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A Linear-quadratic Mean-Field Stochastic Stackelberg Differential Game with Random Exit Time

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.


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]).
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 τ and deal with the emergency. In this case the objective function of the government becomes where y(τ ) represents the discounted value of x(τ ), i.e., the government suffers a loss at time τ ∈ [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.

Problem Formulation
Consider the linear MF-SDE system in the complete probability space (Ω, F, F t , P) satisfying the usual hypothesis as follows: where T > 0 is a finite time duration; τ 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 P 1 and P 2 , respectively; X(t) is the mean-field term with the σ-algebra F = (F t ) t≥0 , generated by the standard one-dimensional Brownian motion B t , is rightcontinuous and increasing; x 0 is an F 0 -measurable and square integrable random variable; F is the smallest right continuous extension in which τ 0 becomes an F-stopping time, i.e., for all s > 0.
Clearly, τ = τ 0 ∧ T ∈ [0, T ] is also an F-stopping time. Noticing that when τ 0 = τ (i.e., the leader stops her action before the game ends), the controller v 0 contributes nothing on the time interval (τ, T ]. For convenience, we set i = 1, 2, j = 0, 1, 2 and L 2 F (0, T ; R) the space of all 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 P i and the leader are given by and respectively. Here Y (t) is the solution to the following MF-SDE with single jump at t = τ .
, while there is a penalty at time τ 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: The followers would like to find the Nash equilibrium point Considering that the followers would take strategy (v * 1 , v * 2 ), the leader expects to find v * 0 ∈ V[0, τ ] such that 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 θ ∈ {a, a, c, c}, ϑ i ∈ {b i , p i , p i , q i , r i }, suppose that θ and ϑ i have the following decompositions.
In addition, suppose that the following conditions hold. (B) p 0 , p 0 , q 0 , r 0 , p i,j , p i,j , q i,j , r i,j are all deterministic, continuous and positive functions on [0, T ]; (D) (Hypothesis (H)) Every càdlàg F -martingale remains an F-martingale; (E) Suppose that there exists an F-predictable (respectively, F -predictable) process γ F (respectively, γ F ) with γ F (s) = I s<τ γ F (s) such that is an F-martingale with jump time τ . The process γ F (respectively, γ F ) is called the F-intensity (respectively, F -intensity) of τ . In addition, we suppose that γ F is upper bounded.

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 ∈ [0, τ ] (the first stage) and t ∈ [τ, T ] (the second stage): 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: and Let According to (5) and Theorem 4.1 in [11], we know that Assume that followers take strategy ( v 1 , 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.

Nash equilibrium for followers at the second stage
In this subsection, we apply the Pontryagin-type maximum principle to find Nash equilibrium point for followers at the second stage. We restate problem (8) as follows.
To solve this subproblem, we define the Hamiltonian function of follower P i at second stage by By the sufficient and necessary maximum principle of MF-SDEs with default (see [11]), we have where P v * i,2 is the solution to corresponding linear BSDE. This leads to the following theorem.
Similarly, ϕ 2 satisfies the following Riccati equation: Repeating the above arguments, we can obtain the existence and uniqueness of solutions ϕ 2,1 and ϕ 2,2 .
Then (16) can be rewritten as 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: Observing that (18) is linear, we set Substituting (19) into (18) yields We also need the following lemma.

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: Similar to Subsection 3.1, we can define the Hamiltonian function of follower P i at first stage by Recalling Remark 3.1 and using the sufficient and necessary maximum principles of MF-SDEs with default (see [11]), we have where P v * i,1 is the solution to corresponding linear BSDE. Now we summarize the above arguments as the following theorem.

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.
where the state process Y (t) is governed by To solve Subproblem 3.3, we define the Hamiltonian function of the leader by setting where (P v0 , Q v0 , R v0 , Z v0 ) is the solution to the following MF-FBSDE: Here µ T 1 and ν T 1 represent the transposed matrixes of µ 1 and ν 1 , respectively. It is easy to show that v 0 becomes an optimal control of Subproblem 3.2 when where ∇ v0 is the Fréchet derivative with respect to v 0 . It remains to verify that, for the following inequality holds: Noticing that Substituting (30) and (32) into (31) obtains 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 Here is a solution to the following linear MF-FBSDE: Remark 3.2. If x 0 ∈ R, then the conditional mean-field term X(t) reduces to the classical mean-field term. If τ > T a.s., then the default event will not happen before time T . Moreover, if x 0 ∈ R and τ > T a.s., then Theorem 3.4 reduces to Theorem 3.4 in [29]. (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. 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.

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 E[a(t)Y (t)E [X(t)]] = E[X(t) [a(t)Y (t)]] when a(t)
is an F t -adapted process. However, by employing the fact that E[a(t)Y (t)E X(t) F s F s ] = E[X(t) a(t)Y (t) F s F s ] (t ≥ 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.