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A Stackelberg reinsuranceinvestment game with derivatives trading
Boundary Value Problems volumeÂ 2023, ArticleÂ number:Â 43 (2023)
Abstract
This paper studies a stochastic Stackelberg differential reinsuranceinvestment game with derivatives trading under a stochastic volatility model. The reinsurer who occupies a monopoly position can price a reinsurance premium and invest her wealth in the financial market consisting of a riskless asset and a stock and derivatives tied to the stock. The insurer, the follower of the Stackelberg game, purchases proportional reinsurance from the reinsurer and invests in the same financial market. The main target of the reinsurer and the insurer is to seek their own optimal strategy to maximize the CARA utility of the relative performance. An explicit equilibrium strategy with derivatives trading is deduced by solving HamiltonJacobiBellman (HJB) equations sequentially. The equilibrium investment strategy demonstrates that the insurer and the reinsurer imitate each otherâ€™s investment strategies, showing a herd effect. In numerical experiments, the sensitivity of the equilibrium strategy to model parameters is analyzed. For the optimal investment strategy, we find that a short position in the derivative may switch to a long position with parameters changing, which provides investors with important decisionmaking information.
1 Introduction
Reinsurance and investment are important means for insurers to manage financial risks and increase returns. In recent years, the research on the optimization of reinsurance and investment under various objectives has received extensive attention. Among them, for example, maximizing the expected utility of terminal wealth (Li etÂ al. [20], Huang etÂ al. [17], Zhao and Rong [28], etc.), minimizing the probability of bankruptcy (Browne [9], Chen etÂ al. [12], Li etÂ al. [19], etc.), and the related research under meanvariance criterion (Bi etÂ al. [8], Zhou etÂ al. [29], Bai etÂ al. [3], etc.).
Most of the above studies are from the perspective of insurer to investigate the optimal reinsuranceinvestment strategy. However, since the reinsurance contract is signed by both the reinsurer and the insurer, unilateral reinsurance optimization for the insurer may not be practical. In reality, the interests of both parties should be considered in the reinsurance contract. Under Stackelberg differential game framework, the reinsurer that occupies a monopoly position is usually regarded as the leader, and the insurer is the follower. In this paper, we mainly study the premium pricing, the optimal investment strategy of reinsurers, and the optimal reinsuranceinvestment strategy of insurers the under stochastic volatility model.
Nowadays, more and more attention has been focused on the game in the insurance market. Many experts have studied the zerosum stochastic differential reinsuranceinvestment game (among them, Zeng [27], Taksar and Zeng [24], Li etÂ al. [18]) and nonzerosum stochastic differential reinsuranceinvestment game (among them, Bensoussan etÂ al. [7], Meng etÂ al. [23], Guan and Liang [15], Yan etÂ al. [26], Deng etÂ al. [14]). This kind of literature mostly studies the competitive game among insurers. In recent years, some experts have studied the stochastic Stackelberg differential game between the insurer and the reinsurer from the perspective of both parties. A stochastic Stackelberg differential reinsurance game is proposed by Chen and Shen [10] to describe the leaderfollower relationship between the reinsurer and the insurer. Chen and Shen [11] further investigated a stochastic Stackelberg differential reinsurance game problem with timeinconsistent mean variance. The above literature on stochastic Stackelberg differential game only considers the reinsurance game, not the investment game. In view of the fact that investment can increase the companyâ€™s earnings for stable operation, based on the CEV model, Bai etÂ al. [4] and Bai etÂ al. [5] studied the bilateral Stackelberg reinsuranceinvestment game and the multiparty hybrid reinsuranceinvestment game under different economic environment, respectively.
A basic assumption of the above research is that there is no opportunity to trade derivatives. Generally, the insurer and the reinsurer invest in stocks with stochastic volatility that is not completely related to the stock price, so the risk can not be hedged fully. Nowadays, derivatives trading is increasingly popular in the financial market (Ahn etÂ al. [2], Bakshi and Madan [6], Liu and Pan [22], Liu etÂ al. [21]). Many studies have shown that derivatives can complete the financial market and improve efficiency. Derivatives trading can provide differential exposure to the imperfect instantaneous correlation between stock returns and volatility. Recently, Xue etÂ al. [25] studied the optimal strategy for the insurer with a constant absolute risk aversion (CARA) who manages its business risk using not only stock investment and proportional reinsurance but also trading derivatives.
Based on this intuition, this paper investigates a stochastic Stackelberg differential reinsuranceinvestment game problem with derivatives trading under stochastic volatility models. The insurer and the reinsurer manage the risk by means of investing in stocks and proportional reinsurance and trading options. The reinsurer who occupies a monopoly position can determine the reinsurance premium pricing and its asset allocation strategy invested in the stock and the option. The insurer, the follower of the Stackelberg game, can determine the proportion of reinsurance according to the price of reinsurance premium and its asset allocation strategy. The explicit Stackelberg equilibrium strategies are deduced by maximizing the CARA utility of relative performance. From the form of investment strategies of the insurer and the reinsurer, we find that they imitate each otherâ€™s investment strategies, showing a herd effect. Furthermore, in numerical simulations, we analyze the sensitivity of the Stackelberg equilibrium strategy to model parameters.
This paper contributes to the existing research from at least the following two aspects. First, the price process of a risky asset described by Hestonâ€™s stochastic volatility model is considered to study the equilibrium strategies in the stochastic Stackelberg differential reinsurance and investment game. Second, the derivatives trading is considered in the Stackelberg game model, and we calculate the optimal asset allocation strategy of both parties of the game.
The remaining paper is constructed as follows. In Sect.Â 2, we describe the stochastic Stackelberg differential reinsurance and investment game model with derivatives trading and stochastic volatility. In Sect.Â 3, the equilibrium reinsurance and investment strategy is deduced. In Sect.Â 4, the numerical simulation is conducted. SectionÂ 5 is the conclusion.
2 Model setup
In this paper, we suppose the insurance market where there exists one reinsurer and one insurer. Denote the finite horizon by \([0,T]\) over which the investment and reinsurance behavior occurs. \((\Omega ,\mathcal{F},P )\) is a complete probability space, where \(\mathcal{F}= \{\mathcal{F}_{t} \}_{0\leq t\leq T}\) is a filtration.
2.1 Modeling the surplus processes
Referring to Browne [9], the insurerâ€™s risk claim process \(\{R_{F}(t), 0\leq t\leq T\}\) satisfies
where \(\lambda _{F}>0\) depicts the claim intensity, \(0<\mu _{F}<+\infty \) and \((\tilde{\sigma}_{F})^{2}<+\infty \). \(\{W_{F}(t),t\geq 0\}\) denotes the PBrownian process.
The insurerâ€™s premium rate is denoted as \(c_{F}\), and its calculation method adopts the expected value premium principle. Then, \(c_{F}=(1+\theta _{F})\lambda _{F}\mu _{F}\), where \(\theta _{F}>0\) is the insurerâ€™s safety loading. The insurerâ€™s reinsurance strategy is described by \(\{q(t),t\geq 0\}\) with \(q(t)\in [0,1]\). Then, the reinsurer will cover \((1q(t))100\%\) of the claims, and the insurer will cover the remaining. \(p(t)\in [c_{F},\bar{c}]\) denotes the price of the reinsurance premium at time t, where \(\bar{c}=(1+\bar{\theta})\lambda _{F}\mu _{F}\), and Î¸Ì„ is an upper bound of the reinsurerâ€™s relative safety loading.
Then, the insurerâ€™s surplus process, \(\{X_{F}(t), 0\leq t\leq T\}\) is as follows
where the insurerâ€™s initial surplus denoted by \(x_{F}^{0}>0\), \(a=\lambda _{F}\mu _{F}\), \(\sigma _{F}=\sqrt{\lambda _{F}(\tilde{\sigma}_{F})^{2}}\).
The reinsurerâ€™s surplus process, \(\{X_{L}(t), 0\leq t\leq T\}\) is as follows
where the reinsurerâ€™s initial surplus denoted by \(x_{L}^{0}>0\).
2.2 Modeling the financial asset price
We consider a financial market consisting of one riskless asset, one risky asset, and the derivative security. \(r_{0}\) is the riskless interest rate. The price model of the riskless asset, \(\{S_{0}(t)\}_{t\geq 0}\), is
The risky asset represents the aggregate equity market. Assuming the stock price, \(\{S_{1}(t)\}_{t\geq 0}\), is described by Hestonâ€™s stochastic volatility model (refer to Heston [16]),
where Î·, Îº, vÌ„, and \(\sigma _{1}\) are all positive constants; \(V(t)\) is the instantaneous variance; \(\rho \in (1,1)\) is the correlation between \(S_{1}(t)\) and \(V(t)\); \(\{W_{1}(t),t\geq 0\}\) and \(\{W_{2}(t),t\geq 0\}\) are independent PBrownian processes, and both are independent of \(\{W_{F}(t),t\geq 0\}\). Moreover, we assume \(2\kappa \bar{v}>\sigma _{1}^{2}\) as mentioned in Cox etÂ al. [13] to ensure that \(V(t)\) is almost surely nonnegative.
Denote the derivative price by \(S_{2}(t)=f(S_{1}(t),V(t))\) depending on the underlying stock price \(S_{1}(t)\) and the volatility \(V(t)\). Referring to Xue etÂ al. [25], then, the derivative price satisfies
where Î· and Î¾ are the risk premiums, \(f_{s_{1}}\neq 0\) and \(f_{v}\neq 0\) are the derivatives of f with respect to \(S_{1}\) and V.
2.3 Modeling the wealth processes
Assuming no taxes and transaction fees, shortselling is allowed. \(b_{F1}(t)\) and \(b_{L1}(t)\) are the dollar amount invested in the stock for the insurer and the reinsurer, respectively. \(b_{F2}(t)\) and \(b_{L2}(t)\) are the dollar amount invested in the derivative. And, the remaining wealth \(X_{F}^{\pi _{F}}(t)b_{F1}(t)b_{F2}(t)\) and \(X_{L}^{\pi _{L}}(t)b_{L1}(t)b_{L2}(t)\) are invested in the riskless asset. We write \(\pi _{F}(t)=(q(t),b_{F1}(t),b_{F2}(t))\) and \(\pi _{L}(t)=(p(t),b_{L1}(t),b_{L2}(t))\).
Denote
Thus, we obtain the wealth processes for the insurer and the reinsurer, respectively, as follows.
Next, we proceed to deduce the optimal risk exposures (\(B^{F_{1}}(t)\), \(B^{F_{2}}(t)\), \(B^{L_{1}}(t)\), \(B^{L_{2}}(t)\)) and then deduce the optimal investment strategies for the insurer and the reinsurer by relations (2.7) and (2.8).
2.4 Modeling the stochastic Stackelberg differential game
This paper takes the derivatives trading into account in the stochastic Stackelberg differential reinsuranceinvestment game model. Referring to Chen and Shen [10], Chen and Shen [11], Bai etÂ al. [4] and Bai etÂ al. [5], the main target of the game is to find the Stackelberg equilibrium by solving the leaderâ€™s (reinsurer) and followerâ€™s (insurer) optimization problems sequentially. The game problem can be solved by the following procedure:

StepÂ 1: The reinsurer moves first by announcing its any admissible strategy \((p(\cdot ), b_{L_{1}}(\cdot ),b_{L_{2}}(\cdot ))\);

StepÂ 2: The insurer observes the reinsurerâ€™s strategy and decides on its optimal strategy \(q^{\ast}(\cdot )=\alpha ^{\ast}(\cdot ,p(\cdot ),b_{L_{1}}(\cdot ),b_{L_{2}}( \cdot ))\), \(b_{F_{1}}^{\ast}(\cdot )=\beta _{1}^{\ast}(\cdot ,p(\cdot ),b_{L_{1}}( \cdot ),b_{L_{2}}(\cdot ))\), \(b_{F_{2}}^{\ast}(\cdot ) =\beta _{2}^{ \ast}(\cdot ,p(\cdot ),b_{L_{1}}(\cdot ),b_{L_{2}}(\cdot ))\) by solving its own optimization problem;

StepÂ 3: Observing that the insurer would execute \(\alpha ^{\ast}(\cdot ,p(\cdot ),b_{L_{1}}(\cdot ),b_{L_{2}}(\cdot ))\), \(\beta _{1}^{\ast}(\cdot ,p(\cdot ), b_{L_{1}}(\cdot ),b_{L_{2}}( \cdot ))\) and \(\beta _{2}^{\ast}(\cdot ,p(\cdot ),b_{L_{1}}(\cdot ),b_{L_{2}}( \cdot ))\), the reinsurer then decides on its admissible strategy \((p^{\ast}(\cdot ),b^{\ast}_{L_{1}}(\cdot ), b^{\ast}_{L_{2}}(\cdot ))\) by solving its own optimization problem.
Due to the psychology of comparison, the reinsurer and the insurer, as the two parties of the game, consider not only the expected utility of their own terminal wealth but also the expected utility of the wealth gap between themselves and the other party. That is to say, the target of the reinsurer is to seek the optimal reinsurance premium pricing strategy and investment strategy such that the expected utility of its relative performance is maximized. For insurer, it is similar. For simplicity, let \(\hat{X}_{F}^{\pi _{F}}(t)=X_{F}^{\pi _{F}}(t)k_{1}X_{L}^{\pi _{L}}(t)\), \(\hat{X}_{L}^{\pi _{L}}(t)=X_{L}^{\pi _{L}}(t)k_{2}X_{F}^{\pi _{F}}(t)\). Then, we have
where \(k_{1}\in [0,1]\) describes the sensitivity of the insurer to the reinsurerâ€™s performance, and \(k_{2}\in [0,1]\) describes the sensitivity of the reinsurer to the insurerâ€™s performance.
Let \(X_{L}^{\pi _{L}}(t)=x_{L}\), \(X_{F}^{\pi _{F}}(t)=x_{F}\), \(\hat{X}_{F}^{\pi _{F}}(t)=X_{F}^{\pi _{F}}(t)k_{1}X_{L}^{\pi _{L}}(t)=x_{F}k_{1}x_{L} \doteq \hat{x}_{F}\), \(\hat{X}_{L}^{\pi _{L}}(t)=X_{L}^{\pi _{L}}(t)k_{2}X_{F}^{\pi _{F}}(t)=x_{L}k_{2}x_{F} \doteq \hat{x}_{L}\), at time \(t\in [0,T]\). Let \(V(t)=v\), at time \(t\in [0,T]\). Then, the admissible strategy is as follows.
Definition 1
(Admissible strategy)
\(\pi (\cdot )=\pi _{L}(\cdot )\times \pi _{F}(\cdot )=(p(\cdot ),b_{L1}( \cdot ),b_{L2}(\cdot ))\times (q(\cdot ),b_{F1}(\cdot ), b_{F2}( \cdot ))\) is admissible if

(i)
\(\{\pi _{L}(t)\}_{t\in [0,T]}\) and \(\{\pi _{F}(t)\}_{t\in [0,T]}\) are \(\mathcal{F}\)progressively measurable processes, such that \(p(t)\in [c_{F},\bar{c}]\), \(q(t)\in [0,1]\) for any \(t\in [0,T]\);

(ii)
\(E \{\int _{t}^{T}[(B^{F_{1}}(t))^{2}+(B^{F_{2}}(t))^{2}]V(t)\,d\ell \}<+\infty \), and \(E \{\int _{t}^{T}[(B^{L_{1}}(t))^{2}+(B^{L_{2}}(t))^{2}]V(t)\,d\ell \} <+\infty \), \(\forall \ell \in [t,T]\);

(iii)
the equation (2.11) associated with \(\pi (\cdot )\) has a unique solution \(\hat{X}_{F}^{\pi _{F}}(\cdot )\), which satisfies \(\{E_{t,\hat{x}_{F},v}[\sup \hat{X}_{F}^{\pi _{F}}(\ell )^{2}]\}^{ \frac{1}{2}}<+\infty \), for \(\forall (t,\hat{x}_{F},v)\in [0,T] \times \mathbb{R}\times \mathbb{R}\), \(\forall \ell \in [t,T]\).

(iv)
the equation (2.12) associated with \(\pi (\cdot )\) has a unique solution \(\hat{X}_{L}^{\pi _{L}}(\cdot )\), which satisfies \(\{E_{t,\hat{x}_{L},v}[\sup \hat{X}_{L}^{\pi _{L}}(\ell )^{2}]\}^{ \frac{1}{2}}<+\infty \), for \(\forall (t,\hat{x}_{L},v)\in [0,T]\times \mathbb{R}\times \mathbb{R}\), \(\forall \ell \in [t,T]\).
Denote \(\Pi =\Pi _{L}\times \Pi _{F}\) as the set of all admissible strategies, where \(\Pi _{L}\) is the set of all admissible strategies of the reinsurer, and \(\Pi _{F}\) is that of the insurer. Then, the Stackelberg game problem is described by ProblemÂ 1.
Problem 1
The insurerâ€™s problem can be described by the following optimization problem: for any \(\pi _{L}(\cdot )=(p(\cdot ),b_{L1}(\cdot ),b_{L2}(\cdot ))\in \Pi _{L}( \cdot )\), find a map \(\pi _{F}^{\ast}(\cdot )=(q^{\ast}(\cdot ),b_{F_{1}}^{\ast}(\cdot ),b_{F_{2}}^{ \ast}(\cdot )) =(\alpha ^{\ast}(\cdot ,\pi _{L}(\cdot )), \beta _{1}^{ \ast}(\cdot ,\pi _{L}(\cdot )), \beta _{2}^{\ast}(\cdot ,\pi _{L}( \cdot ))) :[0,T]\times \Omega \times \Pi _{L}\rightarrow \Pi _{F}\) such that the following value function holds:
where \(U_{F}\) is the utility function of the insurer. The reinsurerâ€™s problem can be described by the following optimization problem: find a \(\pi _{L}^{\ast}(\cdot )=(p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ),b_{L2}^{ \ast}(\cdot ))\in \Pi _{L}\) such that the following value function holds:
where \(U_{L}\) represents the reinsurerâ€™s utility function.
Definition 2
The sixtuple \((p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ),b_{L2}^{\ast}(\cdot ), \alpha ^{\ast}(\cdot ,p^{\ast}(\cdot ),b_{L_{1}}^{\ast}(\cdot ),b_{L_{2}}^{ \ast}(\cdot )), \beta _{1}^{\ast}(\cdot ,p^{\ast}(\cdot ),b_{L_{1}}^{ \ast}(\cdot ), b_{L_{2}}^{\ast}(\cdot )), \beta _{2}^{\ast}( \cdot ,p^{\ast}(\cdot ),b_{L_{1}}^{\ast}(\cdot ),b_{L_{2}}^{\ast}( \cdot )))\) is an equilibrium solution to the Stackelberg game problem 1.
3 Equilibrium solution to the Stackelberg game for CARA preference
Compared with individual investors, the insurer and the reinsurer have considerable wealth, and their risk aversion coefficients are relatively stable and can be regarded as constants in the time interval \([0,T]\). The wealth of the insurer and the reinsurer may be negative due to the randomness of the number and size of future claims. Referring to Yan etÂ al. [26] and so on, the positive condition of the wealth processes is very crucial for some common utility functions (for example, the CRRA utility function and the logarithmic utility function), which is hardly guaranteed, especially when considering relative performance under the game framework. In view of these facts, we assume that both the insurer and the reinsurer are constant absolute risk aversion (CARA) agents whose exponential utility functions are given by
where \(\gamma _{F}>0\) is the constant absolute risk aversion coefficients of the insurer, and \(\gamma _{L}>0\) is the constant absolute risk aversion coefficients of the reinsurer.
3.1 Optimal strategy and value function
In this section, we use the dynamic programming method combined with the procedure mentioned in Sect.Â 2.4 to solve the Stackelberg game problem.
StepÂ 1. In the stochastic Stackelberg differential game, the reinsurer takes action first by announcing its any admissible strategy \((p(\cdot ),b_{L1}(\cdot ),b_{L2}(\cdot ))\in \Pi _{L}\).
StepÂ 2. Based on the reinsurerâ€™s strategy \((p(\cdot ),b_{L1}(\cdot ),b_{L2}(\cdot ))\in \Pi _{L}\), we solve the insurerâ€™s optimization problem (2.13) under the CARA utility function.
Based on the value function of the insurer, we have
where \(\varphi ^{F}(t)\), \(g^{F}_{1}(t)\) and \(g^{F}_{2}(t)\) are differentiable functions with \(\varphi ^{F}(T)=1\), \(g^{F}_{1}(T)=0\) and \(g^{F}_{2}(T)=0\). Thus, for the insurer, the HJB equation is as follows
where \(J^{F}_{t}\), \(J^{F}_{\hat{x}_{F}}\), and \(J^{F}_{v}\) are the first derivatives of \(J^{F}\) with respect to t, \(\hat{x}_{F}\) and v, respectively. \(J^{F}_{\hat{x}_{F}\hat{x}_{F}}\), \(J^{F}_{vv}\) and \(J^{F}_{\hat{x}_{F}v}\) are the second derivatives of \(J^{F}\) with respect to \(\hat{x}_{F}\) and v, respectively. Substituting these derivatives into the HJB equation (3.4) yields
Based on the first order condition for maximizing the value in (3.5), we have
By solving equations (3.8), we have
From the above equations, it is not difficult to find that the reinsurance strategy is independent of the investment strategy. Therefore,
could be rewritten as \(\alpha ^{\ast}(\cdot ,p(\cdot ))\), \(\beta _{1}^{\ast}(\cdot ,b_{L_{1}}(\cdot ),b_{L_{2}}(\cdot ))\) and \(\beta _{2}^{\ast}(\cdot ,b_{L_{1}}(\cdot ),b_{L_{2}}(\cdot ))\). From the relations of \(B^{F_{1}}(t)\) and \(B^{F_{2}}(t)\) (i.e., (2.7)) and the scope of \(q(t)\), we have
Substituting (3.9) and (3.10) into the HJB equation (3.5) gives that
It is not difficult to find that \(q^{\ast}(t,p(t))\) is independent of variables \(\hat{x}_{F}\) and v. Due to \(\varphi ^{F}(T)=1\), \(g^{F}_{1}(T)=0\) and \(g^{F}_{2}(T)=0\), we deduce that
and
By equation (3.14) and \(p(t)\), we obtain \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}>0\). Next, these situations will be discussed:

Case (Fa) If \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}\geq 1\), then \(q^{\ast}(t,p(t))=1\). Substituting \(q^{\ast}(t,p(t))\) into equation (3.18), we get
$$ g^{F}_{2}(t)=g^{Fa}_{2}(t) \doteq \gamma _{F}\theta _{F}a \int _{T}^{t} \varphi ^{F}(s)\,ds  \frac{(\gamma _{F}\sigma _{F})^{2}}{2} \int _{T}^{t}\bigl( \varphi ^{F}(s)\bigr)^{2}\,ds \kappa \bar{v} \int _{T}^{t}g^{F}_{1}(s)\,ds . $$(3.19) 
Case (Fb) If \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}<1\), then \(q^{\ast}(t,p(t))= \frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}\). Substituting \(q^{\ast}(t,p(t))\) into equation (3.18), we get
$$\begin{aligned} g^{F}_{2}(t)&=g^{Fb}_{2}(t) \\ & \doteq \gamma _{F}\theta _{F}a \int _{T}^{t} \varphi ^{F}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{F}_{1}(s)\,ds \\ &\quad{}  \gamma _{F} \int _{T}^{t}\varphi ^{F}(s) \bigl(p(s)a\bigr)\,ds + \frac{1}{2\sigma _{F}^{2}} \int _{T}^{t}\bigl(p(s)a\bigr)^{2}\,ds . \end{aligned}$$(3.20)
StepÂ 3. Observing the reinsurance and investment strategies of the insurer from equations (3.12), (3.13), and (3.14), the reinsurer decides on the strategy \(\pi _{L}^{\ast}(\cdot )=(p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ), b_{L2}^{ \ast}(\cdot ))\in \Pi _{L}\).
Based on the value function of the reinsurer, we have
where \(\varphi ^{L}(t)\), \(g^{L}_{1}(t)\), and \(g^{L}_{2}(t)\) are differentiable functions with \(\varphi ^{L}(T)=1\), \(g^{L}_{1}(T)=0\), and \(g^{L}_{2}(T)=0\). For the reinsurer, the HJB equation is as follows
In the above, \(J^{L}_{t}\), \(J^{L}_{\hat{x}_{L}}\), \(J^{L}_{\hat{x}_{L}\hat{x}_{L}}\), \(J^{L}_{v}\), \(J^{L}_{vv}\), and \(J^{L}_{\hat{x}_{L}v}\) are partial derivatives of \(J^{L}\). Substituting derivatives into the HJB equation (3.22) yields
Similarly, by equation (3.23), we have
Solving the above equation yields that
From the ranges of \(k_{1}\) and \(k_{2}\), we know that \(k_{1}k_{2}\in [0,1]\). In order to solve the problem, we assume \(k_{1}k_{2}\neq 1\). Substituting the expressions of \(B^{F_{1}}(t)\) and \(B^{F_{2}}(t)\) (i.e., (3.9) and (3.10)) into the above formulas, we obtain
From the relations of \(B^{L_{1}}(t)\) and \(B^{L_{2}}(t)\) (i.e., (2.8)), we can get the optimal investment strategy of the reinsurer:
For the insurer, the optimal investment strategy is as follows:
Substituting (3.24) and (3.25) into the reinsurerâ€™s HJB equation (3.23) yields that
It is not difficult to find that \(p^{\ast}(t)\) is independent of variables \(\hat{x}_{L}\) and v. Due to \(\varphi ^{L}(T)=1\) and \(g^{L}_{1}(T)=0\), we deduce that
and
For simplicity, we write
The premium strategy \(p(t)\) is discussed as follows:

Case (La) When \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}\geq 1\), then \(q^{\ast}(t,p(t))=1\). Substituting \(q^{\ast}(t,p(t))\) into equation (3.33), we get
$$\begin{aligned} 0=\sup_{p(t)\in [c_{F},\bar{c}]} \biggl\{ \frac{dg^{L}_{2}(t)}{dt}+g^{L}_{1}(t) \kappa \bar{v}+\gamma _{L}\varphi ^{L}(t)k_{2} \theta _{F}a + \frac{1}{2}\bigl(k_{2}\sigma _{F}\gamma _{L}\varphi ^{L}(t) \bigr)^{2} \biggr\} . \end{aligned}$$(3.35)It is easy to know \(p^{\ast}(t)=p\), where \(p\in [c_{F},\bar{c}]\). Then, \(q^{\ast}(t,p^{\ast}(t))=1\). \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}\geq 1\) yields that \(N^{\theta _{F}}(t)\geq 1\). From equation (3.35) and \(g^{L}_{2}(T)=0\), we have
$$\begin{aligned} g^{L}_{2}(t)&=g^{La}_{2}(t) \\ &\doteq k_{2}\gamma _{L}\theta _{F}a \int _{T}^{t} \varphi ^{L}(s)\,ds  \frac{1}{2}(k_{2}\sigma _{F}\gamma _{L})^{2} \int _{T}^{t}\bigl( \varphi ^{L}(s)\bigr)^{2}\,ds\kappa \bar{v} \int _{T}^{t}g^{L}_{1}(s)\,ds . \end{aligned}$$(3.36) 
Case (Lb) When \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}<1\), then \(q^{\ast}(t,p(t))= \frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}\). Substituting \(q^{\ast}(t,p(t))\) into equation (3.33), we have
$$\begin{aligned} 0& = \sup_{p(t)\in [c_{F},\bar{c}]} \biggl\{ \frac{dg^{L}_{2}(t)}{dt}+g^{L}_{1}(t) \kappa \bar{v}+\gamma _{L}\varphi ^{L}(t)k_{2}\theta _{F}a + \frac{1}{2}\biggl(\frac{1k_{1}k_{2}}{1+k_{1}} \biggr)^{2}\bigl(\sigma _{F}\gamma _{L} \varphi ^{L}(t)\bigr)^{2} \\ &\quad{}  \bigl(p(t)a\bigr)\frac{(1+k_{2})\gamma _{L}\varphi ^{L}(t)}{1+k_{1}}\biggl[1+ \frac{\gamma _{L}}{\gamma _{F}} \frac{(1k_{1}k_{2})}{(1+k_{1})}\biggr] \\ &\quad{} + \bigl(p(t)a\bigr)^{2} \frac{(1+k_{2})\gamma _{L}}{(1+k_{1})\gamma _{F}\sigma _{F}^{2}}\biggl[1+ \frac{\gamma _{L}}{2\gamma _{F}}\frac{(1+k_{2})}{(1+k_{1})}\biggr] \biggr\} . \end{aligned}$$(3.37)Similarly, we have
$$\begin{aligned} p^{\ast}(t)=\bigl[a+K\gamma _{F}\sigma _{F}^{2}\varphi ^{L}(t)\bigr]\vee c_{F} \wedge \bar{c}. \end{aligned}$$(3.38)
Subcase (Lb1) When \(a+K\gamma _{F}\sigma _{F}^{2}\varphi ^{L}(t)\geq \bar{c}\), then \(p^{\ast}(t)=\bar{c}\), \(q^{\ast}(t,p^{\ast}(t))=\frac{1}{1+k_{1}}(N^{\bar{\theta}}(t)+k_{1})\). From \(K<1\) and \(\varphi ^{L}(t)=\varphi ^{F}(t)\), then \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}<1\) yields that \(\frac{1}{1+k_{1}}(N^{\bar{\theta}}(t)+k_{1})<1\). Then, by equation (3.37) and \(g^{L}_{2}(T)=0\), we have
$$\begin{aligned} g^{L}_{2}(t)&=g^{Lb1}_{2}(t) \\ & \doteq k_{2}\gamma _{L}\theta _{F}a \int _{T}^{t}\varphi ^{L}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{L}_{1}(s)\,ds \\ &\quad{} + \gamma _{L}\bar{\theta}a(1+k_{2}) \int _{T}^{t}\varphi ^{L}(s) \biggl[1 \frac{1}{1+k_{1}}\bigl(N^{\bar{\theta}}(s)+k_{1} \bigr)\biggr]\,ds \\ &\quad{}  \frac{(\gamma _{L}\sigma _{F})^{2}}{2} \int _{T}^{t}\bigl(\varphi ^{L}(s) \bigr)^{2}\biggl[1 \frac{1+k_{2}}{1+k_{1}}\bigl(N^{\bar{\theta}}(s)+k_{1} \bigr)\biggr]^{2}\,ds . \end{aligned}$$(3.39)And, equation (3.20) becomes
$$\begin{aligned} g^{F}_{2}(t)=g^{Fb1}_{2}(t) \doteq \gamma _{F}(\theta _{F} \bar{\theta})a \int _{T}^{t}\varphi ^{F}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{F}_{1}(s)\,ds \frac{(\bar{\theta}a)^{2}(Tt)}{2\sigma _{F}^{2}}. \end{aligned}$$(3.40) 
Subcase (Lb2) When \(a+K\gamma _{F}\sigma _{F}^{2}\varphi ^{L}(t)\leq c_{F}\), then \(p^{\ast}(t)=c_{F}\), \(q^{\ast}(t,p^{\ast}(t))=\frac{1}{1+k_{1}}(N^{\theta _{F}}(t)+k_{1})\). And \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}<1\) becomes \(\frac{1}{1+k_{1}}(N^{\theta _{F}}(t)+k_{1})<1\). Then, by equation (3.37) and \(g^{L}_{2}(T)=0\), we can get that
$$\begin{aligned} g^{L}_{2}(t)&=g^{Lb2}_{2}(t) \\ & \doteq k_{2}\gamma _{L}\theta _{F}a \int _{T}^{t}\varphi ^{L}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{L}_{1}(s)\,ds \\ &\quad{} + \gamma _{L}\theta _{F}a(1+k_{2}) \int _{T}^{t}\varphi ^{L}(s) \biggl[1 \frac{1}{1+k_{1}}\bigl(N^{\theta _{F}}(s)+k_{1} \bigr)\biggr]\,ds \\ &\quad{}  \frac{(\gamma _{L}\sigma _{F})^{2}}{2} \int _{T}^{t}\bigl(\varphi ^{L}(s) \bigr)^{2}\biggl[1 \frac{1+k_{2}}{1+k_{1}}\bigl(N^{\theta _{F}}(s)+k_{1} \bigr)\biggr]^{2}\,ds . \end{aligned}$$(3.41)And, equation (3.20) becomes
$$\begin{aligned} g^{F}_{2}(t)=g^{Fb2}_{2}(t) & \doteq \kappa \bar{v} \int _{T}^{t}g^{F}_{1}(s)\,ds  \frac{(\theta _{F}a)^{2}(Tt)}{2\sigma _{F}^{2}}. \end{aligned}$$(3.42) 
Subcase (Lb3) When \(\theta _{F}a< K\gamma _{F}\sigma _{F}^{2}\varphi ^{L}(t)<\bar{\theta}a\), then, \(p^{\ast}(t)=a+K\gamma _{F}\sigma _{F}^{2}\varphi ^{L}(t)\). \(q^{\ast}(t,p^{\ast}(t))=\frac{1}{1+k_{1}}(K+k_{1})\). And \(\frac{(p(t)a)}{\gamma _{F}\varphi ^{F}(t)\sigma _{F}^{2}(1+k_{1})}+ \frac{k_{1}}{1+k_{1}}<1\) becomes \(\frac{1}{1+k_{1}}(K+k_{1})<1\). From equation (3.37) and \(g^{L}_{2}(T)=0\), we have
$$\begin{aligned} g^{L}_{2}(t)&=g^{Lb3}_{2}(t) \\ & \doteq k_{2}\gamma _{L}\theta _{F}a \int _{T}^{t} \varphi ^{L}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{L}_{1}(s)\,ds \\ &\quad{} + \biggl[\gamma _{L}\gamma _{F}\sigma _{F}^{2} \frac{(1+k_{2})(1K)K}{1+k_{1}} \\ &\quad{}  \frac{(\gamma _{L}\sigma _{F})^{2}}{2} \biggl(1 \frac{(1+k_{2})(K+k_{1})}{1+k_{1}}\biggr)^{2} \biggr] \int _{T}^{t}\bigl(\varphi ^{L}(s) \bigr)^{2}\,ds . \end{aligned}$$(3.43)And, equation (3.20) becomes
$$\begin{aligned} g^{F}_{2}(t)&=g^{Fb3}_{2}(t) \\ & \doteq \gamma _{F}\theta _{F}a \int _{T}^{t} \varphi ^{F}(s)\,ds  \kappa \bar{v} \int _{T}^{t}g^{F}_{1}(s)\,ds \\ &\quad{} + K\gamma _{F}^{2}\sigma _{F}^{2} \biggl(1+\frac{K}{2}\biggr) \int _{T}^{t}\bigl( \varphi ^{L}(s)\bigr)^{2}\,ds . \end{aligned}$$(3.44)

Therefore, the theorem is as follows.
Theorem 1
Assuming \(k_{1}k_{2}<1\). The equilibrium strategy of the Stackelberg game problem 1is \((p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ),b_{L2}^{\ast}(\cdot ),q^{ \ast}(t),b_{F1}^{\ast}(\cdot ),b_{F2}^{\ast}(\cdot ))\), where \(b_{L1}^{\ast}(\cdot )\) and \(b_{L2}^{\ast}(\cdot )\) are given by (3.26) and (3.27), respectively; \(b_{F1}^{\ast}(\cdot )\) and \(b_{F2}^{\ast}(\cdot )\) are given by (3.28) and (3.29), respectively; \(p^{\ast}(t)\) and \(q^{\ast}(t)\) under various cases are represented in TableÂ 1; where K, \(N^{\theta _{F}}(t)\) and \(N^{\bar{\theta}}(t)\) satisfy equation (3.34).
For the reinsurer and the insurer, the value functions are respectively
and
where \(\varphi ^{F}(t)\) and \(\varphi ^{L}(t)\) are given by equations (3.16) and (3.31); \(g^{F}_{1}(t)\) and \(g^{L}_{1}(t)\) are given by equations (3.17) and (3.32); \(g^{L}_{2}(t)\) and \(g^{F}_{2}(t)\) under different cases are as follows
\(g^{La}_{2}(t)\), \(g^{Lb1}_{2}(t)\), \(g^{Lb2}_{2}(t)\), \(g^{Lb3}_{2}(t)\), \(g^{Fa}_{2}(t)\), \(g^{Fb1}_{2}(t)\), \(g^{Fb2}_{2}(t)\), and \(g^{Fb3}_{2}(t)\) are given by equations (3.36), (3.39), (3.41), (3.43), (3.19), (3.40), (3.42), and (3.44).
TheoremÂ 1 demonstrates the equilibrium reinsuranceinvestment strategy does not depend on the current wealth. Moreover, for the insurer, their investment strategy does not depend on its reinsurance strategy. Our conclusions are consistent with that of most existing related research, among them, Bensoussan etÂ al. [7], Yan etÂ al. [26], A etÂ al. [1] and Deng etÂ al. [14], etc. In addition, from the form of investment strategies of the insurer and the reinsurer, we can see that they imitate each otherâ€™s investment strategies, showing a herd effect.
Remark 1
When the Stackelberg equilibrium is achieved in Case (4) of TheoremÂ 1, the optimal reinsurance premium is given by the variance premium principle. More precisely, for per unit of risk, the total instantaneous reinsurance premium associated with the ceded proportion \((1q^{\ast}(t))100\%\) is
3.2 Special cases
Special case 1
If there is no opportunity to trade derivatives, i.e., \(b_{F2}(t)=b_{L2}(t)\equiv 0\), and we have
According to the procedure mentioned in Sect.Â 2.4, the optimal reinsurance strategy of the insurer, the optimal reinsurance premium strategy of the reinsurer, and their optimal investment strategies in the stock can be obtained successively. The optimal reinsurance strategy and the optimal reinsurance premium strategy are consistent with TableÂ 1, and the optimal investment strategies are as follow:
where \(\varphi ^{L}(t)=\varphi ^{F}(t)=\exp \{r_{0}(Tt)\}\doteq \varphi (t)\), \(g^{L}_{1}(t)=g^{F}_{1}(t)\doteq g_{1}(t)\) satisfies
Then, the optimal investment strategies can be simplified to:
Special case 2
When \(k_{1}=k_{2}=0\), there is no wealth gap between the insurer and reinsurer. The equilibrium strategy is \((p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ),b_{L2}^{\ast}(\cdot ), q^{ \ast}(\cdot ),b_{F1}^{\ast}(\cdot ),b_{F2}^{\ast}(\cdot ))\), where \(p^{\ast}(t)\) and \(q^{\ast}(t)\) under various cases are represented in TableÂ 2; where \(K^{0}=\frac{\gamma _{F}+\gamma _{L}}{2\gamma _{F}+\gamma _{L}}\). And, \(b_{L1}^{\ast}(\cdot )\), \(b_{L2}^{\ast}(\cdot )\), \(b_{F1}^{\ast}(\cdot )\), \(b_{F2}^{ \ast}(\cdot )\) are as follows
The form of the optimal investment strategies in this special case is consistent with the results in the literature Xue etÂ al. [25]. From equations (3.16), (3.31), (3.17), and (3.32), we get \(\varphi ^{F}(t)=\varphi ^{L}(t)\) and \(g^{F}_{1}(t)=g^{L}_{1}(t)\). Then, we have
We find that for the optimal investment strategies, the insurer and the reinsurer imitate each other, which shows a herd effect.
4 Numerical analysis
This section conducts some numerical examples and analyzes the sensitivity of the equilibrium strategy \((p^{\ast}(\cdot ),b_{L1}^{\ast}(\cdot ),b_{L2}^{\ast}(\cdot ),q^{ \ast}(\cdot ),b_{F1}^{\ast}(\cdot ),b_{F2}^{\ast}(\cdot ))\) to model parameters.
Referring to Heston [16] and Xue etÂ al. [25], the pricing formula of a European call option with stochastic volatility is
where Ï„ is the expiration, and \(\mathbb{P}\) is the strike price. Then, we have
and
where
with \(a=y(1y)\), \(b=\sigma _{1}\rho y\kappa ^{*}\), \(q=\sqrt{b^{2}+a\sigma _{1}^{2}}\), \(\kappa ^{*}=\kappa \sigma _{1}(\rho \eta +\sqrt{1\rho ^{2}}\xi )\) and \(\bar{v}^{*}=\frac{\kappa \bar{v}}{\kappa ^{*}}\). Here, \(\kappa ^{*}\) and \(\bar{v}^{*}\) represent the riskneutral meanreversion rate and longrun mean, respectively; i denotes the imaginary unit.
To be consistent with Xue etÂ al. [25], we use the same parameter values given in TableÂ 3 and TableÂ 4.
4.1 Sensitivity analysis of the equilibrium investment strategy
According to the above parameter settings, this subsection analyzes the sensitivity of the optimal investment strategies to model parameters, including the risk aversion coefficients and sensitivity coefficients of the insurer and the reinsurer, the volatility parameter and mean reversion rate parameter in Hestonâ€™s stochastic volatility model and option pricing model, etc.
FigureÂ 1 shows the effects of the risk aversion coefficients \(\gamma _{L}\) and \(\gamma _{F}\) on optimal investment strategies, respectively. The larger \(\gamma _{L}\) and \(\gamma _{F}\), the more riskaverse the reinsurer and the insurer, and the smaller both the long position in stocks and the short position in options. For reinsurer, the optimal investment strategy is sensitivity to both \(\gamma _{L}\) and \(\gamma _{F}\). For insurer, the conclusion is similar.
FigureÂ 2 shows the effects of sensitivity parameters of the insurer and the reinsurer on optimal investment strategies. As \(k_{1}\) increases, both the reinsurer and the insurer invest more in stocks and short more options. For the sensitivity parameter \(k_{2}\), the conclusion is similar.
FigureÂ 3 shows the effects of parameter Î· on the optimal investment strategy of the reinsurer and the insurer. As Î· increases, the reinsurer invests more in the stocks, and the short position in options first decreases and then increases. And we can get the similar conclusion for the insurer.
FigureÂ 4 shows the sensitivity of the optimal investment strategy to the meanreversion rate Îº that depicts the persistency of volatility. As Îº increases, both the reinsurer and the insurer invest more in the stocks and short more options.
FigureÂ 5 shows the sensitivities of the optimal investment strategy to \(\sigma _{1}\), i.e., the volatility of volatility. As \(\sigma _{1}\) increases, the reinsurer invests less in the stocks and shorts less the option. For the insurer, we can get the similar conclusion.
FigureÂ 6 shows the sensitivity of optimal investment strategy to the volatility \(\sqrt{V_{0}}\). The larger the volatility, the more volatile the finance market, the reinsurer invests more in the stocks and shorts more the option to achieve the optimal volatility exposure. For the insurer, we can get the similar conclusion, which is consistent with the sensitivity analysis in Xue etÂ al. [25].
FigureÂ 7 shows the sensitivity of the optimal investment strategy to the premium Î¾ of volatility risk. With Î¾ increasing, the reinsurer first takes a long (short) position and then switches to a short (long) position in the stock (option). A short position in the derivative may switch to a long position with parameters changing, indicating that the insurer and reinsurer can flexibly adjust the position in the derivative to manage the market uncertainty risk. More precisely, the absolute value of the amount invested in the stock and the option firstly decreases and then increases with Î¾ increasing. For the insurer, we also get a similar conclusion, which is consistent with the sensitivity analysis in Xue etÂ al.Â [25].
4.2 Sensitivity analysis of the equilibrium reinsurance strategy
FigureÂ 8 shows the change of the optimal premium strategy of the reinsurer and the optimal reinsurance strategy of the insurer over time t. Meanwhile, the corresponding numerical results are represented in TableÂ 5. The results show that condition \(N^{\bar{\theta}}(t)\leq K\) is satisfied when \(t\leq 1\), which is corresponding to CaseÂ 2 in TheoremÂ 1; condition \(N^{\theta _{F}}(t)< K< N^{\bar{\theta}}(t)\) is satisfied when \(t\geq 2\), which is corresponding to CaseÂ 4 in TheoremÂ 1. For more intuitively investigating the effects of model parameters on the optimal premium strategy and the optimal reinsurance strategy, we fully discuss the strategies in Case (4) in which all the optimal strategies fall within the feasible range. Thus, we give the strategies at \(t=4\) for sensitivity analysis.
FigureÂ 9 shows the effect of risk aversion coefficients of the insurer and the reinsurer on optimal reinsurance premium strategy and optimal reinsurance strategy. The graph shows that the more risk averse the insurer is, the lower its own risk reserve level will be, and the higher the premium price charged by the reinsurer will be. In addition, the more risk averse the reinsurer is, the more inclined it is to set a higher premium price to avoid risks. At this time, the insurer tends to purchase reinsurance contracts with a lower proportion to avoid more expenses. This is consistent with the phenomenon of the market economy.
FigureÂ 10 shows the effects of the sensitivity parameters of the insurer and the reinsurer on the optimal reinsurance premium price strategy and the optimal reinsurance strategy. The graph shows that the larger \(k_{1}\) is, the higher \(q^{*}(4)\) is, and the lower \(p^{*}(4)\) is. That is to say, the more the insurer cares about the performance of the reinsurer, the more inclined it is to buy fewer reinsurance contracts, and the more willing it is to take more claims risks, the risk of more claims, which leads to the lower price of the reinsurance premium. The more the reinsurer cares about the performance level of the insurer, the more it tends to attract a higher proportion of reinsurance contracts by reducing the reinsurance premium price, which leads to the reduction of the risk reservation level of the insurer. In general, the optimal premium price strategy of the reinsurer and the optimal reinsurance strategy of the insurer are not only affected by their own sensitivity parameter but also by the sensitivity of the other player in the game.
5 Conclusion
In the paper, we investigate a stochastic Stackelberg differential reinsuranceinvestment game problem with derivative trading under a stochastic volatility model. The reinsurer who occupies a monopoly position can determine the price of the reinsurance premium and its asset allocation strategy invested in the stock and the derivative. The insurer, the follower of the Stackelberg game, can determine the proportion of reinsurance according to the price of reinsurance premium and its asset allocation strategy. The target of the reinsurer and the insurer is to find their own optimal strategy that maximizes the CARA utility of relative performance. The explicit equilibrium strategy for the game problem is deduced by solving HJB equations sequentially. The equilibrium investment strategy demonstrates that the insurer and the reinsurer imitate each otherâ€™s investment strategies, showing a herd effect. The numerical experiment represents the sensitivity of the equilibrium strategy to model parameters. We find that a short position in the derivative may switch to a long position with parameters changing, which provides investors with important decisionmaking information.
The differential game in the insurance market is a hot topic in the current economic and financial field, and its future research can be very extensive and interesting, such as to study the Stackelberg reinsuranceinvestment game problem with derivatives trading under meanvariance criterion, to consider regime switching to better model market randomness and to consider parameter uncertainty in the stochastic Stackelberg differential game model.
Availability of data and materials
The data used in this paper is available upon the request.
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Funding
This research is supported by the Natural Science Foundation of Shandong Province (Grant No.Â ZR2022QG060, Grant No.Â ZR2021QG036) and the National Natural Science Foundation of China (Grant No.Â 72171133).
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Rui Gao: Methodology, Software, Writingâ€”original draft, Funding acquisition. Yanfei Bai: Writingâ€”review, Formal analysis, Funding acquisition. All authors reviewed the manuscript
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Gao, R., Bai, Y. A Stackelberg reinsuranceinvestment game with derivatives trading. Bound Value Probl 2023, 43 (2023). https://doi.org/10.1186/s13661023017314
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DOI: https://doi.org/10.1186/s13661023017314