# A Linear-quadratic Mean-Field Stochastic Stackelberg Differential Game with
Random Exit Time††thanks: This work was supported by the National Natural
Science Foundation of China (11471230, 11671282).

Zhun Goua, Nan-jing Huanga111Corresponding author. E-mail addresses:
<EMAIL_ADDRESS><EMAIL_ADDRESS>and Ming-hui Wangb
a. Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064,
P.R. China
b. Department of Economic Mathematics, Southwestern University of Finance and
Economics,
Chengdu, Sichuan 610074, P.R. China

Abstract. In this paper, we investigate a new model of a linear-quadratic
mean-field stochastic Stackelberg differential game with one leader and two
followers, in which the leader is allowed to stop her strategy at a random
time. Our overarching goal is to find the Stackelberg solution of the leader
and followers for such a model. By employing the backward induction method,
the state equation is divided into two-stage equations. Moreover, by using the
maximum principle and the verification theorem, the Stackelberg solution is
obtained for such a model.

Keywords: Stochastic Stackelberg differential game; Mean-field stochastic
differential equation; Random exit time; Stackelberg solution.

2020 Mathematics Subject Classification: 49N80, 60H20, 60J76, 93E20.

##  1 Introduction

####

The study of Stackelberg game was pioneered by Stackelberg [27], where there
are two players in the game. One player acts as the leader (she) while the
other behaves as the follower (he). First, the leader announces her strategy
and the follower reacts to it by optimizing his objective function in
accordance with the leader’s announced strategy. Then, the leader would like
to seek a strategy to optimize her cost function based on the follower’s best
response. The best strategy of the leader together with the best response of
the follower is known as a Stackelberg solution. Since then, Stackelberg game
has been studied extensively by many authors because of its wide application
in various fields, including economics and finance, management and decision,
transportation and evolutionary biology (see, for example, [8, 19, 21, 22,
33]). We would like to point out that many existed literatures only
investigated the Stackelberg game with one leader and one follower. However,
it is important to consider the Stackelberg game with multiple followers,
which is more complicate but obviously more suitable for describing some
practical problems. Different from the Stackelberg game with one leader and
one follower, the strategies of all the players in the Stackelberg game with
multiple followers should be impacted by all the other players’ strategies
(see, for instance, [2, 10, 15]).

As an extension of classical Stackelberg game, the mean-field stochastic
Stackelberg differential game (MF-SSDG), which is described by mean-field
stochastic differential equations (MF-SDEs), has attracted much attention
recently [6, 5, 12, 17, 18, 20, 26, 28]. A significant feature of the game is
that not only the state variable and the controls but also their expectations
are involved in the state equation and objective functions. Such a feature
originates from the mean-field theory, which was developed to study the
collective behaviors resulting from individuals mutual interactions in various
physical and sociological dynamical systems. The current paper focuses on the
study of MF-SSDG with multi-follower.

On the other hand, it is well known that default risk, which is closely
related to default events, naturally appears in financial markets [1, 7, 13].
The standard approach to model default risk is to use the theory of
enlargement of filtration, which has been extensively studied by many authors
(see, for example, [1, 14, 24, 25]). Besides, for practical reasons, one may
require considering default risk in economic models, such as the retailer-
supplier uncooperative replenishment model [30] and supply chain with one
retailer and several suppliers [3]. In [3, 30], both of models were described
by employing the stochastic Stackelberg differential game (SSDG). Moreover, in
[24], the problem of zero-sum stochastic differential game with default risk
was investigated, which can be seen as a utility maximization problem under
the model uncertainty. Thus, it is necessary to consider the default risk in
stochastic differential games. Especially in some real situations, the leader
may stop her action at a ”surprising” time (usually a random time), which
cannot be read or predicted from the reference observation. For such a
”surprising” time, we call it a random exit time which can be regarded as a
default time. Let us illustrate this case with the following example.

######  Example 1.1.

Following [23], we assume that the dynamics of the resource stock $x(t)$ can
be written as

$\displaystyle dx(t)=[rx(t)-h(u(t),v_{1}(t),v_{2}(t))]dt,\quad x(0)=x_{0}>0,$

which does not only depend on the intensity of attacks from two terror
organizations (the followers), but it is also influenced by the counterterror
measures from the government (the leader). Here $x_{0}$ denotes the initial
stock of resources for terrorists; $u(t)$ is the counterterror measures of the
government; $v_{1}(t)$ and $v_{2}(t)$ are the intensity of attacks from
terrorists. Moreover, we assume that along a trajectory the following non-
negativity constraint applies:

$x(t)\geq 0,\quad\mbox{for all $t\geq 0$}.$

Some suitable conditions are imposed on $h(u(t),v_{1}(t),v_{2}(t))$. The
objective function of the government is

$J(u(t),v_{1}(t),v_{2}(t))=\int_{0}^{T}e^{-\rho t}[\varpi
h(u(t),v_{1}(t),v_{2}(t))-cx(t)-k_{1}v_{1}(t)-k_{2}v_{2}(t)-au(t)]dt+e^{-\rho
T}kx(T),$

and the objective function of the terror organization is

$J_{i}(v_{1}(t),v_{2}(t))=\int_{0}^{T}e^{-\rho_{i}t}[\sigma_{i}x(t)+\beta_{i}v_{i}(t)]dt+e^{-\rho_{i}T}c_{i}x(T),$

where $\rho,\rho_{i},c,c_{i},k_{i},\sigma_{i},\beta_{i}$ ($i=1,2$) are all
positive constants. First, the government announce her counterterror measures
$u(t)$. Then, a Nash game is considered for the two terror organizations,
i.e., they would like to maximize their objective functions by

$\begin{cases}J_{1}(v^{*}_{1}(t),v^{*}_{2}(t))=\sup\limits_{v_{1}(t)\geq
0}J_{1}(v_{1}(t),v^{*}_{2}(t)),\\\
J_{2}(v^{*}_{1}(t),v^{*}_{2}(t))=\sup\limits_{v_{2}(t)\geq
0}J_{2}(v^{*}_{1}(t),v_{2}(t)).\end{cases}$

Finally, considering that the terror organizations would take strategy
$(v^{*}_{1},v^{*}_{2})$, the government would like to maximize her objective
function such that

$J(u^{*}(t),v^{*}_{1}(t),v^{*}_{2}(t))=\sup_{u(t)\geq
0}J(u(t),v^{*}_{1}(t),v^{*}_{2}(t)).$

Obviously, this is a problem of the Stackelberg game with one leader and two
followers. Nevertheless, when an emergency happens and is made a top priority,
the government has to stop her counterterror measures at a random time $\tau$
and deal with the emergency. In this case the objective function of the
government becomes

$J(u(t),v_{1}(t),v_{2}(t))=\int_{0}^{\tau}e^{-\rho t}[\varpi
h(u(t),v_{1}(t),v_{2}(t))-cx(t)-k_{1}v_{1}(t)-k_{2}v_{2}(t)-au(t)]dt+e^{-\rho\tau}ky(\tau),$

where $y(\tau)$ represents the discounted value of $x(\tau)$, i.e., the
government suffers a loss at time $\tau\in[0,T]$. As a result, the Stackelberg
game with one leader and two followers mentioned above is nothing but a form
MF-SSDG with one leader and two followers in which the leader is allowed to
exit at a random time (for details, see Problem 2.1 in Section 2).

Example 1.1 tells us that, in some practical situations, it is necessary and
important to consider the problem of MF-SSDG with random exit time. The
purpose of this paper is to investigate a new model of MF-SSDG with random
exit time, which has never been yet explored in the previous literature. The
main features of this paper can be summarized as follows: (i) Based on the
technique of progressive enlargement of filtration, we construct a new model
of a linear-quadratic mean-field stochastic Stackelberg differential game (LQ-
MF-SSDG) with one leader and two followers, in which the leader is allowed to
stop her strategy at a random time; (ii) The state process in the model is
governed by an MF-SDE divided into two-stage subequations by random exit time,
in which the coefficients of subequations are allowed to be different for
describing practical problems; (iii) By employing the backward induction
method, the Stackelberg solution is obtained for LQ-MF-SSDG with random exit
time, even though random coefficient is involved in the mean-field term.

The rest of this paper is structured as follows. In the next section, we give
the formulation of LQ-MF-SSDG with random exit time. In Section 3, we derive
the Stackelberg solution for the leader and followers of LQ-MF-SSDG with
random exit time. Finally, we make some concluding remarks in Section 4.

##  2 Problem Formulation

Consider the linear MF-SDE system in the complete probability space
$(\Omega,\mathfrak{F},\mathfrak{F}_{t},\mathbb{P})$ satisfying the usual
hypothesis as follows:

$\begin{cases}dX(t)=\left[a(t)X(t)+\overline{a}(t)\overline{X}(t)+b_{1}(t)v_{1}(t)+b_{2}(t)v_{2}(t)+b(t)v_{0}(t)\right]dt\\\
\qquad\qquad+\left[c(t)X(t)+\overline{c}(t)\overline{X}(t)\right]dB_{t},\quad
t\in[0,T],\\\ X(0)=x_{0},\end{cases}$ (1)

where $T>0$ is a finite time duration; $\tau$ is a non-negative random
variable, which represents the possible exit time of the leader; $X(t)$ is the
state process; $v_{0}(t)$ is the the control process of the leader; $v_{1}(t)$
and $v_{2}(t)$ are the control processes of the followers $\mathcal{P}_{1}$
and $\mathcal{P}_{2}$, respectively; $\overline{X}(t)$ is the mean-field term
with

$\overline{X}(t)=\mathbb{E}\left[X(t)\Big{|}\mathfrak{F}_{0}\right]\mathbb{I}_{t\in[0,\tau)}+\mathbb{E}\left[X(t)\Big{|}\mathfrak{F}_{\tau}\right]\mathbb{I}_{t\in[\tau,T]};$

the $\sigma$-algebra $\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0}$, generated by
the standard one-dimensional Brownian motion $B_{t}$, is right-continuous and
increasing; $x_{0}$ is an $\mathcal{F}_{0}$-measurable and square integrable
random variable; $\mathfrak{F}$ is the smallest right continuous extension in
which $\tau_{0}$ becomes an $\mathfrak{F}$-stopping time, i.e.,
$\mathfrak{F}_{s}=\mathcal{F}_{s}\vee\sigma\left(\mathbb{I}_{\tau_{0}\leq
u},u\in[0,s]\right)$, for all $s>0$.

Clearly, $\tau=\tau_{0}\wedge T\in[0,T]$ is also an $\mathfrak{F}$-stopping
time. Noticing that when $\tau_{0}=\tau$ (i.e., the leader stops her action
before the game ends), the controller $v_{0}$ contributes nothing on the time
interval $(\tau,T]$. Thus, it is natural to require that
$b(t)=b_{0}(t)\mathbb{I}_{t\in[0,\tau]}$.

For convenience, we set $i=1,2$, $j=0,1,2$ and
$\mathcal{L}_{\mathfrak{F}}^{2}(0,T;\mathbb{R})$ the space of all
$\mathfrak{F}$-adapted and square integrable processes throughout this paper.
Now we introduce the following objective functions.

######  Definition 2.1.

The objective functions of the follower $\mathcal{P}_{i}$ and the leader are
given by

$J_{i}(v_{1},v_{2},v_{0})=-\frac{1}{2}\mathbb{E}\left[\int_{0}^{T}p_{i}(t)X^{2}(t)+\overline{p}_{i}(t)\overline{X}^{2}(t)+q_{i}(t)v_{i}^{2}(t)dt+r_{i}(T)X^{2}(T)\Big{|}\mathfrak{F}_{0}\right],$
(2)

and

$J_{0}(v_{1},v_{2},v_{0})=-\frac{1}{2}\mathbb{E}\left[\int_{0}^{\tau}p_{0}(t)Y^{2}(t)+\overline{p}_{0}(t)\overline{Y}^{2}(t)+q_{0}(t)v_{0}^{2}(t)dt+r_{0}(\tau)Y^{2}(\tau)\Big{|}\mathfrak{F}_{0}\right],$
(3)

respectively. Here $Y(t)$ is the solution to the following MF-SDE with single
jump at $t=\tau$.

$\begin{cases}dY(t)=\left[a(t)Y(t)+\overline{a}(t)\overline{Y}(t)+b_{1}(t)v_{1}(t)+b_{2}(t)v_{2}(t)+b_{0}(t)v_{0}(t)\right]dt\\\
\qquad\qquad+\left[c(t)Y(t)+\overline{c}(t)\overline{Y}(t)\right]dB_{t}+\left[d_{0}(t)Y(t)+\overline{d}_{0}(t)\overline{Y}(t)\right]d\mathbb{I}_{\tau\leq
t},\quad t\in(0,\tau],\\\ Y(0)=x_{0}.\end{cases}$ (4)

######  Remark 2.1.

It is easy to show that $Y(t)=X(t)$ a.s. for $t\in[0,\tau)$, while there is a
penalty at time $\tau$ for the terminal gain of the leader.

Next we introduce the admissible control sets and the optimal strategy
problem.

######  Problem 2.1.

Consider the admissible control sets:

$\displaystyle\mathcal{V}[0,T]=$
$\displaystyle\left\\{v:[0,T]\times\Omega\rightarrow\mathbb{R}\Big{|}v(t)\;\mbox{is
$\mathfrak{F}$-progressively measurable
with}\;\mathbb{E}\left[\int_{0}^{T}|v(t)|^{2}dt\Big{|}\mathfrak{F}_{0}\right]<\infty\;\mbox{a.s.}\right\\};$
$\displaystyle\mathcal{V}[0,\tau]=$
$\displaystyle\left\\{v:[0,\tau]\times\Omega\rightarrow\mathbb{R}\Big{|}v(t)\;\mbox{is
$\mathfrak{F}$-progressively measurable
with}\;\mathbb{E}\left[\int_{0}^{\tau}|v(t)|^{2}dt\Big{|}\mathfrak{F}_{0}\right]<\infty\;\mbox{a.s.}\right\\};$
$\displaystyle\mathcal{V}[\tau,T]=$
$\displaystyle\left\\{v:[\tau,T]\times\Omega\rightarrow\mathbb{R}\Big{|}v(t)\;\mbox{is
$\mathfrak{F}$-progressively measurable
with}\;\mathbb{E}\left[\int_{\tau}^{T}|v(t)|^{2}dt\Big{|}\mathfrak{F}_{\tau}\right]<\infty\;\mbox{a.s.}\right\\}.$

The followers would like to find the Nash equilibrium point
$(v^{*}_{1},v^{*}_{2})\in\mathcal{V}[0,T]\times\mathcal{V}[0,T]$ such that

$\begin{cases}J_{1}(v^{*}_{1},v^{*}_{2},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{1}\in\mathcal{V}[0,T]}J_{1}(v_{1},v^{*}_{2},v_{0}),\\\
J_{2}(v^{*}_{1},v^{*}_{2},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{2}\in\mathcal{V}[0,T]}J_{2}(v^{*}_{1},v_{2},v_{0}).\end{cases}$
(5)

Considering that the followers would take strategy $(v^{*}_{1},v^{*}_{2})$,
the leader expects to find $v^{*}_{0}\in\mathcal{V}[0,\tau]$ such that

$J_{0}(v^{*}_{1},v^{*}_{2},v^{*}_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{0}\in\mathcal{V}[0,\tau]}J_{0}(v^{*}_{1},v^{*}_{2},v_{0}).$
(6)

In order to obtain our main results, we make the following assumptions for MF-
SDEs (1), (4), and objective functions (2), (3) throughout this paper.

######  Assumption 2.1.

For $\theta\in\\{a,\overline{a},c,\overline{c}\\}$,
$\vartheta_{i}\in\\{b_{i},p_{i},\overline{p}_{i},q_{i},r_{i}\\}$, suppose that
$\theta$ and $\vartheta_{i}$ have the following decompositions.

$\theta(t)=\theta_{1}(t)\mathbb{I}_{t\in[0,\tau)}+\theta_{2}(t)\mathbb{I}_{t\in[\tau,T]},\quad\vartheta_{i}(t)=\vartheta_{i,1}(t)\mathbb{I}_{t\in[0,\tau)}+\vartheta_{i,2}(t)\mathbb{I}_{t\in[\tau,T]}.$

In addition, suppose that the following conditions hold.

  1. ($A$)

$a_{i},\overline{a}_{i},b_{0},b_{i,1},b_{i,2},c_{i},\overline{c}_{i},d_{0},\overline{d}_{0}$
are all deterministic, uniformly bounded and real-valued functions on $[0,T]$;

  2. ($B$)

$p_{0},\overline{p}_{0},q_{0},r_{0},p_{i,j},\overline{p}_{i,j},q_{i,j},r_{i,j}$
are all deterministic, continuous and positive functions on $[0,T]$;

  3. ($C$)

$q^{-1}_{1,i}b_{1,i}^{2}=q^{-1}_{2,i}b_{2,i}^{2}=l_{i}$;

  4. ($D$)

(Hypothesis $(\mathcal{H})$) Every càdlàg $\mathcal{F}$-martingale remains an
$\mathfrak{F}$-martingale;

  5. ($E$)

Suppose that there exists an $\mathfrak{F}$-predictable (respectively,
$\mathcal{F}$-predictable) process $\gamma^{\mathfrak{F}}$ (respectively,
$\gamma^{\mathcal{F}}$) with
$\gamma^{\mathfrak{F}}(s)=\mathbb{I}_{s<\tau}\gamma^{\mathcal{F}}(s)$ such
that

$A(t)=\mathbb{I}_{\tau\leq
t}-\int_{0}^{t}\gamma^{\mathfrak{F}}(s)ds=\mathbb{I}_{\tau\leq
t}-\int_{0}^{t\wedge\tau}\gamma^{\mathcal{F}}(s)ds\quad(t\in[0,T])$ (7)

is an $\mathfrak{F}$-martingale with jump time $\tau$. The process
$\gamma^{\mathfrak{F}}$ (respectively, $\gamma^{\mathcal{F}}$) is called the
$\mathfrak{F}$-intensity (respectively, $\mathcal{F}$-intensity) of $\tau$. In
addition, we suppose that $\gamma^{\mathfrak{F}}$ is upper bounded.

######  Remark 2.2.

Concerning assumptions $(A)$-$(E)$, some remarks are listed as follows: (i)
$(A)$ implies that for any
$(v_{0},v_{1},v_{2})\in\mathcal{V}[0,\tau]\times\mathcal{V}[0,T]\times\mathcal{V}[0,T]$,
(1) admits a unique solution
$X(t)\in\mathcal{L}_{\mathfrak{F}}^{2}(0,T;\mathbb{R})$; (ii) $(B)$ ensures
the concavity of the objective functionals; (iii) $(C)$ ensures the well-
posedness of related Riccati equations for followers; (iv) $(D)$ and $(E)$ are
classical assumptions in the theory of progressive enlargement.

##  3 Main results

In this section, we focus on searching the Stackelberg solution of LQ-MF-SSDG
with random exit time, which is closely related to the optimal control for
SDEs with default (with respect to the study of SDEs with default, we refer
the reader to [4, 16, 24]). Motivated by the backward induction method (see,
for example, [11, 25]), we first decompose the state equation into the
following two-stage equations on $t\in[0,\tau]$ (the first stage) and
$t\in[\tau,T]$ (the second stage):

$\begin{cases}dX_{1}(t)=\left[a_{1}(t)X_{1}(t)+\overline{a}_{1}(t)\overline{X}_{1}(t)+b_{1,1}(t)v_{1,1}(t)+b_{2,1}(t)v_{2,1}(t)+b_{0}(t)v_{0}(t)\right]dt\\\
\qquad\qquad\;+\left[c_{1}(t)X_{1}(t)+\overline{c}_{1}(t)\overline{X}_{1}(t)\right]dB_{t},\quad
t\in(0,\tau],\\\
dX_{2}(t)=\left[a_{2}(t)X_{2}(t)+\overline{a}_{2}(t)\overline{X}_{2}(t)+b_{1,2}(t)v_{1,2}(t)+b_{2,2}(t)v_{2,2}(t)\right]dt\\\
\qquad\qquad\;+\left[c_{2}(t)X_{2}(t)+\overline{c}_{2}(t)\overline{X}_{2}(t)\right]dB_{t},\quad
t\in(\tau,T],\\\ X_{1}(0)=x_{0},\quad X_{2}(\tau)=X_{1}(\tau).\end{cases}$

We note that Lemma 2.1 of [25] ensures the rationality of the above
decomposition. Further, we decompose the controls of the followers as follows:

$v_{i}(t)=v_{i,1}(t)\mathbb{I}_{t\in[0,\tau)}+v_{i,2}(t)\mathbb{I}_{t\in[\tau,T]}.$

Next, for fixed $v_{0}\in\mathcal{V}[0,\tau]$, we aim to find
$(v^{*}_{1,2},v^{*}_{2,2})\in\mathcal{V}[\tau,T]\times\mathcal{V}[\tau,T]$
such that

$\begin{cases}\quad\;J_{1,2}(v^{*}_{1,2},v^{*}_{2,2})=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,2}\in\mathcal{V}[\tau,T]}J_{1,2}(v_{1,2},v^{*}_{2,2})\\\
=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,2}\in\mathcal{V}[\tau,T]}\Big{\\{}-\frac{1}{2}\mathbb{E}\left[\int_{\tau}^{T}p_{1,2}(t)X_{2}^{2}(t)+\overline{p}_{1,2}(t)\overline{X}_{2}^{2}(t)+q_{1,2}(t)v_{1,2}^{2}(t)dt+r_{1}(T)X_{2}^{2}(T)\Big{|}\mathfrak{F}_{\tau}\right]\Big{\\}},\\\
\quad\;J_{2,2}(v^{*}_{1,2},v^{*}_{2,2})=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,2}\in\mathcal{V}[\tau,T]}J_{2,2}(v^{*}_{1,2},v_{2,2})\\\
=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,2}\in\mathcal{V}[\tau,T]}\Big{\\{}-\frac{1}{2}\mathbb{E}\left[\int_{\tau}^{T}p_{2,2}(t)X_{2}^{2}(t)+\overline{p}_{2,2}(t)\overline{X}_{2}^{2}(t)+q_{2,2}(t)v_{2,2}^{2}(t)dt+r_{2}(T)X_{2}^{2}(T)\Big{|}\mathfrak{F}_{\tau}\right]\Big{\\}},\end{cases}$
(8)

and $(v^{*}_{1,1},v^{*}_{2,1})\in\mathcal{V}[0,\tau]\times\mathcal{V}[0,\tau]$
such that

$\begin{cases}\quad\;J_{1,1}(v^{*}_{1,1},v^{*}_{2,1},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,1}\in\mathcal{V}[0,\tau]}J_{1,1}(v_{1,1},v^{*}_{2,1},v_{0})\\\
=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,1}\in\mathcal{V}[0,\tau]}\mathbb{E}\Big{\\{}\int_{0}^{\tau}-\frac{1}{2}\left[p_{1,1}(t)X_{1}^{2}(t)+\overline{p}_{1,1}(t)\overline{X}_{2}^{2}(t)+q_{1,1}(t)v_{1,1}^{2}(t)\right]dt+J_{1,2}(X_{2}^{*}(t),v^{*}_{1,2}(t))\Big{|}\mathfrak{F}_{0}\Big{\\}},\\\
\quad\;J_{2,1}(v^{*}_{1,1},v^{*}_{2,1},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,1}\in\mathcal{V}[0,\tau]}J_{2,1}(v^{*}_{1,1},v_{2,1},v_{0})\\\
=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,1}\in\mathcal{V}[0,\tau]}\mathbb{E}\Big{\\{}\int_{0}^{\tau}-\frac{1}{2}\left[p_{2,1}(t)X_{1}^{2}(t)+\overline{p}_{2,1}(t)\overline{X}_{2}^{2}(t)+q_{2,1}(t)v_{2,1}^{2}(t)\right]dt+J_{2,2}(X_{2}^{*}(t),v^{*}_{2,2}(t))\Big{|}\mathfrak{F}_{0}\Big{\\}}.\end{cases}$
(9)

Let

$\widetilde{v}_{i}(t)=v^{*}_{i,1}(t)\mathbb{I}_{t\in[0,\tau)}+v^{*}_{i,2}(t)\mathbb{I}_{t\in[\tau,T]}.$

According to (5) and Theorem 4.1 in [11], we know that
$(\widetilde{v}_{1},\widetilde{v}_{2})\in\mathcal{V}[0,T]\times\mathcal{V}[0,T]$
is the global Nash equilibrium point, i.e.,
$(\widetilde{v}_{1},\widetilde{v}_{2})=(v_{1}^{*},v_{2}^{*})$. Moreover,

$J_{i}(v^{*}_{1},v^{*}_{2},v_{0})=J_{i,1}(v^{*}_{1,1},v^{*}_{2,1},v_{0}).$

Assume that followers take strategy $(\widetilde{v}_{1},\widetilde{v}_{2})$,
we proceed to find the optimal strategy $v_{0}^{*}$ for the leader. In the
following subsections, we solve $(v^{*}_{1,2},v^{*}_{2,2})$,
$(v^{*}_{1,1},v^{*}_{2,1})$ and $v_{0}^{*}$ in the sequence.

###  3.1 Nash equilibrium for followers at the second stage

In this subsection, we apply the Pontryagin-type maximum principle to find
Nash equilibrium point $(v^{*}_{1,2},v^{*}_{2,2})$ for followers at the second
stage. We restate problem (8) as follows.

######  Subroblem 3.1.

Find
$(v^{*}_{1,2},v^{*}_{2,2})\in\mathcal{V}[\tau,T]\times\mathcal{V}[\tau,T]$
such that

$\displaystyle J_{1,2}(v^{*}_{1,2},v^{*}_{2,2})$
$\displaystyle=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,2}\in\mathcal{V}[\tau,T]}J_{1,2}(v_{1,2},v^{*}_{2,2}),$
$\displaystyle J_{2,2}(v^{*}_{1,2},v^{*}_{2,2})$
$\displaystyle=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,2}\in\mathcal{V}[\tau,T]}J_{2,2}(v^{*}_{1,2},v_{2,2}).$

To solve this subproblem, we define the Hamiltonian function of follower
$\mathcal{P}_{i}$ at second stage by

$\displaystyle H_{i,2}$
$\displaystyle:[\tau,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R},$
$\displaystyle H_{i,2}$
$\displaystyle=H_{i,2}(t,X_{2},\overline{X}_{2},v_{1,2},v_{2,2},P_{v_{i,2}},Q_{v_{i,2}})$
$\displaystyle=-\frac{1}{2}\left[p_{i,2}X_{2}^{2}+\overline{p}_{i,2}\overline{X}_{2}^{2}+q_{i,2}v_{i,2}^{2}\right]+\left[a_{2}X_{2}+\overline{a}_{2}\overline{X}_{2}+b_{1,2}v_{1,2}+b_{2,2}v_{2,2}\right]P_{v_{i,2}}$
$\displaystyle\quad\mbox{}\,+\left[c_{2}X_{2}+\overline{c}_{2}\overline{X}_{2}\right]Q_{v_{i,2}}.$

By the sufficient and necessary maximum principle of MF-SDEs with default (see
[11]), we have

$0=\frac{\partial H_{i,2}}{\partial
v_{i,2}}\Big{|}_{v_{i,2}=v^{*}_{i,2}}=-q_{i,2}v^{*}_{i,2}+b_{i,2}P_{v^{*}_{i,2}}\quad\Rightarrow\quad
v^{*}_{i,2}=q^{-1}_{i,2}b_{i,2}P_{v^{*}_{i,2}},$

where $P_{v^{*}_{i,2}}$ is the solution to corresponding linear BSDE. This
leads to the following theorem.

######  Theorem 3.1.

$(v^{*}_{1,2},v^{*}_{2,2})$ is a Nash equilibrium point for Subproblem 3.1 if
and only if

$(v^{*}_{1,2},v^{*}_{2,2})=(q^{-1}_{1,2}b_{1,2}P_{v^{*}_{1,2}},q^{-1}_{2,2}b_{2,2}P_{v^{*}_{2,2}}),$
(10)

where the triple $(X_{2}^{*},P_{v^{*}_{i,2}},Q_{v^{*}_{i,2}})$ satisfies the
following mean-field forward-backward stochastic differential equation (MF-
FBSDE):

$\begin{cases}dX^{*}_{2}=\left[a_{2}X^{*}_{2}+\overline{a}_{2}\overline{X}^{*}_{2}+q^{-1}_{1,2}b_{1,2}^{2}P_{v^{*}_{1,2}}+q^{-1}_{2,2}b_{2,2}^{2}P_{v^{*}_{2,2}}\right]dt+\left[c_{2}X^{*}_{2}+\overline{c}_{2}\overline{X}^{*}_{2}\right]dB_{t},\quad
t\in(\tau,T],\\\
dP_{v^{*}_{i,2}}=\left[p_{i,2}X^{*}_{2}+\overline{p}_{i,2}\overline{X}^{*}_{2}-a_{2}P_{v^{*}_{i,2}}-\overline{a}_{2}\overline{P}_{v^{*}_{i,2}}-c_{2}Q_{v^{*}_{i,2}}-\overline{c}_{2}\overline{Q}_{v^{*}_{i,2}}\right]dt+Q_{v^{*}_{i,2}}dB_{t},\quad
t\in[\tau,T),\\\ X^{*}_{2}(\tau)=X_{1}(\tau),\quad
P_{v^{*}_{i,2}}(T)=-r_{i}(T)X^{*}_{2}(T).\end{cases}$ (11)

According to (9), before solving the Nash equilibrium point for followers at
the first stage, we need to compute the optimal objective subfunctions
$J_{1,2}(X_{2}^{*}(t),v^{*}_{1,2}(t))$ and
$J_{2,2}(X_{2}^{*}(t),v^{*}_{2,2}(t))$. First, we focus on the feedback
representation of the solutions to (11).

Observing the terminal condition of $P_{v^{*}_{i,2}}$, we set

$P_{v^{*}_{i,2}}(t)=\varphi_{i,1}(t)X^{*}_{2}(t)+\varphi_{i,2}(t)\overline{X}^{*}_{2}(t)+\varphi_{i,0}(t),$
(12)

where $\varphi_{i,1}$, $\varphi_{i,2}$ are deterministic functions,
$\varphi_{i,0}$ is an $\mathfrak{F}$-adapted process satisfying

$d\varphi_{i,0}(t)=\varphi_{i,3}(t)dt+\varphi_{i,4}(t)dB_{t},\quad\varphi_{0}(T)=0.$

Taking conditional expectation with respect to $\mathfrak{F}_{\tau}$ in (11),
one has

$d\overline{X}^{*}_{2}=\left[(a_{2}+\overline{a}_{2})\overline{X}^{*}_{2}+q^{-1}_{1,2}b_{1,2}^{2}\overline{P}_{v^{*}_{1,2}}+q^{-1}_{2,2}b_{2,2}^{2}\overline{P}_{v^{*}_{2,2}}\right]dt.$

Computing the differential of $P_{v^{*}_{i,2}}$, we have

$\displaystyle dP_{v^{*}_{i,2}}$
$\displaystyle=\varphi^{\prime}_{i,1}X^{*}_{2}dt+\varphi_{i,1}dX^{*}_{2}+\varphi^{\prime}_{i,2}\overline{X}^{*}_{2}dt+\varphi_{i,2}d\overline{X}^{*}_{2}+\varphi^{\prime}_{i,0}dt$
$\displaystyle=\left[\varphi^{\prime}_{i,1}+\varphi_{i,1}\left(a_{2}+q^{-1}_{1,2}b_{1,2}^{2}\varphi_{1,1}+q^{-1}_{2,2}b_{2,2}^{2}\varphi_{2,1}\right)\right]X^{*}_{2}dt$
$\displaystyle\quad\mbox{}+\Big{\\{}\varphi^{\prime}_{i,2}+\varphi_{i,1}\Big{[}\overline{a}_{2}+q^{-1}_{1,2}b_{1,2}^{2}\varphi_{1,2}+q^{-1}_{2,2}b_{2,2}^{2}\varphi_{2,2}\Big{]}$
$\displaystyle\quad\mbox{}+\varphi_{i,2}\Big{[}a_{2}+\overline{a}_{2}+q^{-1}_{1,2}b_{1,2}^{2}(\varphi_{1,1}+\varphi_{1,2})+q^{-1}_{2,2}b_{2,2}^{2}(\varphi_{2,1}+\varphi_{2,2})\Big{]}\Big{\\}}\overline{X}^{*}_{2}dt$
$\displaystyle\quad\mbox{}+\Big{[}\varphi_{i,3}+\varphi_{i,1}\left(q^{-1}_{1,2}b_{1,2}^{2}\varphi_{1,0}+q^{-1}_{2,2}b_{2,2}^{2}\varphi_{2,0}\right)+\varphi_{i,2}\left(q^{-1}_{1,2}b_{1,2}^{2}\overline{\varphi}_{1,0}+q^{-1}_{2,2}b_{2,2}^{2}\overline{\varphi}_{2,0}\right)\Big{]}dt$
$\displaystyle\quad\mbox{}+\left[\varphi_{i,1}\left(c_{2}X^{*}_{2}+\overline{c}_{2}\overline{X}^{*}_{2}\right)+\varphi_{i,4}\right]dB_{t}.$
(13)

It follows from (3.1) and (11) that

$Q_{v^{*}_{i,2}}=\varphi_{i,1}\left(c_{2}X^{*}_{2}+\overline{c}_{2}\overline{X}^{*}_{2}\right)+\varphi_{i,4}.$

Thus the second equation in (11) can be rewritten as

$\displaystyle dP_{v^{*}_{i,2}}$
$\displaystyle=\left[p_{i,2}-a_{2}\varphi_{i,1}-\varphi_{i,1}c_{2}^{2}\right]X^{*}_{2}dt+\Big{[}\overline{p}_{i,2}-a_{2}\varphi_{i,2}-\bar{a}_{2}\left(\varphi_{i,1}+\varphi_{i,2}\right)-\varphi_{i,1}\left(\overline{c}_{2}^{2}+2c_{2}\overline{c}_{2}\right)\Big{]}\overline{X}_{2}^{*}dt$
$\displaystyle\quad\mbox{}\;-\left(a_{2}\varphi_{i,0}+\overline{a}_{2}\overline{\varphi}_{i,0}+c_{2}\varphi_{i,4}+\overline{c}_{2}\overline{\varphi}_{i,4}\right)dt+\left[\varphi_{i,1}\left(c_{2}X^{*}_{2}+\overline{c}_{2}\overline{X}^{*}_{2}\right)+\varphi_{i,4}\right]dB_{t}.$
(14)

Combining (3.1), (3.1) and Assumption $(C)$, we can see that
$(\varphi_{1,1},\varphi_{1,2},\varphi_{2,1},\varphi_{2,2})$ is a solution to
the following system of ordinary differential equations (ODEs):

$\begin{cases}\varphi^{\prime}_{1,1}+(2a_{2}+c^{2}_{2})\varphi_{1,1}+l_{2}\varphi^{2}_{1,1}+l_{2}\varphi_{1,1}\varphi_{2,1}-p_{1,2}=0,\quad
t\in[\tau,T),\\\
\varphi^{\prime}_{2,1}+(2a_{2}+c^{2}_{2})\varphi_{2,1}+l_{2}\varphi_{1,1}\varphi_{2,1}+l_{2}\varphi^{2}_{2,1}-p_{2,2}=0,\quad
t\in[\tau,T),\\\
\varphi^{\prime}_{1,2}+\left(2a_{2}+2\overline{a}_{2}+2l_{2}\varphi_{1,1}+l_{2}\varphi_{2,1}\right)\varphi_{1,2}+l_{2}\varphi^{2}_{1,2}+l_{2}\varphi_{1,2}\varphi_{2,2}\\\
\quad\;\;\,\mbox{}+l_{2}\varphi_{1,1}\varphi_{2,2}+\varphi_{1,1}\Big{[}2\overline{a}_{2}+\overline{c}_{2}^{2}+2c_{2}\overline{c}_{2}\Big{]}-\overline{p}_{1,2}=0,\quad
t\in[\tau,T),\\\
\varphi^{\prime}_{2,2}+\left(2a_{2}+2\overline{a}_{2}+2l_{2}\varphi_{2,1}+l_{2}\varphi_{1,1}\right)\varphi_{2,2}+l_{2}\varphi^{2}_{2,2}+l_{2}\varphi_{1,2}\varphi_{2,2}\\\
\quad\;\;\,\mbox{}+l_{2}\varphi_{2,1}\varphi_{1,2}+\varphi_{2,1}\Big{[}2\overline{a}_{2}+\overline{c}_{2}^{2}+2c_{2}\overline{c}_{2}\Big{]}-\overline{p}_{2,2}=0,\quad
t\in[\tau,T),\\\
\varphi_{1,1}(T)=-r_{1}(T),\quad\varphi_{1,2}(T)=0,\quad\varphi_{2,1}(T)=-r_{2}(T),\quad\varphi_{2,2}(T)=0.\end{cases}$
(15)

Moreover, $((\varphi_{1,0},\varphi_{1,4}),(\varphi_{2,0},\varphi_{2,4}))$ is a
solution to the following system of mean-field backward stochastic
differential equations (MF-BSDEs):

$\begin{cases}d\varphi_{1,0}=-\Big{[}\left(a_{2}+l_{2}\varphi_{1,1}\right)\varphi_{1,0}+\left(\overline{a}_{2}+l_{2}\varphi_{1,2}\right)\overline{\varphi}_{1,0}+c_{2}\varphi_{1,4}+\overline{c}_{2}\overline{\varphi}_{1,4}\\\
\qquad\quad\,\,\mbox{}+l_{2}\left(\varphi_{1,1}\varphi_{2,0}+\varphi_{1,2}\overline{\varphi}_{2,0}\right)\Big{]}dt+\varphi_{1,4}dB_{t},\quad
t\in[\tau,T),\\\
d\varphi_{2,0}=-\Big{[}\left(a_{2}+l_{2}\varphi_{2,1}\right)\varphi_{2,0}+\left(\overline{a}_{2}+l_{2}\varphi_{2,2}\right)\overline{\varphi}_{2,0}+c_{2}\varphi_{2,4}+\overline{c}_{2}\overline{\varphi}_{2,4}\\\
\qquad\quad\,\,\mbox{}+l_{2}\left(\varphi_{2,1}\varphi_{1,0}+\varphi_{2,2}\overline{\varphi}_{1,0}\right)\Big{]}dt+\varphi_{2,4}dB_{t},\quad
t\in[\tau,T),\\\ \varphi_{1,0}(T)=\varphi_{2,0}(T)=0.\end{cases}$ (16)

Now we show the existence and uniqueness of solutions to (15) and (16),
respectively.

######  Lemma 3.1.

There exists a unique solution
$(\varphi_{1,1},\varphi_{1,2},\varphi_{2,1},\varphi_{2,2})$ to (15).

###### Proof.

Set $\varphi_{j}=\varphi_{1,j}+\varphi_{2,j}$. Then, $\varphi_{1}$ satisfies
the following Riccati equation:

$\begin{cases}\varphi^{\prime}_{1}+(2a_{2}+c^{2}_{2})\varphi_{1}+l_{2}\varphi^{2}_{1}-(p_{1,2}+p_{2,2})=0,\quad
t\in[\tau,T),\\\ \varphi_{1}(T)=-r_{1}(T)-r_{2}(T).\end{cases}$ (17)

By Proposition 7.1 in [31], there exists a unique solution to (17). Therefore,
(15) can be transformed into the following system of linear equations:

$\begin{cases}\varphi^{\prime}_{1,1}+(2a_{2}+c^{2}_{2})\varphi_{1,1}+l_{2}\varphi_{1,1}\varphi_{1}-p_{1,2}=0,\quad
t\in[\tau,T),\\\
\varphi^{\prime}_{2,1}+(2a_{2}+c^{2}_{2})\varphi_{2,1}+l_{2}\varphi_{2,1}\varphi_{1}-p_{2,2}=0,\quad
t\in[\tau,T),\\\
\varphi_{1,1}(T)=-r_{1}(T),\quad\varphi_{2,1}(T)=-r_{2}(T),\end{cases}$

where $\varphi_{1}$ is the unique solution to (17). The existence and
uniqueness of solutions $\varphi_{1,1}$ and $\varphi_{2,1}$ is obtained
immediately.

Similarly, $\varphi_{2}$ satisfies the following Riccati equation:

$\begin{cases}\varphi^{\prime}_{2}+\left(2a_{2}+2\overline{a}_{2}+2l_{2}\varphi_{1}\right)\varphi_{2}+l_{2}\varphi^{2}_{2}+\varphi_{1}\Big{[}2\overline{a}_{2}+\overline{c}_{2}^{2}+2c_{2}\overline{c}_{2}\Big{]}-(\overline{p}_{1,2}+\overline{p}_{2,2})=0,\quad
t\in[\tau,T),\\\ \varphi_{2}(T)=0.\end{cases}$

Repeating the above arguments, we can obtain the existence and uniqueness of
solutions $\varphi_{2,1}$ and $\varphi_{2,2}$. ∎

######  Lemma 3.2.

There exists a unique $\mathfrak{F}_{t}$-adapted solution
$((\varphi_{1,0},\varphi_{1,4}),(\varphi_{2,0},\varphi_{2,4}))$ to (16).

###### Proof.

Set

$\displaystyle\alpha_{2}=\begin{bmatrix}\varphi_{1,0}\\\
\varphi_{2,0}\end{bmatrix},\quad\beta_{2}=\begin{bmatrix}\varphi_{1,4}\\\
\varphi_{2,4}\end{bmatrix},\quad\mu_{2}=\begin{bmatrix}a_{2}+l_{2}\varphi_{1,1}&l_{2}\varphi_{1,1}\\\
l_{2}\varphi_{2,1}&a_{2}+l_{2}\varphi_{2,1}\end{bmatrix},$
$\displaystyle\overline{\alpha}_{2}=\begin{bmatrix}\overline{\varphi}_{1,0}\\\
\overline{\varphi}_{2,0}\end{bmatrix},\quad\overline{\beta}_{2}=\begin{bmatrix}\overline{\varphi}_{1,4}\\\
\overline{\varphi}_{2,4}\end{bmatrix},\quad\nu_{2}=\begin{bmatrix}\overline{a}_{2}+l_{2}\varphi_{1,2}&l_{2}\varphi_{1,2}\\\
l_{2}\varphi_{2,2}&\overline{a}_{2}+l_{2}\varphi_{2,2}\end{bmatrix}.$

Then (16) can be rewritten as

$\begin{cases}d\alpha_{2}=-\left(\mu_{2}\alpha_{2}+\nu_{2}\overline{\alpha}_{2}+c_{2}\beta+\overline{c}_{2}\overline{\beta}_{2}\right)+\beta_{2}dB_{t},\quad
t\in[\tau,T),\\\ \alpha_{2}(T)=0,\end{cases}$

which is a linear 2-dimensional BSDE. Thus the result is obtained directly. ∎

Next, we aim to solve the optimal state $X^{*}_{2}$ at the second stage.
Combining (11) and (12) derives the following MF-SDE:

$\begin{cases}dX^{*}_{2}=\left[(a_{2}+l_{2}\varphi_{1})X^{*}_{2}+(\overline{a}_{2}+l_{2}\varphi_{2})\overline{X}^{*}_{2}+l_{2}\varphi_{0}\right]dt+\left[c_{2}X^{*}_{2}+\overline{c}_{2}\overline{X}^{*}_{2}\right]dB_{t},\quad
t\in[\tau,T),\\\ X^{*}_{2}(\tau)=X_{1}(\tau).\end{cases}$ (18)

Observing that (18) is linear, we set

$X^{*}_{2}(t)=M(t)X^{*}_{2}(\tau)+N(t).$ (19)

Substituting (19) into (18) yields

$\displaystyle X^{*}_{2}(t)$
$\displaystyle=X^{*}_{2}(\tau)+X^{*}_{2}(\tau)\int_{\tau}^{t}(a_{2}(s)+l_{2}(s)\varphi_{1}(s))M(s)+(\overline{a}_{2}(s)+l_{2}(s)\varphi_{2}(s))\overline{M}(s)ds$
$\displaystyle\quad\mbox{}+\int_{\tau}^{t}(a_{2}(s)+l_{2}(s)\varphi_{1}(s))N(s)+(\overline{a}_{2}(s)+l_{2}(s)\varphi_{2}(s))\overline{N}(s)+l_{2}(s)\varphi_{0}(s)ds$
$\displaystyle\quad\mbox{}+X^{*}_{2}(\tau)\int_{\tau}^{t}c_{2}(s)M(s)+\overline{c}_{2}(s)\overline{M}(s)dB_{s}+\int_{\tau}^{t}c_{2}(s)N(s)+\overline{c}_{2}(s)\overline{N}(s)dB_{s}.$

Thus, $M(t)$ and $N(t)$ are solutions to the following system of linear MF-
SDEs:

$\begin{cases}dM=\left[(a_{2}+l_{2}\varphi_{1})M+(\overline{a}_{2}+l_{2}\varphi_{2})\overline{M}\right]dt+\left[c_{2}M+\overline{c}_{2}\overline{M}\right]dB_{t},\quad
t\in(\tau,T],\\\
dN=\left[(a_{2}+l_{2}\varphi_{1})N+(\overline{a}_{2}+l_{2}\varphi_{2})\overline{N}+l_{2}\varphi_{0}\right]dt+\left[c_{2}N+\overline{c}_{2}\overline{N}\right]dB_{t},\quad
t\in(\tau,T],\\\ M(\tau)=1,\quad N(\tau)=0.\end{cases}$ (20)

We also need the following lemma.

######  Lemma 3.3.

At $(v_{1,2},v_{2,2})=(v^{*}_{1,2},v^{*}_{2,2})$, the value of the objective
subfunctions for followers is

$J_{i,2}(v^{*}_{1,2},v^{*}_{2,2})=\frac{1}{2}\left[\left((\varphi_{i,1}+\varphi_{i,2})X^{*2}_{2}+\varphi_{i,0}X^{*}_{2}\right)(\tau)+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}(t)P_{v^{*}_{1,2}}(t)P_{v^{*}_{2,2}}(t)dt\Big{|}\mathfrak{F}_{\tau}\right].$
(21)

###### Proof.

Applying Itô’s formula to $X^{*}_{2}P_{v^{*}_{i,2}}$ and taking conditional
expectation with respect to $\mathfrak{F}_{\tau}$, one has

$\displaystyle\quad\;\mathbb{E}\left[X^{*}_{2}(T)P_{v^{*}_{1,2}}(T)-X^{*}_{2}(\tau)P_{v^{*}_{i,2}}(\tau)\Big{|}\mathfrak{F}_{\tau}\right]$
$\displaystyle=-\mathbb{E}\left[r_{i}(T)X^{*2}_{2}(T)\Big{|}\mathfrak{F}_{\tau}\right]-X^{*}_{2}(\tau)P_{v^{*}_{i,2}}(\tau)$
$\displaystyle=\mathbb{E}\Big{[}\int_{\tau}^{T}\left[q^{-1}_{1,2}b_{1,2}^{2}P_{v^{*}_{1,2}}P_{v^{*}_{i,2}}+q^{-1}_{2,2}b_{2,2}^{2}P_{v^{*}_{2,2}}P_{v^{*}_{i,2}}+P_{v^{*}_{1,2}}X^{*2}_{2}+\overline{P}_{v^{*}_{1,2}}\overline{X}^{*}_{2}{X}^{*}_{2}\right](t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]}.$

For follower $\mathcal{P}_{1}$, it follows from the fact
$q^{-1}_{1,2}b_{1,2}^{2}P_{v^{*}_{1,2}}P_{v^{*}_{1,2}}=q_{1,2}v^{*2}_{1,2}$
that

$\displaystyle\quad\;X^{*}_{2}(\tau)P_{v^{*}_{1,2}}(\tau)$
$\displaystyle=\varphi_{1,1}(\tau)X^{*2}_{2}(\tau)+\varphi_{1,2}(\tau)\overline{X}^{*}_{2}(\tau)X^{*}_{2}(\tau)+\varphi_{1,0}(\tau)X^{*}_{2}(\tau)$
$\displaystyle=(\varphi_{1,1}(\tau)+\varphi_{1,2}(\tau))X^{*2}_{2}(\tau)+\varphi_{1,0}(\tau)X^{*}_{2}(\tau)$
$\displaystyle=-\mathbb{E}\Big{[}\int_{\tau}^{T}\left[q_{1,2}v^{*2}_{1,2}+q^{-1}_{2,2}b_{2,2}^{2}P_{v^{*}_{2,2}}P_{v^{*}_{1,2}}+P_{v^{*}_{1,2}}X^{*2}_{2}+\overline{P}_{v^{*}_{1,2}}\overline{X}^{*2}_{2}\right](t)dt+r_{i}(T)X^{*2}_{2}(T)\Big{|}\mathfrak{F}_{\tau}\Big{]}.$

This implies

$J_{1,2}(v^{*}_{1,2},v^{*}_{2,2})=\frac{1}{2}\left[(\varphi_{1,1}(\tau)+\varphi_{1,2}(\tau))X^{*2}_{2}(\tau)+\varphi_{1,0}(\tau)X^{*}_{2}(\tau)+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}(t)P_{v^{*}_{2,2}}(t)P_{v^{*}_{1,2}}(t)dt\Big{|}\mathfrak{F}_{\tau}\right].$

Analogously, for follower $\mathcal{P}_{2}$, one has

$J_{2,2}(v^{*}_{1,2},v^{*}_{2,2})=\frac{1}{2}\left[(\varphi_{2,1}(\tau)+\varphi_{2,2}(\tau))X^{*2}_{2}(\tau)+\varphi_{2,0}(\tau)X^{*}_{2}(\tau)+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}(t)P_{v^{*}_{1,2}}(t)P_{v^{*}_{2,2}}(t)dt\Big{|}\mathfrak{F}_{\tau}\right].$

This ends the proof. ∎

Now, we are able to give the formulation of the optimal objective subfunctions
at the second stage. Combining (12), (19) and Lemma 3.3, one has

$\displaystyle\mathbb{E}$
$\displaystyle\left[\int_{\tau}^{T}l_{2}(t)P_{v^{*}_{1,2}}(t)P_{v^{*}_{2,2}}(t)dt\Big{|}\mathfrak{F}_{\tau}\right]$
$\displaystyle=\mathbb{E}$
$\displaystyle\Big{[}\int_{\tau}^{T}l_{2}\left[\left(\varphi_{1,1}M+\varphi_{1,2}\overline{M}\right)X^{*}_{2}(\tau)+\left(\varphi_{1,1}N+\varphi_{1,2}\overline{N}+\varphi_{1,0}\right)\right]$
$\displaystyle\times\left[\left(\varphi_{2,1}M+\varphi_{2,2}\overline{M}\right)X^{*}_{2}(\tau)+\left(\varphi_{2,1}N+\varphi_{2,2}\overline{N}+\varphi_{2,0}\right)\right](t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]}$
$\displaystyle=\mathbb{E}$
$\displaystyle\Big{[}\int_{\tau}^{T}l_{2}\left(\varphi_{1,1}M+\varphi_{1,2}\overline{M}\right)\left(\varphi_{2,1}M+\varphi_{2,2}\overline{M}\right)(t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]}\left(X^{*}_{2}(\tau)\right)^{2}+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}\Big{[}\left(\varphi_{1,1}M+\varphi_{1,2}\overline{M}\right)$
$\displaystyle\left(\varphi_{2,1}N+\varphi_{2,2}\overline{N}+\varphi_{2,0}\right)+\left(\varphi_{1,1}N+\varphi_{1,2}\overline{N}+\varphi_{1,0}\right)\left(\varphi_{2,1}M+\varphi_{2,2}\overline{M}\right)\Big{]}(t)\Big{|}\mathfrak{F}_{\tau}\Big{]}X^{*}_{2}(\tau)$
$\displaystyle+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}\left(\varphi_{2,1}N+\varphi_{2,2}\overline{N}+\varphi_{2,0}\right)\left(\varphi_{1,1}N+\varphi_{1,2}\overline{N}+\varphi_{1,0}\right)(t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]}.$
(22)

Substituting (3.1) into (21), we have the following theorem.

######  Theorem 3.2.

For player $\mathcal{P}_{i}$, the optimal objective subfunction is

$J_{i,2}(v^{*}_{1,2},v^{*}_{2,2})=\frac{1}{2}\left[\kappa_{i,1}(\tau)X^{*2}_{2}(\tau)+\kappa_{i,2}(\tau)X^{*}_{2}(\tau)+\kappa_{i,0}(\tau)\right],$

where

$\begin{cases}\kappa_{i,1}(\tau)=\varphi_{i,1}(\tau)+\varphi_{i,2}(\tau)+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}\left(\varphi_{1,1}M+\varphi_{1,2}\overline{M}\right)\left(\varphi_{2,1}M+\varphi_{2,2}\overline{M}\right)(t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]},\\\
\kappa_{i,2}(\tau)=\varphi_{i,0}(\tau)+\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}\Big{[}\left(\varphi_{1,1}M+\varphi_{1,2}\overline{M}\right)\left(\varphi_{2,1}N+\varphi_{2,2}\overline{N}+\varphi_{2,0}\right)\\\
\qquad\quad\;\mbox{}+\left(\varphi_{1,1}N+\varphi_{1,2}\overline{N}+\varphi_{1,0}\right)\left(\varphi_{2,1}M+\varphi_{2,2}\overline{M}\right)\Big{]}(t)\Big{|}\mathfrak{F}_{\tau}\Big{]},\\\
\kappa_{i,0}(\tau)=\mathbb{E}\Big{[}\int_{\tau}^{T}l_{2}\left(\varphi_{2,1}N+\varphi_{2,2}\overline{N}+\varphi_{2,0}\right)\left(\varphi_{1,1}N+\varphi_{1,2}\overline{N}+\varphi_{1,0}\right)(t)dt\Big{|}\mathfrak{F}_{\tau}\Big{]}.\end{cases}$

######  Remark 3.1.

By Theorem 3.8 Chapter IX in [9], it is easy to show that
$\varphi_{i,1}(t)<0$, $\varphi_{i,2}(t)<0$ and $M(t)>0$. Therefore,
$\kappa_{i,1}(\tau)<0$ a.s. and the concavity of
$J_{i,2}(v^{*}_{1,2},v^{*}_{2,2})$ with respect to $X^{*}_{2}(\tau)$ follows.

###  3.2 Nash equilibrium for followers at the first stage

In this subsection, we solve the Nash equilibrium point
$(v^{*}_{1,1},v^{*}_{2,1})$ for followers at the first stage. According to
Theorem 3.2, Subproblem (9) can be rewritten as follows:

######  Subroblem 3.2.

Find
$(v^{*}_{1,1},v^{*}_{2,1})\in\mathcal{V}[0,\tau]\times\mathcal{V}[0,\tau]$
such that

$\begin{cases}J_{1,1}(v^{*}_{1,1},v^{*}_{2,1},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{1,1}\in\mathcal{V}[0,\tau]}\mathbb{E}\Big{\\{}\int_{0}^{\tau}-\frac{1}{2}\left[p_{1,1}(t)X_{1}^{2}(t)+\overline{p}_{1,1}(t)\overline{X}_{2}^{2}(t)+q_{1,1}(t)v_{1,1}^{2}(t)\right]dt\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\mbox{}+\frac{1}{2}\left[\kappa_{1,1}(\tau)X^{*2}_{1}(\tau)+\kappa_{1,2}(\tau)X^{*}_{1}(\tau)+\kappa_{1,0}(\tau)\right]\Big{|}\mathfrak{F}_{0}\Big{\\}},\\\
J_{2,1}(v^{*}_{1,1},v^{*}_{2,1},v_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{2,1}\in\mathcal{V}[0,\tau]}\mathbb{E}\Big{\\{}\int_{0}^{\tau}-\frac{1}{2}\left[p_{2,1}(t)X_{1}^{2}(t)+\overline{p}_{2,1}(t)\overline{X}_{2}^{2}(t)+q_{2,1}(t)v_{2,1}^{2}(t)\right]dt\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\mbox{}+\frac{1}{2}\left[\kappa_{2,1}(\tau)X^{*2}_{1}(\tau)+\kappa_{2,2}(\tau)X^{*}_{1}(\tau)+\kappa_{2,0}(\tau)\right]\Big{|}\mathfrak{F}_{0}\Big{\\}}.\end{cases}$

Similar to Subsection 3.1, we can define the Hamiltonian function of follower
$\mathcal{P}_{i}$ at first stage by

$\displaystyle H_{i,1}$
$\displaystyle:[0,\tau]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R},$
$\displaystyle H_{i,1}$
$\displaystyle=H_{i,1}(t,X_{1},\overline{X}_{1},v_{0},v_{1,1},v_{2,1},P_{v_{i,1}},Q_{v_{i,1}}),$
$\displaystyle=-\frac{1}{2}\left[p_{i,1}X_{1}^{2}+\overline{p}_{i,1}\overline{X}_{1}^{2}+q_{i,1}v_{i,1}^{2}\right]+\left[a_{1}X_{1}+\overline{a}_{1}\overline{X}_{1}+b_{1,1}v_{1,1}+b_{2,1}v_{2,1}+b_{0}v_{0}\right]P_{v_{i,1}}$
$\displaystyle\quad\,\mbox{}+\left[c_{1}X_{1}+\overline{c}_{1}\overline{X}_{1}\right]Q_{v_{i,1}}.$

Recalling Remark 3.1 and using the sufficient and necessary maximum principles
of MF-SDEs with default (see [11]), we have

$0=\frac{\partial H_{i,1}}{\partial
v_{i,1}}\Big{|}_{v_{i,1}=v^{*}_{i,1}}=-q_{i,1}v^{*}_{i,1}+b_{i,1}P_{v^{*}_{i,1}}\quad\Rightarrow\quad
v^{*}_{i,1}=q^{-1}_{i,1}b_{i,1}P_{v^{*}_{i,1}},$

where $P_{v^{*}_{i,1}}$ is the solution to corresponding linear BSDE.

Now we summarize the above arguments as the following theorem.

######  Theorem 3.3.

$(v^{*}_{1,1},v^{*}_{2,1})$ is a Nash equilibrium point for Subproblem 3.2 if
and only if

$(v^{*}_{1,1},v^{*}_{2,1})=(q^{-1}_{1,1}b_{i,1}P_{v^{*}_{1,1}},q^{-1}_{2,1}b_{2,1}P_{v^{*}_{2,1}}),$
(23)

where the optimal triple $(X_{1}^{*},P_{v^{*}_{i,1}},Q_{v^{*}_{i,1}})$ enjoys
the following MF-FBSDE:

$\begin{cases}dX^{*}_{1}=\left[a_{1}X^{*}_{1}+\overline{a}_{1}\overline{X}^{*}_{1}+q^{-1}_{1,1}b_{1,1}^{2}P_{v^{*}_{1,1}}+q^{-1}_{2,1}b_{2,1}^{2}P_{v^{*}_{2,1}}+b_{0}v_{0}\right]dt+\left[c_{1}X^{*}_{1}+\overline{c}_{1}\overline{X}^{*}_{1}\right]dB_{t},\quad
t\in(0,\tau],\\\
dP_{v^{*}_{i,1}}=\left[p_{i,1}X^{*}_{1}+\overline{p}_{i,1}\overline{X}^{*}_{1}-a_{1}P_{v^{*}_{i,1}}-\overline{a}_{1}\overline{P}_{v^{*}_{i,1}}-c_{1}Q_{v^{*}_{i,1}}-\overline{c}_{1}\overline{Q}_{v^{*}_{i,1}}\right]dt+Q_{v^{*}_{i,1}}dB_{t},\quad
t\in[0,\tau),\\\ X_{1}^{*}(\tau)=x_{0},\quad
P_{v^{*}_{i,1}}(\tau)=\kappa_{i,1}(\tau)X^{*}_{2}(\tau)+\kappa_{i,2}(\tau).\end{cases}$
(24)

Next, we put

$P_{v^{*}_{i,1}}(t)=\psi_{i,1}(t)X^{*}_{1}(t)+\psi_{i,2}(t)\overline{X}^{*}_{1}(t)+\psi_{i,0}(t),$
(25)

where $\psi_{i,1}$, $\psi_{i,2}$ are deterministic functions, $\psi_{i,0}$ is
an $\mathfrak{F}$-adapted process such that

$d\psi_{i,0}(t)=\psi_{i,3}(t)dt+\psi_{i,4}(t)dB_{t},\;\psi_{0}(T)=0.$

Moreover, $\psi_{j}=\psi_{1,j}+\psi_{2,j}$.

Then, repeating the arguments in Subsection 3.1, we can obtain the following
system of ODEs:

$\begin{cases}\psi^{\prime}_{1,1}+(2a_{1}+c^{1}_{2})\psi_{1,1}+l_{1}\psi^{2}_{1,1}+l_{1}\psi_{1,1}\psi_{2,1}-p_{1,1}=0,\quad
t\in[0,\tau),\\\
\psi^{\prime}_{2,1}+(2a_{1}+c^{1}_{2})\psi_{2,1}+l_{1}\psi_{1,1}\psi_{2,1}+l_{1}\psi^{2}_{2,1}-p_{2,1}=0,\quad
t\in[0,\tau),\\\
\psi^{\prime}_{1,2}+\left(2a_{1}+2\overline{a}_{1}+2l_{1}\psi_{1,1}+l_{1}\psi_{2,1}\right)\psi_{1,2}+l_{1}\psi^{2}_{1,2}+l_{1}\psi_{1,2}\psi_{2,2}\\\
\quad\;\;\,\mbox{}+l_{1}\psi_{1,1}\psi_{2,2}+\psi_{1,1}\Big{[}2\overline{a}_{1}+\overline{c}_{1}^{2}+2c_{1}\overline{c}_{1}\Big{]}-\overline{p}_{1,2}=0,\quad
t\in[0,\tau),\\\
\psi^{\prime}_{2,2}+\left(2a_{1}+2\overline{a}_{1}+2l_{1}\psi_{2,1}+l_{1}\psi_{1,1}\right)\psi_{2,2}+l_{1}\psi^{2}_{2,2}+l_{1}\psi_{1,2}\psi_{2,2}\\\
\quad\;\;\,\mbox{}+l_{1}\psi_{2,1}\psi_{1,2}+\psi_{2,1}\Big{[}2\overline{a}_{1}+\overline{c}_{1}^{2}+2c_{1}\overline{c}_{1}\Big{]}-\overline{p}_{2,2}=0,\quad
t\in[0,\tau),\\\
\psi_{1,1}(\tau)=\kappa_{1,1}(\tau),\quad\psi_{1,2}(\tau)=0,\quad\psi_{2,1}(\tau)=\kappa_{2,1}(\tau),\quad\psi_{2,2}(\tau)=0\end{cases}$
(26)

for $(\psi_{1,1},\psi_{1,2},\psi_{2,1},\psi_{2,2})$, and the following system
of MF-BSDEs:

$\begin{cases}d\psi_{1,0}=-\Big{[}\psi_{1,1}b_{0}v_{0}+\psi_{1,2}b_{0}\overline{v}_{0}+\left(a_{1}+l_{1}\psi_{1,1}\right)\psi_{1,0}+\left(\overline{a}_{1}+l_{1}\psi_{1,2}\right)\overline{\psi}_{1,0}+c_{1}\psi_{1,4}+\overline{c}_{1}\overline{\psi}_{1,4}\\\
\qquad\quad\,\,\mbox{}+l_{1}\left(\psi_{1,1}\psi_{2,0}+\psi_{1,2}\overline{\psi}_{2,0}\right)\Big{]}dt+\psi_{1,4}dB_{t},\quad
t\in[0,\tau),\\\
d\psi_{2,0}=-\Big{[}\psi_{2,1}b_{0}v_{0}+\psi_{2,2}b_{0}\overline{v}_{0}+\left(a_{1}+l_{1}\psi_{2,1}\right)\psi_{2,0}+\left(\overline{a}_{1}+l_{1}\psi_{2,2}\right)\overline{\psi}_{2,0}+c_{1}\psi_{2,4}+\overline{c}_{1}\overline{\psi}_{2,4}\\\
\qquad\quad\,\,\mbox{}+l_{1}\left(\psi_{2,1}\psi_{1,0}+\psi_{2,2}\overline{\psi}_{1,0}\right)\Big{]}dt+\psi_{2,4}dB_{t},\quad
t\in[0,\tau),\\\
\psi_{1,0}(\tau)=\kappa_{1,2}(\tau),\quad\psi_{2,0}(\tau)=\kappa_{2,2}(\tau)\end{cases}$
(27)

for $((\psi_{1,0},\psi_{1,4}),(\psi_{2,0},\psi_{2,4}))$. Analogously, we know
that both (26) and (27) have unique solutions. By setting

$\displaystyle\alpha_{1}=\begin{bmatrix}\psi_{1,0}\\\
\psi_{2,0}\end{bmatrix},\quad$
$\displaystyle\beta_{1}=\begin{bmatrix}\psi_{1,4}\\\
\psi_{2,4}\end{bmatrix},\quad$
$\displaystyle\mu_{1}=\begin{bmatrix}a_{1}+l_{1}\psi_{1,1}&l_{1}\psi_{1,1}\\\
l_{1}\psi_{2,1}&a_{1}+l_{1}\psi_{2,1}\end{bmatrix},$
$\displaystyle\overline{\alpha}_{1}=\begin{bmatrix}\overline{\psi}_{1,0}\\\
\overline{\psi}_{2,0}\end{bmatrix},\quad$
$\displaystyle\overline{\beta}_{1}=\begin{bmatrix}\overline{\psi}_{1,4}\\\
\overline{\psi}_{2,4}\end{bmatrix},\quad$
$\displaystyle\nu_{1}=\begin{bmatrix}\overline{a}_{1}+l_{1}\psi_{1,2}&l_{1}\psi_{1,2}\\\
l_{1}\psi_{2,2}&\overline{a}_{1}+l_{1}\psi_{2,2}\end{bmatrix},$
$\displaystyle\delta_{1}=\begin{bmatrix}\psi_{1,1}b_{0}\\\
\psi_{2,1}b_{0}\end{bmatrix},\quad$
$\displaystyle\overline{\delta}_{1}=\begin{bmatrix}\psi_{1,2}b_{0}\\\
\psi_{2,2}b_{0}\end{bmatrix},\quad$
$\displaystyle\iota(\tau)=\begin{bmatrix}\kappa_{1,2}(\tau)\\\
\kappa_{2,2}(\tau)\end{bmatrix},$

(27) can be rewritten as

$\begin{cases}d\alpha_{1}=-\left(\mu_{1}\alpha_{1}+\nu_{1}\overline{\alpha}_{1}+c_{1}\beta_{1}+\overline{c}_{1}\overline{\beta}_{1}+\delta_{1}v_{0}+\overline{\delta}_{1}\overline{v}_{0}\right)+\beta_{1}dB_{t},\quad
t\in[0,\tau),\\\ \alpha_{1}(\tau)=\iota(\tau).\end{cases}$ (28)

Thus, at the first stage, the optimal state $X^{*}_{1}(t)$ at Nash equilibrium
point $(v^{*}_{1,1},v^{*}_{2,1})$ is uniquely determined by the following
system:

$\begin{cases}dX^{*}_{1}=\left[(a_{1}+l_{1}\psi_{1})X^{*}_{1}+(\overline{a}_{1}+l_{1}\psi_{2})\overline{X}^{*}_{1}+(l_{1},l_{1})\alpha_{1}+b_{0}v_{0}\right]dt\\\
\qquad\quad\,\mbox{}+\left[c_{1}X^{*}_{1}+\overline{c}_{1}\overline{X}^{*}_{1}\right]dB_{t},\quad
t\in(0,\tau],\\\
d\alpha_{1}=-\left(\mu_{1}\alpha_{1}+\nu_{1}\overline{\alpha}_{1}+c_{1}\beta_{1}+\overline{c}_{1}\overline{\beta}_{1}+\delta_{1}v_{0}+\overline{\delta}_{1}\overline{v}_{0}\right)+\beta_{1}dB_{t},\quad
t\in[0,\tau),\\\
X^{*}_{1}(0)=x_{0},\quad\alpha_{1}(\tau)=\iota(\tau).\end{cases}$ (29)

###  3.3 Stackelberg solution of LQ-MF-SSDG with random exit time

In this subsection, we focus on finding the optimal strategy $v^{*}_{0}$ for
the leader and then deriving optimal strategies of followers by setting
$v_{0}=v^{*}_{0}$ in (24). Combining (4), (7), (6) and (29), we obtain the
following optimal strategy problem for the leader, which is indeed an optimal
control problem for MF-SDEs with default.

######  Subroblem 3.3.

Find $v^{*}_{0}\in\mathcal{V}[0,\tau]$ such that

$\displaystyle\quad\;J_{0}(v^{*}_{1},v^{*}_{2},v^{*}_{0})=\mathop{\operatorname{ess\,sup}}\limits_{v_{0}\in\mathcal{V}[0,\tau]}J_{0}(v^{*}_{1},v^{*}_{2},v_{0})$
$\displaystyle=\mathop{\operatorname{ess\,sup}}\limits_{v_{0}\in\mathcal{V}[0,\tau]}\left\\{-\frac{1}{2}\mathbb{E}\left[\int_{0}^{\tau}p_{0}(t)Y^{2}(t)+\overline{p}_{0}(t)\overline{Y}^{2}(t)+q_{0}(t)v_{0}^{2}(t)dt+r_{0}(\tau)Y^{2}(\tau)\Big{|}\mathfrak{F}_{0}\right]\right\\},$

where the state process $Y(t)$ is governed by

$\begin{cases}dY=\left[(a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0})Y+(\overline{a}_{1}+l_{1}\psi_{2}+\gamma^{\mathfrak{F}}\overline{d}_{0})\overline{Y}+(l_{1},l_{1})\alpha_{1}+b_{0}v_{0}\right]dt\\\
\qquad\;\,\>\mbox{}+\left[c_{1}Y+\overline{c}_{1}\overline{Y}\right]dB_{t}+\left[d_{0}Y+\overline{d}_{0}\overline{Y}\right]dA_{t},\quad
t\in(0,\tau],\\\
d\alpha_{1}=-\left(\mu_{1}\alpha_{1}+\nu_{1}\overline{\alpha}_{1}+c_{1}\beta_{1}+\overline{c}_{1}\overline{\beta}_{1}+\delta_{1}v_{0}+\overline{\delta}_{1}\overline{v}_{0}\right)dt+\beta_{1}dB_{t},\quad
t\in[0,\tau);\\\ Y(0)=x_{0},\quad\alpha_{1}(\tau)=\iota(\tau).\end{cases}$

To solve Subproblem 3.3, we define the Hamiltonian function of the leader by
setting

$\displaystyle H_{0}$
$\displaystyle:[0,\tau]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R},$
$\displaystyle H_{0}$
$\displaystyle=H_{0}(t,Y,\overline{Y},v_{0},\alpha_{1},P_{v_{0}},Q_{v_{0}},R_{v_{0}},Z_{v_{0}})$
$\displaystyle=-\frac{1}{2}\left[p_{0}Y^{2}+\overline{p}_{0}\overline{Y}^{2}+q_{0}v_{0}^{2}\right]+\left\langle
P_{v_{0}},(a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0})Y+(\overline{a}_{1}+l_{1}\psi_{2}+\gamma^{\mathfrak{F}}\overline{d}_{0})\overline{Y}+(l_{1},l_{1})\alpha_{1}+b_{0}v_{0}\right\rangle$
$\displaystyle\quad\,\mbox{}+\left\langle
Q_{v_{0}},c_{1}Y+\overline{c}_{1}\overline{Y}\right\rangle+\left\langle
R_{v_{0}},(d_{0}Y+\overline{d}_{0}\overline{Y})\gamma^{\mathcal{F}}\right\rangle+\left\langle
Z_{v_{0}},\mu_{1}\alpha_{1}+\nu_{1}\overline{\alpha}_{1}+c_{1}\beta_{1}+\overline{c}_{1}\overline{\beta}_{1}+\delta_{1}v_{0}+\overline{\delta}_{1}\overline{v}_{0}\right\rangle,$

where $(P_{v_{0}},Q_{v_{0}},R_{v_{0}},Z_{v_{0}})$ is the solution to the
following MF-FBSDE:

$\begin{cases}dP_{v_{0}}=\Big{\\{}p_{0}Y+\overline{p}_{0}\overline{Y}-(a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0})P_{v_{0}}-(\overline{a}_{1}+l_{1}\psi_{2})\overline{P}_{v_{0}}-\overline{d}_{0}\mathbb{E}\left[\gamma^{\mathfrak{F}}P_{v_{0}}\Big{|}\mathfrak{F}_{0}\right]\\\
\qquad\;\,\>\mbox{}-(c_{1}Q_{v_{0}}+\overline{c}_{1}\overline{Q}_{v_{0}})-(R_{v_{0}}\gamma^{\mathfrak{F}}d_{0}+\overline{d}_{0}\mathbb{E}\left[\gamma^{\mathfrak{F}}R_{v_{0}}\Big{|}\mathfrak{F}_{0}\right])\Big{\\}}dt+Q_{v_{0}}dB_{t}+R_{v_{0}}dA_{t},\quad
t\in(0,\tau],\\\ dZ_{v_{0}}=\left(\begin{bmatrix}l_{1}\\\
l_{1}\end{bmatrix}P_{v_{0}}+\mu_{1}^{T}Z_{v_{0}}+\nu_{1}^{T}\overline{Z}_{v_{0}}\right)dt+\left(c_{1}Z_{v_{0}}+\overline{c}_{1}\overline{Z}_{v_{0}}\right)dB_{t},\quad
t\in[0,\tau),\\\ P_{v_{0}}(\tau)=-r_{0}(\tau)Y(\tau),\quad
Z_{v_{0}}(0)=\begin{bmatrix}0\\\ 0\end{bmatrix}.\end{cases}$

Here $\mu_{1}^{T}$ and $\nu_{1}^{T}$ represent the transposed matrixes of
$\mu_{1}$ and $\nu_{1}$, respectively. It is easy to show that $v_{0}$ becomes
an optimal control of Subproblem 3.2 when

$0=\frac{\partial H_{0}}{\partial
v_{0}}+\mathbb{E}\left[\nabla_{\overline{v}_{0}}H_{0}\Big{|}\mathfrak{F}_{0}\right]\Big{|}_{v_{0}=v^{*}_{0}}=-q_{0}v^{*}_{0}+b_{0}P_{v^{*}_{0}}+\langle\delta_{1},Z_{v^{*}_{0}}\rangle+\langle\overline{\delta}_{1},\overline{Z}_{v^{*}_{0}}\rangle,$

where $\nabla_{\overline{v}_{0}}$ is the Fréchet derivative with respect to
$\overline{v}_{0}$. It remains to verify that, for

$v_{0}^{*}=q_{0}^{-1}(b_{0}P_{v^{*}_{0}}+\langle\delta_{1},Z_{v^{*}_{0}}\rangle+\langle\overline{\delta}_{1},\overline{Z}_{v^{*}_{0}}\rangle),$
(30)

the following inequality holds:

$J_{0}(v^{*}_{1},v^{*}_{2},v^{*}_{0})\leq
J_{0}(v^{*}_{1},v^{*}_{2},v_{0}),\quad\forall v_{0}\in\mathcal{V}[0,\tau].$

Setting $\widetilde{Y}=Y^{*}-Y$,
$\widetilde{\alpha}_{1}=\alpha_{1}^{*}-\alpha_{1}$ and
$\widetilde{v}_{0}=v_{0}^{*}-v_{0}$, one has

$\displaystyle\quad\;J_{0}(v^{*}_{1},v^{*}_{2},v^{*}_{0})-J_{0}(v^{*}_{1},v^{*}_{2},v_{0})$
$\displaystyle=-\frac{1}{2}\mathbb{E}\left[\int_{0}^{\tau}p_{0}(t)\widetilde{Y}^{*2}(t)+\overline{p}_{0}(t)\widetilde{\overline{Y}}^{*2}(t)+q_{0}(t)\widetilde{v}_{0}^{*2}(t)dt+r_{0}(\tau)\widetilde{Y}^{2}(\tau)\Big{|}\mathfrak{F}_{0}\right]$
$\displaystyle\quad\,\mbox{}-\mathbb{E}\left[\int_{0}^{\tau}p_{0}(t)Y^{*}(t)\widetilde{Y}(t)+\overline{p}_{0}(t)\overline{Y}^{*}(t)\widetilde{\overline{Y}}(t)+v^{*}_{0}(t)\widetilde{v}_{0}(t)dt+r_{0}(\tau)Y^{*}(\tau)\widetilde{Y}(\tau)\Big{|}\mathfrak{F}_{0}\right]$
$\displaystyle\leq-\mathbb{E}\left[\int_{0}^{\tau}p_{0}(t)Y^{*}(t)\widetilde{Y}(t)+\overline{p}_{0}(t)\overline{Y}^{*}(t)\widetilde{\overline{Y}}(t)+q_{0}(t)v^{*}_{0}(t)\widetilde{v}_{0}(t)dt-\widetilde{Y}(\tau)P_{v^{*}_{0}}(\tau)\Big{|}\mathfrak{F}_{0}\right].$
(31)

Noticing that
$\widetilde{Y}(0)=\langle\widetilde{\alpha}_{1}(0),Z_{v^{*}_{0}}(0)\rangle=\langle\widetilde{\alpha}_{1}(T),Z_{v^{*}_{0}}(T)\rangle$=0,
we have

$\displaystyle\quad\;\mathbb{E}\left[\widetilde{Y}(\tau)P_{v^{*}_{0}}(\tau)\Big{|}\mathfrak{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[\left(\widetilde{Y}(\tau)P_{v^{*}_{0}}(\tau)-\widetilde{Y}(0)P_{v^{*}_{0}}(0)\right)-\left(\langle\widetilde{\alpha}_{1}(T),Z_{v^{*}_{0}}(T)\rangle-\langle\widetilde{\alpha}_{1}(0),Z_{v^{*}_{0}}(0)\rangle\right)\Big{|}\mathfrak{F}_{0}\right]$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{\tau}\widetilde{Y}(t)dP_{v^{*}_{0}}(t)+P_{v^{*}_{0}}(t)d\widetilde{Y}(t)+\langle
d\widetilde{Y}(t),dP_{v^{*}_{0}}(t)\rangle-\langle\widetilde{\alpha}_{1}(t),dZ_{v^{*}_{0}}(t)\rangle$
$\displaystyle\quad\;\mbox{}-\langle
Z_{v^{*}_{0}}(t),d\widetilde{\alpha}_{1}(t)\rangle-\langle
d\widetilde{\alpha}_{1}(t),dZ_{v^{*}_{0}}(t)\rangle\Big{|}\mathfrak{F}_{0}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{\tau}\left[p_{0}(t)\widetilde{Y}(t)+\overline{p}_{0}(t)\widetilde{\overline{Y}}(t)\right]Y^{*}(t)-\left[(l_{1},l_{1})\widetilde{\alpha}_{1}+b_{0}\widetilde{v}_{0}\right]P_{v^{*}_{0}}(t)+\langle\delta_{1}(t),Z_{v^{*}_{0}}(t)\rangle\widetilde{v}_{0}(t)$
$\displaystyle\quad\;\mbox{}+\left\langle\overline{\delta}_{1}(t),\overline{Z}_{v^{*}_{0}}(t)\right\rangle\widetilde{v}_{0}(t)+\langle\widetilde{\alpha}_{1}(t),\begin{bmatrix}l_{1}\\\
l_{1}\end{bmatrix}P_{v^{*}_{0}}(t)\rangle dt\Big{|}\mathfrak{F}_{0}\Big{]}.$
(32)

Substituting (30) and (3.3) into (3.3) obtains

$\displaystyle\quad\;J_{0}(v^{*}_{1},v^{*}_{2},v^{*}_{0})-J_{0}(v^{*}_{1},v^{*}_{2},v_{0})$
$\displaystyle\leq\mathbb{E}\Big{[}\int_{0}^{\tau}q_{0}(t)v^{*}_{0}(t)\widetilde{v}_{0}(t)+\left[(l_{1},l_{1})(t)\widetilde{\alpha}_{1}(t)-b_{0}(t)\widetilde{v}_{0}\right]P_{v^{*}_{0}}(t)+\langle\delta_{1}(t),Z_{v^{*}_{0}}(t)\rangle\widetilde{v}_{0}(t)$
$\displaystyle\quad\;\mbox{}-\langle\overline{\delta}_{1}(t),\overline{Z}_{v^{*}_{0}}(t)\rangle\widetilde{v}_{0}(t)-\langle\widetilde{\alpha}_{1}(t),\begin{bmatrix}l_{1}\\\
l_{1}\end{bmatrix}(t)P_{v^{*}_{0}}(t)\rangle dt\Big{|}\mathfrak{F}_{0}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{\tau}\left(q_{0}(t)v^{*}_{0}(t)-b_{0}(t)P_{v^{*}_{0}}(t)-\langle\delta_{1}(t),Z_{v^{*}_{0}}(t)\rangle-\langle\overline{\delta}_{1}(t),\overline{Z}_{v^{*}_{0}}(t)\rangle\right)\widetilde{v}_{0}(t)dt\Big{|}\mathfrak{F}_{0}\Big{]}$
$\displaystyle=0,$

which is the desired result.

According to the above arguments, we have the following theorem.

######  Theorem 3.4.

The optimal strategy of the leader for Subproblem 3.3 is

$v_{0}^{*}=q_{0}^{-1}(b_{0}P_{v^{*}_{0}}+\langle\delta_{1},Z_{v^{*}_{0}}\rangle+\langle\overline{\delta}_{1},\overline{Z}_{v^{*}_{0}}\rangle).$
(33)

Here
$(Y^{*},\alpha^{*}_{1},\beta^{*},P_{v^{*}_{0}},Q_{v^{*}_{0}},R_{v^{*}_{0}},Z_{v^{*}_{0}})$
is a solution to the following linear MF-FBSDE:

$\begin{cases}dY^{*}=\left[(a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0})Y^{*}+(\overline{a}_{1}+l_{1}\psi_{2}+\gamma^{\mathfrak{F}}\overline{d}_{0})\overline{Y}^{*}+(l_{1},l_{1})\alpha^{*}_{1}+b_{0}v^{*}_{0}\right]dt\\\
\qquad\;\,\>\mbox{}+\left[c_{1}Y^{*}+\overline{c}_{1}\overline{Y}^{*}\right]dB_{t}+\left[d_{0}Y^{*}+\overline{d}_{0}\overline{Y}^{*}\right]dA_{t},\quad
t\in(0,\tau],\\\
d\alpha^{*}_{1}=-\left(\mu_{1}\alpha^{*}_{1}+\nu_{1}\overline{\alpha}^{*}_{1}+c_{1}\beta^{*}_{1}+\overline{c}_{1}\overline{\beta}^{*}_{1}+\delta_{1}v^{*}_{0}+\overline{\delta}_{1}\overline{v}^{*}_{0}\right)dt+\beta^{*}_{1}dB_{t},\quad
t\in[0,\tau),\\\
dP_{v^{*}_{0}}=\Big{\\{}p_{0}Y^{*}+\overline{p}_{0}\overline{Y}^{*}-(a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0})P_{v^{*}_{0}}-(\overline{a}_{1}+l_{1}\psi_{2})\overline{P}_{v^{*}_{0}}-\overline{d}_{0}\mathbb{E}\left[\gamma^{\mathfrak{F}}P_{v^{*}_{0}}\Big{|}\mathfrak{F}_{0}\right]\\\
\qquad\;\,\>\mbox{}-(c_{1}Q_{v^{*}_{0}}+\overline{c}_{1}\overline{Q}_{v^{*}_{0}})-(R_{v^{*}_{0}}\gamma^{\mathfrak{F}}d_{0}+\overline{d}_{0}\mathbb{E}\left[\gamma^{\mathfrak{F}}R_{v^{*}_{0}}\Big{|}\mathfrak{F}_{0}\right])\Big{\\}}dt+Q_{v^{*}_{0}}dB_{t}+R_{v^{*}_{0}}dA_{t},\quad
t\in(0,\tau],\\\
dZ_{v^{*}_{0}}=\left((l_{1},l_{1})^{T}P_{v^{*}_{0}}+\mu_{1}^{T}Z_{v^{*}_{0}}+\nu_{1}^{T}\overline{Z}_{v^{*}_{0}}\right)dt+\left(c_{1}Z_{v^{*}_{0}}+\overline{c}_{1}\overline{Z}_{v^{*}_{0}}\right)dB_{t},\quad
t\in[0,\tau),\\\ Y^{*}(0)=x_{0},\quad\alpha^{*}_{1}(\tau)=\iota(\tau),\quad
P_{v^{*}_{0}}(\tau)=-r_{0}(\tau)Y^{*}(\tau),\quad
Z_{v^{*}_{0}}(0)=(0,0)^{T}.\end{cases}$ (34)

######  Remark 3.2.

If $x_{0}\in\mathbb{R}$, then the conditional mean-field term
$\overline{X}(t)$ reduces to the classical mean-field term. If $\tau>T$ a.s.,
then the default event will not happen before time $T$. Moreover, if
$x_{0}\in\mathbb{R}$ and $\tau>T$ a.s., then Theorem 3.4 reduces to Theorem
3.4 in [29].

By choosing $v_{0}=v_{0}^{*}$ in (24), $(P_{v^{*}_{1,1}},P_{v^{*}_{2,1}})$ is
uniquely determined and so is $(v^{*}_{1,1},v^{*}_{2,1})$. Combining Theorems
3.1, 3.3 and 3.4, we have the following results.

######  Theorem 3.5.

The Stackelberg solution of Problem 2.1 is given by (10), (23) and (30).

In practical situations, the feedback representation of the Stackelberg
solution is very useful. However, it is hard to find the feedback
representation of the Stackelberg solution of Problem 2.1 since random
coefficients and conditional expectation are involved in (34). Nevertheless,
the following example shows that we can obtain the feedback representation of
the Stackelberg solution in some special cases.

######  Example 3.1.

When $\overline{a}_{1}=\overline{c}_{1}=\overline{d}_{0}\equiv 0$, (33) and
(34) reduce respectively to

$v_{0}^{*}=q_{0}^{-1}(b_{0}P_{v^{*}_{0}}+\langle\delta_{1},Z_{v^{*}_{0}}\rangle)$

and

$\begin{cases}d\mathcal{Y}=\left[\mathcal{L}_{1}\mathcal{Y}+\mathcal{L}_{2}\mathcal{P}\right]dt+c_{1}\mathcal{Y}dB_{t}+\mathcal{L}_{3}\mathcal{Y}dA_{t},\quad
t\in(0,\tau],\\\
d\mathcal{P}=-\left(\mathcal{M}_{1}\mathcal{P}+\mathcal{M}_{2}\mathcal{Y}+\mathcal{M}_{3}\mathcal{R}+c_{1}\mathcal{Q}\right)dt+\mathcal{Q}dB_{t}+\mathcal{R}dA_{t},\quad
t\in[0,\tau),\\\
\mathcal{Y}(0)=(x_{0},0,0)^{T},\quad\mathcal{P}(\tau)=\mathcal{G}(\tau)\mathcal{Y}(\tau).\end{cases}$

Here $k^{\mathfrak{F}}=a_{1}+l_{1}\psi_{1}+\gamma^{\mathfrak{F}}d_{0}$ and

$\mathcal{Y}=\begin{bmatrix}Y^{*}\\\
Z_{v^{*}_{0}}\end{bmatrix},\quad\mathcal{P}=\begin{bmatrix}P_{v^{*}_{0}}\\\
\alpha^{*}_{1}\end{bmatrix},\quad\mathcal{Q}=\begin{bmatrix}\beta_{1}^{*}\\\
Q_{v^{*}_{0}}\end{bmatrix},\quad\mathcal{R}=\begin{bmatrix}0\\\
R_{v^{*}_{0}}\end{bmatrix},$
$\mathcal{L}_{1}=\begin{bmatrix}k^{\mathfrak{F}}&q_{0}^{-1}b_{0}\delta_{1}^{T}\\\
0&\mu_{1}^{T}\end{bmatrix},\quad\mathcal{L}_{2}=\begin{bmatrix}q_{0}^{-1}b^{2}_{0}&(l_{1},l_{1})\\\
(l_{1},l_{1})^{T}&0\end{bmatrix},\quad\mathcal{M}_{1}=\begin{bmatrix}k^{\mathfrak{F}}&0\\\
\delta_{1}q_{0}^{-1}b_{0}&\mu_{1}\end{bmatrix},\quad\mathcal{M}_{2}=\begin{bmatrix}0&q_{0}^{-1}\delta_{1}\delta^{T}_{1}\\\
-p_{0}&0\end{bmatrix},$ $\mathcal{L}_{3}=\begin{bmatrix}d_{0}&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix},\quad\mathcal{M}_{3}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\
0&0&\gamma^{\mathfrak{F}}d_{0}\end{bmatrix},\quad\mathcal{G}(\tau)=\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&-r_{0}(\tau)\end{bmatrix}.$

Obviously, $\mathcal{L}_{1}$ and $\mathcal{M}_{1}$ are random matrixes with
$\mathcal{L}_{1}=\mathcal{M}_{1}^{T}$. Analogously by the method of
undetermined coefficients as in Subsection 3.1, suppose that

$\mathcal{P}(t)=\widetilde{G}(t)\mathcal{Y}(t),$

where $d\mathcal{G}(t)=\varpi_{1}dt+\varpi_{2}dB_{t}+\varpi_{3}dA_{t}$. Then,
we are able to conclude that $\widetilde{G}(t)$ is a solution to the following
stochastic Riccati equation:

$\begin{cases}d\widetilde{G}=-\left[\widetilde{G}\mathcal{L}_{1}+\widetilde{G}\mathcal{L}_{2}\widetilde{G}+\mathcal{L}_{1}^{T}\widetilde{G}+\mathcal{M}_{2}+\mathcal{M}_{3}\widetilde{G}\mathcal{L}_{3}+c_{1}^{2}\widetilde{G}+c_{1}\varpi_{2}+\mathcal{M}_{3}\varpi_{3}\right]dt\\\
\qquad\;\;\,+\varpi_{2}dB_{t}+\varpi_{3}\mathcal{Y}dA_{t},\quad
t\in[0,\tau),\\\ \widetilde{G}(\tau)=\mathcal{G}(\tau).\end{cases}$ (35)

Note that (35) is indeed a matrix-valued nonlinear BSDE with single jump.
Thus, by using Theorem 4.1 in [32], the solution to stochastic Riccati
equation (35) has an feedback representation and so does $v_{0}^{*}$. By
setting $v_{0}=v_{0}^{*}$ in (28), $(v^{*}_{1,1},v^{*}_{2,1})$ is uniquely
determined via (23) and (25). Thus, the feedback representation of Stackelberg
solution is obtained.

##  4 Conclusions

This paper focuses on the study of LQ-MF-SSDG with random exit time in the
framework of progressive enlargement of filtration, in which the leader is
allowed to stop her strategy at a random time. By employing the backward
induction method, we obtain the Stackelberg solution of LQ-MF-SSDG with random
exit time. We would like to point out that in the classical mean-field
setting, it is hard to derive BSDE for LQ-MF-SSDG because
$\mathbb{E}[a(t)Y(t)\mathbb{E}\left[X(t)\right]]\neq\mathbb{E}[X(t)\left[a(t)Y(t)\right]]$
when $a(t)$ is an $\mathfrak{F}_{t}$-adapted process. However, by employing
the fact that
$\mathbb{E}[a(t)Y(t)\mathbb{E}\left[X(t)\Big{|}\mathfrak{F}_{s}\right]\Big{|}\mathfrak{F}_{s}]=\mathbb{E}[X(t)\left[a(t)Y(t)\Big{|}\mathfrak{F}_{s}\right]\Big{|}\mathfrak{F}_{s}]\;(t\geq
s)$, we can obtain BSDE for LQ-MF-SSDG in the conditional mean-field setting
and so the Pontryagin maximum principle remains valid.

Recall that the problem of the feedback representation of $v_{0}^{*}$ in (34)
is unsolved. Therefore, how to find the feedback representation of Stackelberg
solution for LQ-MF-SSDG with random exit time is a problem worth
investigating. We also mention that the backward induction method becomes
invalid for solving the Stackelberg solution when the followers are allowed to
stop their strategies at a random time. This problem is closely related to the
multi-stage Stackelberg game, which is usually known as a very difficult
problem to be solved. Thus, it would be interesting to derive the Stackelberg
solution of LQ-MF-SSDG when one of the followers is allowed to exit the game
at a random time. We will consider these problems in future research.

## References

  * [1] Aksamit A. and Jeanblanc M.  Enlargement of Filtration with Finance in View.  Springer, Cham, 2017.
  * [2] Askar S. S.  Tripoly Stackelberg game model: one leader versus two followers.  Appl. Math. Comput., 328 (2018), 301–311.
  * [3] Babich V., Burnetas A. N. and Ritchken, P. H.  Competition and diversification effects in supply chains with supplier default risk.  M$\&$som-manuf. Serv. Op., 9 (2007), no. 2, 123–146.
  * [4] Bachir Cherif K., Agram N. and Dahl K.  Stochastic maximum principle with default.  arXiv: 2001.01535, 2020.
  * [5] Bensoussan A., Cass T., Chau M. H. M. and Yam S. C. P.  Mean field games with parametrized followers.  IEEE Trans. Automat. Control, 65 (2020), no. 1, 12–27.
  * [6] Bensoussan A., Chau M. H. M., Lai Y., and Yam S. C. P.  Linear-quadratic mean field Stackelberg games with state and control delays.  SIAM J. Control Optim., 55 (2017), no. 4, 2748–2781.
  * [7] Calvia A. and Gianin E R.  Risk measures and progressive enlargement of filtration: A BSDE approach.  SIAM J. Financial Math., 11 (2020), no. 3, 815–848.
  * [8] Carreño N. and Santos, M. C.  Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation.  J. Differential Equations, 266 (2019), no. 9, 6068–6108.
  * [9] Revuz D. and Yor M.  Continuous Martingales and Brownian Motion (Third edition).  Springer, Berlin, 1999.
  * [10] Fang H., Xu L. and Wang X.  Coordinated multiple-relays based physical-layer security improvement: A single-leader multiple-followers Stackelberg game scheme.  IEEE T. Inf. Foren. Sec., 13 (2017), no. 1, 197–209.
  * [11] Gou Z., Huang N. J., Wang M. H.  Optimal control problems governed by MF-SDEs with multi-defaults.  arXiv: 2010.13608, 2020.
  * [12] Huang J., Si K. and Wu Z.  Linear-quadratic mixed Stackelberg-Nash stochastic differential game with major-minor agents.  Appl. Math. Optim., https://doi.org/10.1007/s00245-020-09713-z, 3 2020\.
  * [13] Jeanblanc M. and Li L.  Characteristics and constructions of default times.  Financial Math., 11 (2020), no. 3, 720–749.
  * [14] Jeanblanc M., Yor M. and Chesney M.  Mathematical Methods for Financial Markets.  Springer, London, 2009.
  * [15] Jiang J. and Liu X.  Multi-objective Stackelberg game model for water supply networks against interdictions with incomplete information.  European J. Oper. Res., 266 (2018), no. 3, 920–933.
  * [16] Kharroubi I. and Lim T.  Progressive enlargement of filtrations and backward stochastic differential equations with jumps.  J. Theoret. Probab., 27 (2014), no. 3, 683–724.
  * [17] Lin Y.  Feedback Stackelberg strategies for the discrete-time mean-field stochastic systems in infinite horizon.  J. Franklin Inst., 356 (2019), no. 10, 5222–5239.
  * [18] Lin Y., Jiang X. and Zhang W.  An open-loop Stackelberg strategy for the linear quadratic mean-field stochastic differential game.  IEEE Trans. Automat. Contr., 64 (2019), no. 1, 97–110.
  * [19] Megahed E.  The Stackelberg differential game for counter-terrorism.  Qual. Quant., 53 (2019), 207–220.
  * [20] Moon J. and Başar T.  Linear quadratic mean field Stackelberg differential games.  Automatica, 97 (2018), 200–213.
  * [21] Moon J. and Başar T.  Linear-quadratic stochastic Stackelberg differential games for jump-diffusion systems.  SIAM J. Control Optim., 59 (2021), no. 2, 954–976.
  * [22] Mu Y.  Stackelberg-Nash equilibrium, social welfare and optimal structure in hierarchical continuous Public Goods game.  Systems Control Lett., 112 (2018), 1–8.
  * [23] Novak A.J., Feichtinger G. and Leitmann G.  A differential game related to terrorism: Nash and Stackelberg strategies.  J. Optim. Theory Appl., 144 (2010), no. 3, 533–555.
  * [24] Peng S. and Xu X.  BSDEs with random default time and their applications to default risk.  arXiv preprint arXiv:0910.2091, 2009.
  * [25] Pham H.  Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management.  Stochastic Process. Appl., 120 (2010), no. 9, 1795–1820.
  * [26] Si K. and Wu Z.  Backward-forward linear-quadratic mean-field Stackelberg games.  Adv. Difference Equ., https://doi.org/10.1186/s13662-021-03236-9, 73 (2021).
  * [27] Stackelberg H. V.  The Theory of the Market Economy.  Oxford University Press, London, 1952.
  * [28] Wang G., Wang Y. and Zhang S.  An asymmetric information mean-field type linear-quadratic stochastic Stackelberg differential game with one leader and two followers.  Optim. Control Appl. Meth., 41 (2020), no. 4, 1034–1051.
  * [29] Wang G. and Zhang S.  A mean-field linear-quadratic stochastic Stackelberg differential game with one leader and two followers.  J. Syst. Sci. Complex., 33 (2020), no. 5, 1383–1401.
  * [30] Wu, C., Zhao, Q. and Xi, M.  A retailer-supplier supply chain model with trade credit default risk in a supplier-Stackelberg game.  Comput. Ind. Eng., 112 (2017), no. 5, 568–575.
  * [31] Yong J. M. and Zhou X. Y.  Stochastic Controls: Hamiltonian Systems and HJB Equations.  Springer, New York, 1999.
  * [32] Zhang F., Dong Y. and Meng Q.  Backward stochastic riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients.  arXiv: 1808.08336, 2018.
  * [33] Zou S., Ma Z., Peng W. and Liu X.  A retailer-supplier supply chain model with trade credit default risk in a supplier-Stackelberg game.  Int. J. Control, 93 (2020), no. 12, 2804–2813.