Kawasaki like disease in SARS-CoV-2 infected children – a key role for neutrophil and macrophage extracellular traps

1.   Introduction

Time inconsistency in dynamic decision making is often observed in social systems and daily life. Motivated by practical applications, especially in mathematical economics and finance, time-inconsistency control problems have recently attracted considerable research interest and efforts attempting to seek equilibrium, instead of optimal controls. At a conceptual level, the idea is that a decision made by the controller at every instant of time is considered as a game against all the decisions made by the future incarnations of the controller. An "equilibrium" control is therefore one such that any deviation from it at any time instant will be worse off. The study on time inconsistency by economists can be dated back to Stroz ^[1] and Phelps ^[2,3] in models with discrete time (see ^[4] and ^[5] for further developments), and adapted by Karp ^[6,7], and by Ekeland and Lazrak ^{[8,9,10,11,12,13]} to the case of continuous time. In the LQ control problems, Yong ^[14] studied a time-inconsistent deterministic model and derived equilibrium controls via some integral equations.

It is natural to study time inconsistency in the stochastic models. Ekeland and Pirvu ^[15] studied the non-exponential discounting which leads to time inconsistency in an agent's investment-consumption policies in a Merton model. Grenadier and Wang ^[16] also studied the hyperbolic discounting problem in an optimal stopping model. In a Markovian systems, Björk and Murgoci ^[17] proposed a definition of a general stochastic control problem with time inconsistent terms, and proposed some sufficient condition for a control to be solution by a system of integro-differential equations. They constructed some solutions for some examples including an LQ one, but it looks very hard to find not-to-harsh condition on parameters to ensure the existence of a solution. Björk, Murgoci and Zhou ^[18] also constructed an equilibrium for a mean-variance portfolio selection with state-dependent risk aversion. Basak and Chabakauri ^[19] studied the mean-variance portfolio selection problem and got more details on the constructed solution. Hu, Jin and Zhou ^[20,21] studied the general LQ control problem with time inconsistent terms in a non-Markovian system and constructed an unique equilibrium for quite general LQ control problem, including a non-Markovian system.

To the best of our knowledge, most of the time-inconsistent problems are associated with the control problems though we use the game formulation to define its equilibrium. In the problems of game theory, the literatures about time inconsistency is little ^[22,23]. However, the definitions of equilibrium strategies in the above two papers are based on some corresponding control problems like before. In this paper, we formulate a general stochastic LQ differential game, where the objective functional of each player include both a quadratic term of the expected state and a state-dependent term. These non-standard terms each introduces time inconsistency into the problem in somewhat different ways. We define our equilibrium via open-loop controls. Then we derive a general sufficient condition for equilibrium strategies through a system of forward-backward stochastic differential equations (FBSDEs). An intriguing feature of these FBSDEs is that a time parameter is involved; so these form a flow of FBSDEs. When the state process is scalar valued and all the coefficients are deterministic functions of time, we are able to reduce this flow of FBSDEs into several Riccati-like ODEs. Comparing to the ODEs in ^[20], though the state process is scalar valued, the unknowns are matrix-valued because of two players. Therefore, such ODEs are harder to solve than those of ^[20]. Under some more stronger conditions, we obtain explicitly an equilibrium strategy, which turns out to be a linear feedback. We also prove that the equilibrium strategy we obtained is unique.

The rest of the paper is organized as follows. The next section is devoted to the formulation of our problem and the definition of equilibrium strategy. In Section 3, we apply the spike variation technique to derive a flow of FBSEDs and a sufficient condition of equilibrium strategies. Based on this general results, we solve in Section 4 the case when the state is one dimensional and all the coefficients are deterministic. The uniqueness of such equilibrium strategy is also proved in this section.

2.   Problem setting

Let $T > 0$ be the end of a finite time horizon, and let $(W_t)_{0\le t\le{T}} = (W_t^1, ..., W_t^d)_{0\le t\le{T}}$ be a $d$-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Denote by $(\mathcal{F}_t)$ the augmented filtration generated by $(W_t)$.

Let $\mathbb{S}^n$ be the set of symmetric $n\times n$ real matrices; $L_{\mathcal{F}}^2(\Omega, \mathbb{R}^l)$ be the set of square-integrable random variables; $L_{\mathcal{F}}^2(t, T;\mathbb{R}^n)$ be the set of $\{\mathcal{F}_s\}_{s\in[t, T]}$-adapted square-integrable processes; and $L_{\mathcal{F}}^2(\Omega; C(t, T;\mathbb{R}^n))$ be the set of continuous $\{\mathcal{F}_s\}_{s\in[t, T]}$-adapted square-integrable processes.

We consider a continuous-time, $n$-dimensional nonhomogeneous linear controlled system:

Here $A$ is a bounded deterministic function on $[0, T]$ with value in $\mathbb{R}^{n\times n}$. The other parameters $B_1, B_2, C, D_1, D_2$ are all essentially bounded adapted processes on $[0, T]$ with values in $\mathbb{R}^{l\times n}, \mathbb{R}^{l\times n}, \mathbb{R}^{n\times n}, \mathbb{R}^{n\times l}, \mathbb{R}^{n\times l}$, respectively; $b$ and $\sigma^j$ are stochastic processes in $L^2_{\mathcal{F}}(0, T;\mathbb{R}^n)$. The processes $u_i\in L^2_{\mathcal{F}}(0, T;\mathbb{R}^l), \ i = 1, 2$ are the controls, and $X$ is the state process valued in $\mathbb{R}^{n}$. Finally, $x_0\in\mathbb{R}^{n}$ is the initial state. It is obvious that for any controls $u_i\in L^2_{\mathcal{F}}(0, T;\mathbb{R}^l), \ i = 1, 2$, there exists a unique solution $X\in L^2_{\mathcal{F}}(\Omega, C(0, T;\mathbb{R}^n))$.

As time evolves, we need to consider the controlled system starting from time $t\in[0, T]$ and state $x_t\in L^2_{\mathcal{F}_t}(\Omega; \mathbb{R}^n)$:

For any controls $u_i\in L^2_{\mathcal{F}}(0, T;\mathbb{R}^l), \ i = 1, 2$, there exists a unique solution $X^{t, x_t, u_1, u_2}\in L^2_{\mathcal{F}}(\Omega, C(0, T;\mathbb{R}^n))$.

We consider a two-person differential game problem. At any time $t$ with the system state $X_t = x_t$, the $i$-th ($i = 1, 2$) person's aim is to minimize her cost (if maximize, we can times the following function by $-1$):

over $u_1, u_2\in L_{\mathcal{F}}^2(t, T;\mathbb{R}^l)$, where $X = X^{t, x_t, u_1, u_2}$, and $\mathbb{E}_t[\cdot] = \mathbb{E}[\cdot|\mathcal{F}_t]$. Here, for $i = 1, 2$, $Q_i$ and $R_i$ are both given essentially bounded adapted process on $[0, T]$ with values in $\mathbb{S}^n$ and $\mathbb{S}^l$, respectively, $G_i, h_i, \lambda_i, \mu_i$ are all constants in $\mathbb{S}^n$, $\mathbb{S}^n$, $\mathbb{R}^{n\times n}$ and $\mathbb{R}^n$, respectively. Furthermore, we assume that $Q_i, R_i$ are non-negative definite almost surely and $G_i$ are non-negative definite.

Given a control pair $(u_1^*, u_2^*)$. For any $t\in[0, T), \epsilon > 0$, and $v_1, v_2\in L_{\mathcal{F}_t}^2(\Omega, \mathbb{R}^l)$, define

Because each person at time $t > 0$ wants to minimize his/her cost as we claimed before, we have

Definition 2.1. Let $(u_1^*, u_2^*)\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)\times L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)$ be a given strategy pair, and let $X^*$ be the state process corresponding to $(u_1^*, u_2^*)$. The strategy pair $(u_1^*, u_2^*)$ is called an equilibrium if

where $u_i^{t, \epsilon, v_i}, i = 1, 2$ are defined by (2.4), for any $t\in[0, T)$ and $v_1, v_2\in L_{\mathcal{F}_t}^2(\Omega, \mathbb{R}^l)$.

Remark. The above definition means that, in each time $t$, the equilibrium is a static Nash equilibrium in a corresponding game.

3.   Main result

Let $(u_1^*, u_2^*)$ be a fixed strategy pair, and let $X^*$ be the corresponding state process. For any $t\in[0, T)$, as a similar arguments of Theorem 5.1 in pp. 309 of ^[24], defined in the time interval $[t, T]$, there exist adapted processes $(p_i(\cdot; t), (k_i^j(\cdot; t)_{j = 1, 2, ..., d}))\in L_{\mathcal{F}}^2(t, T;\mathbb{R}^n)\times(L_{\mathcal{F}}^2(t, T;\mathbb{R}^n))^d$ and $(P_i(\cdot; t), (K_i^j(\cdot; t)_{j = 1, 2, ..., d}))\in L_{\mathcal{F}}^2(t, T;\mathbb{S}^n)\times(L_{\mathcal{F}}^2(t, T;\mathbb{S}^n))^d$ for $i = 1, 2$ satisfying the following equations:

for $i = 1, 2$. From the assumption that $Q_i$ and $G_i$ are non-negative definite, it follows that $P_i(s; t)$ are non-negative definite for $i = 1, 2$.

Proposition 1. For any $t\in[0, T), \epsilon > 0$, and $v_1, v_2\in L_{\mathcal{F}_t}^2(\Omega, \mathbb{R}^l)$, define $u_i^{t, \epsilon, v_i}, i = 1, 2$ by (2.4). Then

where $\Lambda_i(s; t) = B_{i, s}p_i(s; t)+\sum_{j = 1}^d(D_{i, s}^j)'k_i^j(s; t)+R_{i, s}u_{i, s}^*$ and $H_i(s; t) = R_{i, s}+\sum_{j = 1}^d(D_{i, s}^j)'P_i(s; t)D_{i, s}^j$ for $i = 1, 2$.

Proof. Let $X^{t, \epsilon, v_1, v_2}$ be the state process corresponding to $u_i^{t, \epsilon, v_i}, i = 1, 2$. Then by standard perturbation approach (cf. ^[20,25] or pp. 126-128 of ^[24]), we have

where $Y\equiv Y^{t, \epsilon, v_1, v_2}$ and $Z\equiv Z^{t, \epsilon, v_1, v_2}$ satisfy

Moreover, by Theorem 4.4 in ^[24], we have

With $A$ being deterministic, it follows from the dynamics of $Y$ that, for any $s\in[t, T]$, we have

Hence we conclude that

By these estimates, we can calculate

Recalling that $(p_i(\cdot; t), k_i(\cdot; t))$ and $(P_i(\cdot; t), K_i(\cdot; t))$ solve, respectively, BSDEs (3.1) and (3.2) for $i = 1, 2$, we have

and

Combining (3.11)-(3.13), we have

Take $i = 1$, we let $v_2 = 0$, then $u_2^{t, \epsilon, v_2} = u_2^*$, from (3.14), we obtain

This proves (3.3), and similarly, we obtain (3.4).

Because of $R_{i, s}$ and $P_i(s; t), i = 1, 2$ are non-negative definite, $H_i(s; t), \ i = 1, 2$ are also non-negative definite. In view of (3.3)-(3.4), a sufficient condition for an equilibrium is

By an arguments similar to the proof of Proposition 3.3 in ^[21], we have the following lemma:

Lemma 3.1. For any triple of state and control processes $(X^*, u_1^*, u_2^*)$, the solution to BSDE (3.1) in $L^2(0, T;\mathbb{R}^n)\times (L^2(0, T;\mathbb{R}^n))^d$ satisfies $k_i(s; t_1) = k_i(s; t_2)$ for a.e. $s\ge\max\{t_1, t_2\}, \; i = 1, 2$. Furthermore, there exist $\rho_i\in L^2(0, T;\mathbb{R}^l)$, $\delta_i\in L^2(0, T;\mathbb{R}^{l\times n})$ and $\xi_i\in L^2(\Omega; C(0, T;\mathbb{R}^n))$, such that

Therefore, we have another characterization for equilibrium strategies:

Proposition 2. Given a strategy pair $(u_1^*, u_2^*)\in L^2(0, T;\mathbb{R}^l)\times L^2(0, T;\mathbb{R}^l)$. Denote $X^*$ as the state process, and $(p_i(\cdot; t), (k_i^j(\cdot; t)_{j = 1, 2, ..., d}))\in L_{\mathcal{F}}^2(t, T;\mathbb{R}^n)\times(L_{\mathcal{F}}^2(t, T;\mathbb{R}^n))^d$ as the unique solution for the BSDE (3.1), with $k_i(s) = k_i(s; t)$ according to Lemma 3.1 for $i = 1, 2$ respectively. For $i = 1, 2$, letting

then $u^*$ is an equilibrium strategy if and only if

Proof. From (3.17), we have $\Lambda_1(s; t) = \rho_1(s)+\delta_1(s)\xi_1(t)$. Since $\delta_1$ is essentially bounded and $\xi_1$ is continuous, we have

and hence

Therefore, if (3.19) holds, we have

When $i = 2$, we can prove (3.19) similarly.

Conversely, if (3.16) holds, then $\lim_{\epsilon\downarrow0}{1\over\epsilon}\int_t^{t+\epsilon}\mathbb{E}_t[\Lambda_i(s; s)]ds = 0, i = 1, 2$ leading to (3.19) by virtue of Lemma 3.4 of ^[21].

The following is the main general result for the stochastic LQ differential game with time-inconsistency.

Theorem 3.2. A strategy pair $(u_1^*, u_2^*)\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)\times L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)$ is an equilibrium strategy pair if the following two conditions hold for any time $t$:

(i) The system of SDEs

admits a solution $(X^*, p_1, k_1, p_2, k_2)$;

(ii) $\Lambda_i(s; t) = R_{i, s}u_{i, s}^*+B_{i, s}p_i(s; t)+\sum_{j = 1}^d(D_{i, s}^j)'k_i^j(s; t), i = 1, 2$ satisfy condition (3.19).

Proof. Given a strategy pair $(u_1^*, u_2^*)\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)\times L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)$ satisfying (i) and (ii), then for any $v_1, v_2\in L_{\mathcal{F}_t}^2(\Omega, \mathbb{R}^l)$, define $\Lambda_i, H_i, i = 1, 2$ as in Proposition 1. We have

proving the first condition of Definition 2.1, and the proof of the second condition is similar.

Theorem 3.2 involve the existence of solutions to a flow of FBSDEs along with other conditions. The system (3.20) is more complicated than system (3.6) in ^[20]. As declared in ^[20], "proving the general existence for this type of FBSEs remains an outstanding open problem", it is also true for our system (3.20).

In the rest of this paper, we will focus on the case when $n = 1$. When $n = 1$, the state process $X$ is a scalar-valued rocess evolving by the dynamics

where $A$ is a bounded deterministic scalar function on $[0, T]$. The other parameters $B, C, D$ are all essentially bounded and $\mathcal{F}_t$-adapted processes on $[0, T]$ with values in $\mathbb{R}^l, \mathbb{R}^d, \mathbb{R}^{d\times l}$, respectively. Moreover, $b\in L_{\mathcal{F}}^2(0, T;\mathbb{R})$ and $\sigma\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^d)$.

In this case, the adjoint equations for the equilibrium strategy become

for $i = 1, 2$. For convenience, we also state here the $n = 1$ version of Theorem 3.2:

Theorem 3.3. A strategy pair $(u_1^*, u_2^*)\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)\times L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)$ is an equilibrium strategy pair if, for any time $t\in[0, T)$,

(i) The system of SDEs

admits a solution $(X^*, p_1, k_1, p_2, k_2)$;

(ii) $\Lambda_i(s; t) = R_{i, s}u_{i, s}^*+B_{i, s}p_i(s; t)+(D_{i, s})'k_i(s; t), i = 1, 2$ satisfy condition (3.19).

4.   The deterministic coefficients case

The unique solvability of (3.25) remains a challenging open problem even for the case $n = 1$. However, we are able to solve this problem when the parameters $A, B_1, B_2, C, D_1, D_2, b, \sigma, Q_1, Q_2, R_1$ and $R_2$ are all deterministic functions.

Throughout this section we assume all the parameters are deterministic functions of $t$. In this case, since $G_1, G_2$ have been also assumed to be deterministic, the BSDEs (3.24) turns out to be ODEs with solutions $K_i\equiv0$ and $P_i(s; t) = G_ie^{\int_s^T(2A_u+|C_u|^2)du}+\int_s^Te^{\int_s^T(2A_u+|C_u|^2)du}Q_{i, v}dv$ for $i = 1, 2$. Hence, the equilibrium strategy will be characterized through a system of coupled Riccati-type equations.

4.1.   The uniqueness of the equilibrium strategy

As in classical LQ control, we attempt to look for a linear feedback equilibrium strategy pair. For such purpose, motivated by ^[20], given any $t\in[0, T]$, we consider the following process:

where $M_i, N_i, \Gamma_i, \Phi_i$ are deterministic differentiable functions with $\dot{M}_i = m_i, \dot{N}_i = n_i, \dot{\Gamma}_i = \gamma_i$ and $\dot\Phi_i = \phi_i$ for $i = 1, 2$. The advantage of this process is to separate the variables $X_s^*, \mathbb{E}_t[X_s^*]$ and $X_t^*$ in the solutions $p_i(s; t), i = 1, 2$, thereby reducing the complicated FBSDEs to some ODEs.

For any fixed $t$, applying Ito's formula to (4.1) in the time variable $s$, we obtain, for $i = 1, 2$,

Comparing the $dW_s$ term of $dp_i(s; t)$ in (3.25) and (4.2), we have

Notice that $k(s; t)$ turns out to be independent of $t$.

Putting the above expressions (4.1) and (4.3) of $p_i(s; t)$ and $k_i(s; t), i = 1, 2$ into (3.19), we have

for $i = 1, 2$. Then we can formally deduce

Let $M_s = {{{\text{diag}}}}(M_{1, s}I_l, M_{2, s}I_l), N_s = {{{\text{diag}}}}(N_{1, s}I_l, N_{2, s}I_l), \Gamma_s = {{{\text{diag}}}}(\Gamma_{1, s}I_l, \Gamma_{2, s}I_l), \Phi_s = {{{\text{diag}}}}(\Phi_{1, s}I_l, \Phi_{2, s}I_l)$, $R_s = {{{\text{diag}}}}(R_{1, s}, R_{2, s}), B_s = \left(\begin{matrix}B_{1, s}\\ B_{2, s}\end{matrix}\right), D_s = \left(\begin{matrix}D_{1, s}, \; D_{2, s}\end{matrix}\right)$, $u_s^* = \left(\begin{matrix}u_{1, s}^*\\ u_{2, s}^*\end{matrix}\right), \alpha_s = \left(\begin{matrix}\alpha_{1, s}\\ \alpha_{2, s}\end{matrix}\right)$ and $\beta_s = \left(\begin{matrix}\beta_{1, s}\\ \beta_{2, s}\end{matrix}\right)$. Then from (4.4), we have

and hence

Next, comparing the $ds$ term of $dp_i(s; t)$ in (3.25) and (4.2) (we supress the argument $s$ here), we have

Notice in the above that $X^* = X_s^*$ and $\mathbb{E}_t[X^*] = \mathbb{E}_t[X_s^*]$ due to the omission of $s$. This leads to the following equations for $M_i, N_i, \Gamma_i, \Phi_i$:

Though $M_i, N_i, \Gamma_i, \Phi_i, i = 1, 2$ are scalars, $M, N, \Gamma, \Phi$ are now matrices because of two players. Therefore, the above equations are more complicated than the similar equations (4.5)–(4.8) in ^[20]. Before we solve the equations (4.10)–(4.13), we first prove that, if exist, the equilibrium constructed above is the unique equilibrium. Indeed, we have

Theorem 4.1. Let

and

Suppose all the parameters $A, B_1, B_2, C, D_1, D_2, b, \sigma, Q_1, Q_2, R_1$ and $R_2$ are all deterministic.

When $(M_i, N_i, \Gamma_i, \Phi_i), i = 1, 2$ exist, and for $i = 1, 2$, $(p_i(s; t), k_i(s; t))\in\mathcal{L}_1\times \mathcal{L}_2$, the equilibrium strategy is unique.

Proof. Suppose there is another equilibrium $(X, u_1, u_2)$, then the BSDE (3.1), with $X^*$ replaced by $X$, admits a solution $(p_i(s; t), k_i(s), u_{i, s})$ for $i = 1, 2$, which satisfies $B_{i, s}p_i(s; s)+D_{i, s}'k_i(s)+R_{i, s}u_{i, s} = 0$ for a.e. $s\in[0, T]$. For $i = 1, 2$, define

where $k_i(s) = k_i(s; t)$ by Lemma 3.1.

We define $p(s; t) = {{{\text{diag}}}}(p_1(s; t)I_l, p_2(s; t)I_l)$, $\bar{p}(s; t) = {{{\text{diag}}}}(\bar{p}_1(s; t)I_l, \bar{p}_2(s; t)I_l)$, and $u = \left(\begin{matrix}u_{1, s}\\ u_{2, s}\end{matrix}\right)$. By the equilibrium condition (3.19), we have

Since $R_s+M_sD_s'D_s$ is invertible, we have

and hence for $i = 1, 2$,

where we suppress the subscript $s$ for the parameters, and we have used the equations (4.10)–(4.13) for $M_i, N_i, \Gamma_i, \Phi_i$ in the last equality. From (4.16) and (4.17), we have $(\bar{p}_i, \bar{k}_i)\in\mathcal{L}_1\times\mathcal{L}_2$. Therefore, by Theorem 4.2 of ^[21], we obtain $\bar{p}(s; t)\equiv0$ and $\bar{k}(s)\equiv0$.

Finally, plugging $\bar{p}\equiv\bar{k}\equiv0$ into $u$ of (4.19), we get $u$ being the same form of feedback strategy as in (4.5), and hence $(X, u_1, u_2)$ is the same as $(X^*, u_1^*, u_2^*)$ which defined by (4.5) and (3.25).

4.2.   Existence of the equilibrium strategies

The solutions to (4.12) is

for $i = 1, 2$. Let $\tilde{N} = N_1/ N_2$, from (4.11), we have $\dot{\tilde{N}} = 0$, and hence

Equations (4.10) and (4.11) form a system of coupled Riccati-type equations for $(M_1, M_2, N_1)$:

Finally, once we get the solution for $(M_1, M_2, N_1)$, (4.13) is a simple ODE. Therefore, it is crucial to solve (4.23).

Formally, we define $\tilde{M} = {M_1\over M_2}$ and $J_1 = {M_1\over N_1}$ and study the following equation for $(M_1, \tilde{M}, J_1)$:

where $M = {{{\text{diag}}}}(M_1I_l, {M_1\over\tilde M}I_l), N = {{{\text{diag}}}}({M_1\over J_1}I_l, {h_2\over h_1}{M_1\over J_1}I_l)$ and $\Gamma = {{{\text{diag}}}}(\lambda_1e^{\int_s^T A_tdt}I_l, \lambda_2e^{\int_s^T A_tdt}I_l)$.

By a direct calculation, we have

Proposition 3. If the system (4.24) admits a positive solution $(M_1, \tilde{M}, J_1)$, then the system (4.23) admits a solution $(M_1, M_2, N_1)$.

In the following, we will use the truncation method to study the system (4.24). For convenienc, we use the following notations:

Moreover, for a matrix $M\in\mathbb{R}^{m\times n}$ and a real number $c$, we define

We first consider the standard case where $R-\delta{I}\succeq0$ for some $\delta > 0$. We have

Theorem 4.2. Assume that $R-\delta{I}\succeq0$ for some $\delta > 0$ and $G\ge h > 0$. Then (4.24), and hence (4.23) admit unique solution if

(i) there exists a constant $\lambda\ge0$ such that $B = \lambda D'C$;

(ii) $\frac{|C|^2}{2l}D'D-(\lambda+1)D'CC'D\succeq0$.

Proof. For fixed $c > 0$ and $K > 0$, consider the following truncated system of (4.24):

where $M_{c}^+ = {{{\text{diag}}}}((M_1\vee0)I_l, {{M_1\vee0}\over\tilde M\vee c}I_l)$, $N_c^+ = {{{\text{diag}}}}({{M_1\vee0}\over J_1\vee c}I_l, {h_2\over h_1}{{M_1\vee0}\over J_1\vee c}I_l)$ and

Since $R-\delta I\succeq0$, the above system (4.29) is locally Lipschitz with linear growth, and hence it admits a unique solution $(M_1^{c, K}, \tilde{M}^{c, K}, J_1^{c, K})$. We will omit the superscript $(c, K)$ when there is no confusion.

We are going to prove that $J_1\ge1$ and that $M_1, \tilde{M}\in[L_1, L_2]$ for some $L_1, L_2 > 0$ independent of $c$ and $K$ appearing in the truncation functions. We denote

Then $\lambda^{(2)}$ is bounded, and $M_1$ satisfies

Hence $M_1 > 0$. Similarly, we have $\tilde{M} > 0$.

The equation for $\tilde{M}$ is

hence $\tilde{M}$ admits an upper bound $L_2$ independent of $c$ and $K$. Choosing $K = L_2$ and examining again (4.33), we deduce that there exists $L_1 > 0$ independent of $c$ and $K$ such that $\tilde{M}\ge L_1$. Indeed, we can choose $L_1 = \min_{0\le t\le T}{Q_{1, t}\over Q_{2, t}}\wedge{G1\over G_2}$ and $L_2 = \max_{0\le t\le T}{Q_{1, t}\over Q_{2, t}}\vee{G1\over G_2}$. As a result, choosing $c < L_1$, the terms $M_c^+$ can be replaced by $M = {{{\text{diag}}}}(M_1I_l, {M_1\over\tilde M}I_l)$, respectively, in (4.29) without changing their values.

Now we prove $J\ge1$. Denote $\tilde{J} = J_1-1$, then $\tilde{J}$ satisfies the ODE:

where

with $H = {|C|^2+Q_1/(M_1\vee c)\over2l}(R+D'DM)-(\lambda+1)D'CC'DM$.

When $c$ is small enough such that $R-cD'D\succeq0$, we have

Hence,

and consequently $a^{(1)}\ge tr\{(R+MD'D)^{-1}H\}\ge0$. We then deduce that $\tilde{J}\ge0$, and hence $J_1\ge1$. The boundness of $M_1$ can be proved by a similar argument in the proof of Theorem 4.2 in ^[20].

Similarly, for the singular case $R\equiv0$, we have

Theorem 4.3. Given $G_1\ge h_1\ge1, R\equiv0$, if $B = \lambda D'C$ and $|C|^2-(\lambda+1)C'D(D'D)^{-1}D'C\ge0$, then (4.24) and (4.23) admit a unique positive solution.

Concluding the above two theorems, we can present our main results of this section:

Theorem 4.4. Given $G_1\ge h_1\ge1$ and $B = \lambda D'C$. The (4.23) admits a unique positive solution $(M_1, M_2, N_1)$ in the following two cases:

(i) $R-\delta I\succeq0$ for some $\delta > 0$, $\frac{|C|^2}{2l}D'D-(\lambda+1)D'CC'D\succeq0$;

(ii) $R\equiv0$, $|C|^2-(\lambda+1)C'D(D'D)^{-1}D'C\ge0$.

Proof. Define $p_i(s; t)$ and $k_i(s; t)$ by (4.1) and (4.3), respectively. It is straightforward to check that $(u_1^*, u_2^*, X^*, p_1, p_2, k_1, k_2)$ satisfies the system of SDEs (3.25). Moreover, in the both cases, we can check that $\alpha_{i, s}$ and $\beta_{i, s}$ in (4.5) are all uniformly bounded, and hence $u_i^*\in L_{\mathcal{F}}^2(0, T;\mathbb{R}^l)$ and $X^*\in L^2(\Omega; C(0, T;\mathbb{R}))$.

Finally, denote $\Lambda_i(s; t) = R_{i, s}u_{i, s}^*+p_i(s; t)B_{i, s}+(D_{i, s})'k_i(s; t), i = 1, 2$. Plugging $p_i, k_i, u_i^*$ define in (4.1), (4.3) and (4.5) into $\Lambda_i$, we have

and hence,

Therefore, $\Lambda_i$ satisfies the seond condition in (3.19).

5.   Conclusions

We investigate a general stochastic linear-quadratic differential game, where the objective functional of each player include both a quadratic term of the expected state and a state-dependent term. As discussed in detail in Björk and Murgoci ^[17] and ^[18], the last two terms in each objective functional, respectively, introduce two sources of time inconsistency into the differential game problem. That is to say, the usual equilibrium aspect is not a proper way when the players at 0 cannot commit the players at all intermediate times to implement the decisions they have planed. With the time-inconsistency, the notion "equilibrium" needs to be extended in an appropriate way. We turn to adopt the concept of equilibrium strategy between the players at all different times, which is at any time, an equilibrium "infinitesimally" via spike variation. By applying the spike variation technique, We derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. The unique solvability of such FBSDEs remains a challenging open problem.

For a special case, when the state is one-dimensional and the coefficients are all deterministic, the equilibrium strategy will be characterized through a system of coupled Riccati-type equations. At last, we find an explicit equilibrium strategy, which is also proved be the unique equilibrium strategy.

Acknowledgments

The research of the first author was partially supported by NSFC (No.12171426), the Natural Science Foundation of Zhejiang Province (No. Y19A010020) and the Fundamental Research Funds for the Central Universities (No. 2021FZZX001-01). The research of the second author was partially supported by NSFC (No. 11501325, No.71871129) and the China Postdoctoral Science Foundation (Grant No. 2018T110706, No.2018M642641). The authors would like to thank sincerely the referees and the associate editor for their helpful comments and suggestions.

Conflict of interest

The authors declare there is no conflicts of interest.