-s)}+\frac{1}{2}L+T\|\lambda\|_{\infty}}{\eta_{T}-L(T-s)}\,ds$
$\displaystyle=~{}\int_{T-t}^{\varepsilon}\frac{1}{u-\frac{L}{\eta_{T}}u^{2}}\,du+\left(\frac{1}{2}+\frac{T\|\lambda\|_{\infty}}{L}\right)\int_{T-t}^{\varepsilon}\frac{1}{\frac{\eta_{T}}{L}-u}\,du.$

For the two integrals on the right we then have

$\displaystyle\int_{T-t}^{\varepsilon}\frac{1}{u-\frac{L}{\eta_{T}}u^{2}}\,du=-\ln\left(\frac{T-t}{\varepsilon}\right)+\ln\left(\frac{\frac{\eta_{T}}{L}-(T-t)}{\frac{\eta_{T}}{L}-\varepsilon}\right)\leq-\ln\left(\frac{T-t}{\varepsilon}\right)-\ln\left(1-\frac{L}{\eta_{T}}\varepsilon\right)$

and similarly

$\displaystyle\int_{T-t}^{\varepsilon}\frac{1}{\frac{\eta_{T}}{L}-u}\,du=\ln\left(\frac{\frac{\eta_{T}}{L}-(T-t)}{\frac{\eta_{T}}{L}-\varepsilon}\right)\leq-\ln\left(1-\frac{L}{\eta_{T}}\varepsilon\right).$

Putting together the estimates yields

$\displaystyle
e^{-\int_{T-\varepsilon}^{t}\frac{A_{s}}{\eta_{s}}\,ds}\geqslant
K_{\varepsilon}\varepsilon^{-1}(T-t),\quad\text{with
}K_{\varepsilon}=\left(1-\frac{L}{\eta_{T}}\varepsilon\right)^{\frac{3}{2}+\frac{T\|\lambda\|_{\infty}}{L}}\in(0,1),$

for all $t\in(T-\varepsilon,T]$. This allows us to finally estimate using
again Lemma A.1

$\displaystyle\alpha^{0}_{T}=e^{-\int_{0}^{T-\varepsilon}\frac{A_{s}}{\eta_{s}}\,ds}\left(\lim_{t\rightarrow
T}A_{t}e^{-\int_{T-\varepsilon}^{t}\frac{A_{s}}{\eta_{s}}\,ds}\right)\geq
e^{-\int_{0}^{T-\varepsilon}\frac{A_{s}}{\eta_{s}}\,ds}\varepsilon^{-1}K_{\varepsilon}\|\eta^{-1}\|_{\infty}^{-1}>0.$

∎

######  Lemma A.3.

For any $\delta\in[0,1]$ there exists a unique solution $A^{\delta}$ to the
singular terminal value problem

$\displaystyle-\dot{A}_{t}=$
$\displaystyle~{}-\frac{1}{\eta_{t}}A^{2}_{t}+\delta\frac{\kappa_{t}}{\eta_{t}}A_{t}+\lambda_{t},\qquad\lim_{t\nearrow
T}A_{t}=\infty.$ (A.2)

Furthermore, the solutions satisfy the comparison principle

$A^{\delta_{1}}\leq A^{\delta_{2}},\qquad 0\leq\delta_{1}\leq\delta_{2}\leq 1$

and the map

$[0,1]\mapsto(C^{0}(I;\mathbb{R}),\|\cdot\|_{\infty}),\quad\delta\mapsto
A^{\delta},$ (A.3)

is continuous uniformly on compact subintervals $I\subset[0,T)$.

###### Proof.

For $\delta=0$ the equation (A.2) coincides with (A.1) and we denote
$A=A^{0}$. For $\delta\in(0,1]$ define

$\displaystyle\widetilde{\eta}_{t}:=\eta_{t}e^{\int_{0}^{t}\delta\frac{\kappa_{s}}{\eta_{s}}ds},\qquad\widetilde{\lambda}_{t}:=\lambda_{t}e^{\int_{0}^{t}\delta\frac{\kappa_{s}}{\eta_{s}}ds},\qquad
t\in[0,T].$ (A.4)

Then clearly $\widetilde{\eta}\in C^{1}([0,T],(0,\infty))$ and
$\widetilde{\lambda}$ is bounded and therefore by the above Lemma A.1 there
exists a unique solution $\tilde{A}$ to the singular terminal value problem

$\displaystyle-\frac{d}{dt}\widetilde{A}_{t}=$
$\displaystyle~{}-\frac{1}{\widetilde{\eta}_{t}}\widetilde{A}^{2}_{t}+\tilde{\lambda}_{t},\qquad\lim_{t\nearrow
T}\widetilde{A}_{t}=\infty.$

A simple calculation then shows that
$A^{\delta}:=\tilde{A}e^{-\int_{0}^{\cdot}\delta\frac{\kappa_{r}}{\eta_{r}}dr}$
solves (A.2) (and vice versa for uniqueness). For the forthcoming part of the
proof, note that $A^{\delta}>c\tilde{A}>0$ for a suitable constant $c>0$.

Now let $0\leq\delta_{1}\leq\delta_{2}\leq 1$ and let $A^{\delta_{2}}$ and
$A^{\delta_{2}}$ be the corresponding solutions to (A.2). Then,

$\left(\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\right)^{\prime}=\frac{\kappa_{t}}{\eta_{t}}\left(\frac{\delta_{1}}{A^{\delta_{1}}_{t}}-\frac{\delta_{2}}{A^{\delta_{2}}_{t}}\right)+\lambda_{t}\left(\frac{1}{A^{\delta_{1}}_{t}}+\frac{1}{A^{\delta_{2}}_{t}}\right)\left(\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\right).$
(A.5)

Since it follows from the first part of the proof that $A^{\delta}>0$ we can
estimate

$\frac{d}{dt}\left(\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\right)\leq\left(\frac{\delta_{2}\kappa_{t}}{\eta_{t}}+\lambda_{t}\left(\frac{1}{A^{\delta_{1}}_{t}}+\frac{1}{A^{\delta_{2}}_{t}}\right)\right)\left(\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\right).$

Recalling that $\lim_{t\to T}\frac{1}{A^{\delta}_{t}}=0$ and applying the
Grönwall’s inequality in the backward form we obtain

$\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\geq 0,\quad
t\in[0,T],$

which proves the comparison principle. This being proven, we can estimate in
(A.5) as follows:

$\displaystyle\frac{d}{dt}\left(\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}\right)~{}\geq~{}-\frac{\kappa_{t}}{\eta_{t}}\frac{\delta_{2}-\delta_{1}}{A^{\delta_{2}}_{t}}~{}\geq~{}-\frac{\kappa_{t}}{\eta_{t}}\frac{\delta_{2}-\delta_{1}}{A_{t}}.$

Integrating both sides of the inequality and using Lemma A.1 yields

$\frac{1}{A^{\delta_{1}}_{t}}-\frac{1}{A^{\delta_{2}}_{t}}~{}\leq~{}(\delta_{2}-\delta_{1})\int_{t}^{T}\frac{\kappa_{s}}{\eta_{s}}\frac{1}{A_{s}}\,ds~{}\leq~{}C(\delta_{2}-\delta_{1}),\qquad
t\in[0,T).$

for some constant $C>0$. From the comparison principle we then obtain

$0\leq
A^{\delta_{2}}_{t}-A^{\delta_{1}}_{t}\leq(A^{1}_{t})^{2}C(\delta_{2}-\delta_{1}),\qquad
t\in[0,T).$

The continuity of the map $\delta\mapsto A^{\delta}$ now follows from the
uniform boundedness of the map $A^{1}$ on any compact subset of $[0,T)$. ∎

######  Lemma A.4.

For all $\delta\in[0,1]$ the map $\alpha^{\delta}:t\mapsto
A^{\delta}_{t}e^{-\int_{0}^{t}\frac{A_{r}^{\delta}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
is non-increasing and

$\alpha^{\delta}_{T}=\lim_{t\to
T}A^{\delta}_{t}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}dr}\in(0,\infty)$
(A.6)

Furthermore we have the following convergence

$\lim_{\delta\to 0}\alpha^{\delta}_{T}=\alpha^{0}_{T}.$

###### Proof.

It can be verified directly that

$\frac{d}{dt}\left(A^{\delta}_{t}e^{-\int_{0}^{t}\frac{A_{r}^{\delta}-\delta\kappa_{r}}{\eta_{r}}\,dr}\right)=-\lambda_{t}e^{-\int_{0}^{t}\frac{A_{r}^{\delta}-\delta\kappa_{r}}{\eta_{r}}\,dr}\leq
0.$

From the proof of Lemma A.3 we have that
$A^{\delta}e^{\int_{0}^{\cdot}\delta\frac{\kappa_{r}}{\eta_{r}}dr}$ satisfies
the equation (A.2) with $\delta=0$ and modified coefficients
$\widetilde{\eta}$ and $\widetilde{\lambda}$ defined in (A.4) that satisfy
Assumption 2.1. Therefore the existence and positivity of the limit in (A.6)
follows immediately from Lemma A.2.

Regarding the convergence result, note that on the one hand

$\alpha^{\delta}_{T}=\lim_{t\to
T}A^{\delta}_{t}e^{-\int_{0}^{t}\frac{A_{s}^{\delta}-\delta\kappa_{s}}{\eta_{s}}\,ds}=A^{\delta}_{0}-\int_{0}^{T}\lambda_{s}e^{-\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds.$

From Lemma A.3 it follows that $A^{\delta}$ converges pointwise on $[0,T)$
towards $A$. By the dominated convergence theorem we then have

$\lim_{\delta\to 0}\alpha^{\delta}_{T}=\lim_{\delta\to
0}\left(A^{\delta}_{0}-\int_{0}^{T}\lambda_{s}e^{-\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds\right)=A^{0}_{0}-\int_{0}^{T}\lambda_{s}e^{-\int_{0}^{s}\frac{A^{0}_{r}}{\eta_{r}}\,dr}\,ds.$

The desired convergence result now follows from

$\alpha^{0}_{T}=\lim_{t\to
T}A^{0}_{t}e^{-\int_{0}^{t}\frac{A^{0}_{s}}{\eta_{s}}\,ds}=A^{0}_{0}-\int_{0}^{T}\lambda_{s}e^{-\int_{0}^{s}\frac{A^{0}_{r}}{\eta_{r}}\,dr}\,ds.$

∎

###### Proof of Lemma 2.4.

From Lemma A.4, we have for any $\delta\in[0,1]$

$\begin{split}h^{\delta}_{t}=&~{}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{t}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{2A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds\\\
=&~{}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{t}e^{\int_{0}^{s}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\frac{A^{\delta}_{s}}{\eta_{s}}\frac{1}{\alpha^{\delta}_{s}}\,ds\\\
\leq&~{}\frac{1}{\alpha^{\delta}_{t}}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\left(e^{\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}-1\right)\\\
\leq&~{}\frac{1}{\alpha^{\delta}_{T}},\qquad t\in[0,T).\end{split}$ (A.7)

From Lemma A.4, we have the uniform boundedness of
$\left(\frac{1}{\alpha^{\delta}_{T}}\right)_{0\leq\delta\leq 1}$, which leads
to a uniform upper bound of $h^{\delta}$ for $\delta\in[0,1]$. On the other
hand, note that

$\displaystyle\frac{d}{dt}\left(\frac{e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}}{A^{\delta}_{t}}\right)$
$\displaystyle=~{}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\frac{\lambda_{t}}{(A^{\delta}_{t})^{2}},\qquad
t\in[0,T).$

Therefore using integration by parts we can establish the following lower
bound on $h^{\delta}$

$\begin{split}h^{\delta}_{t}&=e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{t}\frac{1}{\eta_{s}}e^{\int_{0}^{s}\frac{2A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}\,ds\\\
&=e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\int_{0}^{t}\left(\frac{A^{\delta}_{s}}{\eta_{s}}e^{\int_{0}^{s}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\right)\frac{e^{\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}}{A^{\delta}_{s}}\,ds\\\
&=e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\left(e^{\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\frac{e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}}{A^{\delta}_{t}}-\frac{1}{A^{\delta}_{0}}-\int_{0}^{t}e^{\int_{0}^{s}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\frac{\lambda_{s}e^{\int_{0}^{s}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}}{(A^{\delta}_{s})^{2}}\,ds\right)\\\
&=\frac{e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}}{A^{\delta}_{t}}+e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\left(-\int_{0}^{t}\frac{\lambda_{s}e^{\int_{0}^{s}\delta\frac{\kappa_{r}}{\eta_{r}}\,dr}}{(\alpha^{\delta}_{s})^{2}}\,ds-\frac{1}{A^{\delta}_{0}}\right)\\\
&\geq\frac{1}{\alpha^{\delta}_{t}}-Ce^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr},\qquad
t\in[0,T),\end{split}$ (A.8)

where $C>0$ is a constant that by Lemma A.3 and Lemma A.4 can be chosen
independently of $\delta\in[0,1]$. Furthermore combining the estimates (A.7)
and (A.8) and letting $t\nearrow T$ we see that
$h^{\delta}_{T}=1/\alpha^{\delta}_{T}$. Finally, note that $h^{\delta}$ is
continuously differentiable on $[0,T)$. Hence, using the lower bound (A.8) and
once more that $\alpha^{\delta}$ is non-increasing we obtain the following
uniform bound over $\delta\in[0,1]$

$\displaystyle\dot{h}^{\delta}_{t}$
$\displaystyle=~{}-\frac{A^{\delta}_{t}}{\eta_{t}}h^{\delta}_{t}+\frac{1}{\eta_{t}}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle\leq~{}-\frac{A^{\delta}_{t}}{\eta_{t}}\left(\frac{1}{\alpha^{\delta}_{t}}-Ce^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\right)+\frac{1}{\eta_{t}}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle=~{}-\frac{1}{\eta_{t}}\frac{A^{\delta}_{t}}{\alpha^{\delta}_{t}}+C\frac{\alpha_{t}}{\eta_{t}}e^{-\int_{0}^{t}\frac{\kappa_{r}}{\eta_{r}}\,dr}+\frac{1}{\eta_{t}}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle\leq~{}CA^{\delta}_{0}\frac{1}{\eta_{t}}$
$\displaystyle\leq~{}CA^{\delta}_{0}\|1/\eta\|_{\infty},\qquad t\in[0,T].$

On the other hand, using the upper bound (A.7) we obtain

$\displaystyle\dot{h}^{\delta}_{t}$
$\displaystyle\geq~{}-\frac{A^{\delta}_{t}}{\eta_{t}}\frac{1}{\alpha^{\delta}_{t}}\left(1-e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}\right)+\frac{1}{\eta_{t}}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle=~{}-\frac{A^{\delta}_{t}}{\eta_{t}}\frac{1}{\alpha^{\delta}_{t}}+\frac{A^{\delta}_{t}}{\eta_{t}}\frac{1}{\alpha^{\delta}_{t}}e^{-\int_{0}^{t}\frac{A^{\delta}_{r}}{\eta_{r}}\,dr}+\frac{1}{\eta_{t}}e^{\int_{0}^{t}\frac{A^{\delta}_{r}-\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle=~{}\frac{1}{\eta_{t}}e^{-\int_{0}^{t}\frac{\delta\kappa_{r}}{\eta_{r}}\,dr}$
$\displaystyle\geq~{}\frac{1}{\|\eta\|_{\infty}}e^{-\int_{0}^{T}\frac{\delta\kappa_{r}}{\eta_{r}}\,dr},\qquad
t\in[0,T].$

This proves in particular that $h^{\delta}$ is strictly increasing.

Finally, the a.e. convergence $h^{\delta}\rightarrow h^{0}$ and
$\dot{h}^{\delta}\rightarrow\dot{h}^{0}$ can be easily obtained by the
expressions of $h^{\delta}$ and $\dot{h}^{\delta}$, as well as the convergence
of $A^{\delta}\rightarrow A^{0}$ established in Lemma A.3. ∎

The following version of Grönwall’s inequality is frequently used in the main
text, but it is difficult to locate it in the literature, where the function
$\beta$ is usually assumed to be non-negative.

######  Lemma A.5 (Lower Grönwall’s inequality).

Let $u$ be an absolutely continuous function on $[0,T]$, with $u_{T}>0$ and
such that there exists a continuous function $\beta$ such that

$\displaystyle\dot{u}_{t}~{}\leq~{}\beta_{t}u_{t}\qquad\text{ for a.e.
}t\in[0,T].$

Then it holds

$\displaystyle u_{t}\geq
u_{T}\exp\\!\left(-\int_{t}^{T}\beta_{s}\,ds\right),\qquad t\in[0,T].$

###### Proof.

Define the differentiable function
$\nu_{\cdot}:=\exp(\int_{\cdot}^{T}\beta_{s}ds)/u_{T}$. Since $u_{T}>0$ by
assumption it holds $\nu>0$. At any point $t\in[0,T]$ where $u$ is
differentiable we obtain

$\displaystyle\frac{d}{dt}(\nu_{t}u_{t})=$
$\displaystyle~{}\dot{\nu}_{t}u_{t}+\nu_{t}\dot{u}_{t}$ $\displaystyle\leq$
$\displaystyle~{}-\beta_{t}\nu_{t}u_{t}+\beta_{t}\nu_{t}u_{t}=0.$

Since $u$ is almost everywhere differentiable, integration then yields

$\displaystyle
u_{T}\nu_{T}-u_{t}\nu_{t}=\int_{t}^{T}\frac{d}{dt}(\nu_{s}u_{s})\,ds\leq 0,$

and with $\nu_{T}u_{T}=1$ the stated estimate follows. ∎

## References

  * [1] A. Alfonsi, A. Schied, and A. Slynko.  Order book resilience, price manipulation, and the positive portfolio problem.  SIAM Journal on Financial Mathematics, 3(1):511–533, 2012.
  * [2] R. Almgren and N. Chriss.  Optimal execution of portfolio transactions.  Journal of Risk, 3:5–40, 2001.
  * [3] S. Ankirchner, M. Jeanblanc, and T. Kruse.  BSDEs with singular terminal condition and a control problem with constraints.  SIAM Journal on Control and Optimization, 52(2):893–913, 2014.
  * [4] P. Bank and M. Voß.  Linear quadratic stochastic control problems with singular stochastic terminal constraint.  SIAM Journal on Control and Optimization, 56(2), 2018.
  * [5] J.F. Bonnans, J. Gianatti, and L. Pfeiffer.  A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints.  SIAM Journal on Control and Optimization, 61(1):105–134, 2023.
  * [6] L. Campi and M. Burzoni.  Mean field games with absorption and common noise with a model of bank run.  arXiv:2107.00603, 2021.
  * [7] L. Campi and M. Fischer.  $N$-player games and mean-field games with absorption.  Annals of Applied Probability, 28(4):2188–2242, 2018.
  * [8] L. Campi, M. Ghio, and G. Livieri.  $N$-player games and mean-field games with smooth dependence on past absorptions.  Annales de l’Institut Henri Poincaré, 57(4):1901–1939, 2021\.
  * [9] P. Cardaliaguet and C. Lehalle.  Mean field game of controls and an application to trade crowding.  Mathematics and Financial Economics, 12(3):335–363, 2018.
  * [10] P. Casgrain and S. Jaimungal.  Mean field games with partial information for algorithmic trading.  arXiv:1803.04094, 2018.
  * [11] P. Casgrain and S. Jaimungal.  Mean-field games with differing beliefs for algorithmic trading.  Mathematical Finance, 30(3):995–1034, 2020.
  * [12] Riccardo Cesari and Harry Zheng.  Stochastic maximum principle for optimal liquidation with control-dependent terminal time.  Applied Mathematics and Optimization, 85(3), 2022.
  * [13] R. Dumitrescu, M. Leutscher, and P. Tankov.  Control and optimal stopping mean field games: a linear programming approach.  Electronic Journal of Probability, 26:1–49, 2021.
  * [14] D. Evangelista and Y. Thamsten.  On finite population games of optimal trading.  arXiv:2004.00790, 2020.
  * [15] A. Fruth, T. Schöneborn, and M. Urusov.  Optimal trade execution and price manipulation in order books with time-varying liquidity.  Mathematical Finance, 24(4):651–695, 2014.
  * [16] G. Fu, P. Graewe, U. Horst, and A. Popier.  A mean field game of optimal portfolio liquidation.  Mathematics of Operations Research, 46(4):1251–1281, 2021.
  * [17] G. Fu and U. Horst.  Mean-field leader-follower games with terminal state constraint.  SIAM Journal on Control and Optimization, 58(4):2078–2113, 2020\.
  * [18] G. Fu, U. Horst, and X. Xia.  Portfolio liquidation games with self exciting order flow.  Mathematical Finance, 30(4):1020–1065, 2022.
  * [19] J. Gatheral and A. Schied.  Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework.  International Journal of Theoretical and Applied Finance, 14(3):353–368, 2011.
  * [20] J. Gatheral, A. Schied, and Slynko.  Transient linear price impact and Fredholm integral equations.  Mathematical Finance, 22(3):445–474, 2012.
  * [21] P. J. Graber and A. Bensoussan.  Existence and uniqueness of solutions for Bertrand and Cournot mean field games.  Applied Mathematics & Optimization, 77:47–71, 2016.
  * [22] P. Jameson Graber and R. Sircar.  Master equation for Cournot mean field games of control with absorption.  Journal of Differential Equations, 343:816–909, 2023.
  * [23] P. Graewe and U. Horst.  Optimal trade exection with instantaneous price impact and stochastic resilience.  SIAM Journal on Control and Optimization, 55(6):3707–3725, 2017\.
  * [24] P. Graewe, U. Horst, and J. Qiu.  A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions.  SIAM Journal on Control and Optimization, 53(2):690–711, 2015.
  * [25] P. Graewe, U. Horst, and E. Séré.  A maximum principle approach to a deterministic mean field game of control with absorption.  Stochastic Processes and their Applications, 128(3):979–1006, 2018\.
  * [26] P. Graewe, U. Horst, and R. Sircar.  A maximum principle approach to a deterministic mean field game of control with absorption.  SIAM Journal on Control and Optimization, 60(5):3173–3190, 2022\.
  * [27] U. Horst, X. Xia, and C. Zhou.  Portfolio liquidation under factor uncertainty.  Annals of Applied Probability, 32(1):80–123, 2022.
  * [28] X. Huang, S. Jaimungal, and M. Nourian.  Mean-field game strategies for optimal execution.  Applied Mathematical Finance, 26:153–185, 2019.
  * [29] P. Kratz.  An explicit solution of a nonlinear-quadratic constrained stochastic control problem with jumps: Optimal liquidation in dark pools with adverse selection.  Mathematics of Operations Research, 39(4):1198–1220, 2014.
  * [30] T. Kruse and A. Popier.  Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting.  Stochastic Processes and their Applications, 126(9):2554–2592, 2016\.
  * [31] A. Obizhaeva and J. Wang.  Optimal trading strategy and supply/demand dynamics.  Journal of Financial Markets, 16(1):1–32, 2013.
  * [32] A. Popier and C. Zhou.  Second order BSDE under monotonicity condition and liquidation problem under uncertainty.  Annals of Applied Probability, 29(3), 2019.
  * [33] A. Schied, E. Strehle, and T. Zhang.  High-frequency limit of Nash equilibria in a market impact game with transient price impact.  SIAM Journal on Financial Mathematics, 8(1):589–634, 2017.
  * [34] A. Schied and T. Zhang.  A market impact game under transient price impact.  Mathematics of Operations Research, 44(1):102–121, 2019.
  * [35] G. Teschl.  Ordinary Differential Equations and Dynamical Systems. AMS, 2016.