Investment and reinsurance are two significant issues for the insurers. Extensive studies have been conducted to find the optimal investment-reinsurance strategies under different objectives, such as minimizing the probability of ruin (Browne [1], Zhang et al. [2]), maximizing the expected utility of terminal wealth (Chen and Yang [3], Bai and Guo [4]) and the mean-variance (MV) criterion (Shen and Zeng [5], Zeng [6]).

The CEV risky asset model has received extensive attentions due to the empirical advantage and mathematical tractability. The study in this area includes but is not limited to the following work. For example, Gu et al. [7] considered the excess-of-loss reinsurance and investment problem for an insurer. Zheng et al. [8] investigated a robust optimal portfolio and reinsurance problem under a Cramér-Lundberg risk model. Wang et al. [9] derived the optimal investment strategies for both insurer and reinsurer. Li et al. [10] focused on a stochastic differential game between two insurers.

Most of the above-mentioned research did not take the default risk into consideration. However, the large market share of defaultable bonds make itself one of the most important investment products. Recently, many authors considered the financial market with a risk-free asset, a risky asset and a defaultable bond (see, e.g., Zhao et al. [11], Sun et al. [12] and Wang et al. [13]).

In reality, there exist many insurers playing the roles of both partners and competitors. The non-zero sum game theory is a kind of idea to reflect the interaction among several insurers. Bensoussan et al. [14] formulated a non-zero-sum stochastic differential game between two insurers and obtained explicit solutions for optimal strategies. Deng et al. [15] discussed this problem between two competitive insurers facing the default risk. Zhang et al. [16] invested the same problem under the CEV model. Yang et al. [17] further investigated the optimal reinsurance-investment strategy for $ n $ mean-variance insurers.

In our paper, a non-zero-sum stochastic differential game theory is applied to solve the optimal investment reinsurance problem for $ n $ competitive insurers under the exponential utility function. In the financial market, we assume the risky asset price process is governed by the CEV model. As well as the risky assets and the risky-free asset the insurers are allowed to invest in a defaultable bond. The explicit expressions of Nash equilibrium strategies and equilibrium value functions under pre-default case and post-default case are derived, respectively. We also analyze the impact of volatility skew on each insurer's equilibrium strategy through numerical examples. Last but not least, we find an interesting conclusion, the number of insurers is also one of the factors that affect insurer's trading strategy.

The rest of this paper is organized as follows. Assumption and problem formulation are described in Section 2. In Section 3, we obtain the optimal strategies and the corresponding value functions under pre-default case and post-default case, respectively. Section 4 provides some numerical studies to analyze our results. Section 5 concludes this paper.

Let $ \left(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\in[0, T]}, P\right) $ is a complete probability space, where the filtration $ \{\mathcal{F}_{t}\}_{t\in[0, T]} $ is right-continuous and $ P $-complete; $ [0, T] $ is a fixed and finite time horizon. $ \mathcal{F}_{t} $ stands for the information available until time $ t $, which is generated by $ 2n $ mutually independent standard Brownian motions $ \{W_{1_0}(t)\} $, $ \{W_{2_0}(t)\} $, $ \cdots $, $ \{W_{n_0}(t)\} $, $ \{W_{1}(t)\} $, $ \{W_{2}(t)\} $, $ \cdots $, $ \{W_{n}(t)\} $.

Suppose that there are $ n $ insurers in the insurance market, named insurer $ 1, 2, \cdots, n $. The $ i $th insurer's claim process $ \{Z_i(t), t\geq0\} $ is described by the drifted Brownian motion

$ \begin{eqnarray*} dZ_i(t)=a_{i}dt-\sigma_{i_0} dW_{i_0}(t), \end{eqnarray*} $

where $ a_{i} $ is the expectation of claim, $ \sigma_{i_0} $ is the volatility. According to the expected value premium principle, the $ i $th insurer's premium rate is $ c_i=(1+\eta_{i_1})a_i $, where $ \eta_{i_1} $ is the $ i $th insurer's relative safety loading.

The insurer can buy proportional reinsurance or acquire new business to reduce the claim risk. Let $ q_i(t)(0\leq q_i(t)\leq1) $ be the proportional reinsurance level, that is, when a claim occurs, the reinsurer pays $ 100(1-q_i(t))\% $ of the claim while the insurer pays $ 100q_i(t)\% $. In this case, the insurer should distribute part of the premium to the reinsurer at the rate of $ (1+\eta_{i_2})a_i(1-q_i(t)) $, where $ \eta_{i_2}(>\eta_{i_1}) $ refers to the safety loading of the reinsurer. It is worth to notice that $ q_i(t)\in(1, +\infty) $ means acquiring new business. With the reinsurance strategy $ q_i(t) $, $ t\in[0, T] $, the $ i $th insurer's surplus process can be expressed by

$ \begin{eqnarray*} dR_i(t)=a_i(\theta_i+\eta_{i_2}q_i(t))dt+\sigma_{i_0} q_i(t)dW_{i_0}(t), \end{eqnarray*} $

where $ \theta_i=\eta_{i_1}-\eta_{i_2} $. We assume that the financial market contains a risk-free asset, $ n $ risky assets and a defaultable bond. The risk-free asset's price process $ B(t) $ is described by $ dB(t)=rB(t)dt, \; B(0)=b_0>0, $ where $ r>0 $ is the risk-free asset interest rate. The price process of the $ j $th risky asset $ S_j(t) $ follows the CEV model $ dS_j(t)=S_j(t)[\mu_jdt+\sigma_jS_j^\beta(t) dW_j(t)], \; S_j(0)=s_{j_0}, $ where $ \mu_j $ is the expected instantaneous return rate of the $ j $th risky asset; $ \sigma_j $ is a positive constant; $ \beta $ is called the elasticity parameter; $ \sigma_jS_j(t)^\beta $ is the instantaneous volatility. As usual, we assume that $ \mu_j>r $ and $ \beta $ satisfies the general condition $ \beta\geq0 $. Let $ T_1>T $ denotes the maturity time of the defaultable bond, and a nonnegative random variable $ \tau $ represents the default time of the corporate issuing this bond. The dynamics of the defaultable bond under measure $ P $ can be described as

$ \begin{eqnarray*} dp(t, T_1)=p(t-, T_1)[rdt+(1-H(t))(1-\Delta)\delta dt-(1-H(t-))\zeta dM^p(t)], \end{eqnarray*} $

where, the default process $ {H(t)} $ has a constant default intensity $ h^p $, $ 1/\Delta\geq1 $ is the default risk premium, $ \zeta\in[0, 1] $ denotes the constant loss rate when a default happens, 1-$ \zeta $ is the default recovery rate and $ \delta $ represents the risk neutral credit spread. Let $ \mathcal{G}_{t}=\mathcal{F}_{t}\vee \sigma\{H(s):0\leq s \leq t\} $ such that $ \mathbb{G} :=\{\mathcal{G}_t\}_{t\in[0, T]} $ is the smallest filtration. The martingale jump process $ M^p(t) $ is given by $ M^p(t) :=H(t)-\int_{0}^{t}(1-H(u-))h^p du $, which is a ($ \mathbb{G}, P $)-martingale.

Suppose that each insurer is allowed to invest his surplus in financial market described above. Let $ \pi_{i1}(t) $, $ \pi_{i2}(t) $, $ \cdots $, $ \pi_{in}(t) $ and $ \pi_{ip}(t) $ represent the money amounts that the $ i $th insurer invests in risky assets $ 1 , 2, \cdots, n $ and the defaultable bond at time $ t $, the rest of the surplus is then invested in the risk-free asset. We denote the whole reinsurance-investment strategy by $ \pi_{i}(t)=(q_i(t), \pi_{i1}(t), \pi_{i2}(t), \cdots, \pi_{in}(t), \pi_{ip}(t)) $. Based on the trading strategy $\{ \pi_{i}(t), t\in[0, T] \}$, the $ i $th insurer's wealth process $ X_{i}^{\pi_{i}}(t) $ is presented by

$ \begin{align} \label{1} dX_{i}^{\pi_{i}}(t)=&dR_i(t)+\frac{X_{i}^{\pi_{i}}(t)-\sum \limits_{j=1}^{n}\pi_{ij}(t)-\pi_{ip}(t)}{B(t)}dB(t)+\sum \limits_{j=1}^{n}\frac{\pi_{ij}(t)}{S_j(t)}dS_j(t)+\frac{\pi_{ip}(t)}{p(t-, T_1)}dp(t-, T_1)\\ =&[rX_{i}^{\pi_{i}}(t)+a_i(\theta_i+\eta_{i_2}q_i(t))+\sum \limits_{j=1}^{n}(\mu_j-r)\pi_{ij}(t)+(1-H(t))(1-\Delta)\delta\pi_{ip}(t)]dt\\ &+\sum \limits_{j=1}^{n}\pi_{ij}(t)\sigma_jS_j^\beta(t) dW_j(t)+\sigma_{i_0} q_i(t)dW_{i_0}(t)-(1-H(t-))\zeta\pi_{ip}(t)dM^p(t), \end{align} $

(2.1)

where $ X_{i}^{\pi_{i}}(0)=x_{i_{0}} $ is the $ i $th insurer's initial wealth.

Definition 2.1  For any fixed $ t\in[0, T] $, a trading strategy $ \pi_i(t)=(q_i(t), \pi_{i1}(t), \pi_{i2}(t), $ $ \cdots, \pi_{in}(t), \pi_{ip}(t)) $ is said to be admissible if it satisfies

(1) $ \pi_{i}(t) $ is $ \mathcal{G} $-progressively measurable;

(2) $ \forall t\in[0, T], q_i(t)\in[0, +\infty) $ and $ E_{P}\{\int_{0}^{T}[\sum \limits_{j=1}^{n}\pi_{ij}^2(t)S_{j}^{2\beta}(t)+\pi_{ip}^2(t)+q_i^2(t)]dt\}<\infty $;

(3) For $ \forall x_{i_{0}}\in\mathbb{R} $, the stochastic differential equation (2.1) with respect to $ \pi_i(t) $ has a pathwise unique solution $ X_{i}^{\pi_{i}}(t) $ satisfying $ {\rm E}_{P}[{\rm exp}(-\gamma_{i}X_{i}^{\pi_{i}}(t))]<\infty $.

Let $ \Pi_i $ denote the set of all admissible strategies of the $ i $th insurer. In a competitive market, each insurer considers the relative performance compared with the others. Following Espinosa and Touzi [18], we define the relative performance for $ n>2 $ insurers as the difference between the performance of an insurer and the average performance of his competitors. For the $ i $th insurer, the average performance of his competitors is defined as

$ \begin{align*} \bar X_i^{(\pi_m)_{m\neq i}}(t) :=\frac{1}{n-1}\sum \limits_{m=1, m\neq i}^{n}X_m^{\pi_m}(t)=\frac{1}{n-1}\sum \limits_{m\neq i}X_m^{\pi_m}(t), \end{align*} $

here $ (\pi_m)_{m\neq i} :=(\pi_1(t), \pi_2(t), \cdots, \pi_{i-1}(t), \pi_{i+1}(t), \cdots, \pi_n(t)) $. We use $ \hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(t) $ to represent the $ i $th insurer's relative wealth process, which is denoted by

$ \begin{align} \hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(t)=(1-\tau_i)X_i^{\pi_i}(t)+\tau_i(X_i^{\pi_i}(t)-\bar X_i^{(\pi_m)_{m\neq i}}(t)), \end{align} $

(2.2)

here the constant $ \tau_i\in[0, 1) $ measures the $ i $th insurer's sensitivity to the average performance of his competitors and reflects the degree of competition among $ n $ insurers. Combining equations (2.1) and (2.2), the $ i $th insurer's relative wealth process can be written as

$ \begin{align} &d\hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(t)\\ =&\big[r\hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(t)+a_i(\theta_i+\eta_{i_2}q_i(t))-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_m(t))\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj})(t)+(1-H(t-))(1-\Delta)\delta f(\pi_{ip}, \pi_{mp})(t)\big]dt+\sigma_{i_0}q_i(t)dW_{i_0}(t)\\ &+\sum \limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj})(t)\sigma_jS_j^\beta(t)dW_j(t)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\sigma_{m_0}q_m(t)dW_{m_0}(t)\\ &-(1-H(t-))\zeta f(\pi_{ip}, \pi_{mp})(t)dM^p(t), \end{align} $

(2.3)

here $ f(\pi_{ij}, \pi_{mj})(t):=\pi_{ij}(t)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\pi_{mj}(t) $, $ f(\pi_{ip}, \pi_{mp})(t):=\pi_{ip}(t)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\pi_{mp}(t) $.

In this paper, we assume the $ i $th insurer has an exponential utility function, i.e.,

$ \begin{eqnarray} U_i(x)=-\frac{1}{\gamma_i}e^{-\gamma_ix}, \end{eqnarray} $

(2.4)

with the constant risk aversion parament $ \gamma_i>0 $. The $ i $th insurer looks for the optimal reinsurance and investment strategy to maximize the objective function

$ \begin{equation} \begin{aligned} &J_{i}^{\pi_i, (\pi_m)_{m\neq i}}(t, \hat x_i, \bar s, z)={\rm E}_{t, \hat x_i, \bar s, z}\left[U_i\left(\hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(T)\right)\right] \end{aligned} \end{equation} $

(2.5)

where $ {\rm E}_{t, \hat x_i, \bar s, z}\left[\cdot\right]={\rm E}_{t, \hat x_i, \bar s, z}[\cdot|\hat X_i^{\pi_i, (\pi_m)_{m\neq i}}(t)=\hat x_i, \bar S(t)=\bar s, H(t)=z] $. $ \bar s=(s_1, s_2, \cdots, s_n) $ and $ \bar S(t)=(S_1(t), S_2(t), \cdots, S_n(t)) $ are two row vectors.

Problem 1  The classical non-zero-sum stochastic differential game between $ n $ competing insurers is to find a Nash equilibrium $ (\pi_{1}^{*}, \pi_{2}^{*}, \cdots, \pi_{n}^{*})\in\Pi_1\times\Pi_2\times\cdots\times\Pi_n $ such that for any $ (\pi_{1}, \pi_{2}, \cdots, \pi_{n})\in\Pi_1\times\Pi_2\times\cdots\times\Pi_n $, we have

$ \begin{equation} \begin{aligned} &J_{1}^{\pi_{1}^{*}, \pi_{2}^{*}, \ldots , \pi_{n}^{*}}(t, \hat x_1, \bar s, z)\geq J_{1}^{\pi_{1}, \pi_{2}^{*}, \ldots , \pi_{n}^{*}}(t, \hat x_1, \bar s, z), \\ &J_{2}^{\pi_{1}^{*}, \pi_{2}^{*}, \ldots , \pi_{n}^{*}}(t, \hat x_2, \bar s, z)\geq J_{2}^{\pi_{1}^{*}, \pi_{2}, \ldots , \pi_{n}^{*}}(t, \hat x_2, \bar s, z), \\ &\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \vdots\\ &J_{n}^{\pi_{1}^{*}, \pi_{2}^{*}, \ldots , \pi_{n}^{*}}(t, \hat x_n, \bar s, z)\geq J_{1}^{\pi_{1}^{*}, \pi_{2}^{*}, \ldots , \pi_{n}}(t, \hat x_n, \bar s, z). \end{aligned} \end{equation} $

(2.6)

In this section, we derive the equilibrium strategies and corresponding equilibrium value functions in the post-default case($ z=1 $) and the pre-default case($ z=0 $), respectively.

By the stochastic control theory, we define the $ i $th insurer's value function $ V^i(t, \hat x_i, \bar s, z) $ by

$ \begin{eqnarray} V^i(t, \hat x_i, \bar s, z)=\sup\limits_{\pi_{i}\in\Pi_i}{\rm E}_{t, \hat x_i, \bar s, z}\left[U\left(\hat X_i^{\pi_i, (\pi_{m}^{*})_{m\neq i}}(T)\right)\right] \end{eqnarray} $

(3.1)

with $ V^i(T, \hat x_i, \bar s, z)=U_i(\hat x_i) $.

Before giving the Hamilton-Jacobi-Bellman(HJB) equation, we first denote $ C^{1, 2, 2, \cdots, 2}([0, T]\times \mathbb{R}\times \mathbb{R^+}\times\cdots\times\mathbb{R^+}\times\{0, 1\}):=\big\{\omega(t, x, s_1, \cdots, s_n, z)|\omega(t, x, s_1, \cdots, s_n, z) $ is once continuously differentiable in $ t $, and twice continuously differentiable in $ s_1 $, $ s_2 $, $ \cdots $, $ s_n $ and $ x \big\}$. In addition, we omit the function parameters for notational simplicity in the following paragraph.

For any functions $ V^i(t, \hat x_i, s_1, \cdots, s_n, 1) $, $ V^i(t, \hat x_i, s_1, \cdots, s_n, 0) $ $ \in C^{1, 2, 2, \cdots, 2}([0, T]\times \mathbb{R}\times \mathbb{R^+}\times\cdots\times\mathbb{R^+}\times\{0, 1\}) $, denote the variational operator $ \mathcal{A}^{\pi_{i}, (\pi_{m})_{m\neq i}}V^i $ as follows:

$ \begin{eqnarray} \begin{split} \mathcal{A}^{\pi_{i}, (\pi_{m})_{m\neq i}}V^i =&V_t^i+[r\hat x_i+a_i(\theta_i+\eta_{i_2}q_i)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_m)\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj})+(1-z)\delta f(\pi_{ip}, \pi_{mp})]V_{\hat x_i}^i+\sum \limits_{j=1}^{n}\mu_js_jV_{s_j}^i\\ &+\frac{1}{2}[\sum \limits_{j=1}^{n}f^2(\pi_{ij}, \pi_{mj})\sigma_j^2s_j^{2\beta}+\sigma_{i_0}^2q_i^2+\frac{\tau_i^2}{(n-1)^2}\sum\limits_{m\neq i}\sigma_{m_0}^2q_m^2]V_{\hat x_i\hat x_i}^i\\ &+\sum\limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj})\sigma_j^2s_j^{2\beta+1}V_{\hat x_is_j}^i+\frac{1}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}V_{s_js_j}^i\\ &+h^p[V^i(t, \bar s, \hat x_i-\zeta f(\pi_{ip}, \pi_{mp}), 1)-V^i(t, \bar s, \hat x_i, 0)](1-z), \end{split} \end{eqnarray} $

(3.2)

where $ V_t^i, V_{\hat x_i}^i, V_{s_i}^i, V_{\hat x_i\hat x_i}^i, V_{\hat x_is_j}^i $ and $ V_{s_js_j}^i $ are the partial derivative of function $ V^i(t, \hat x_i, s_1, \cdots, s_n, z) $.

According to the dynamic programming principle, the HJB equation can be derived as

$ \begin{eqnarray} \sup\limits_{\pi_{i}\in\Pi_i}\{\mathcal{A}^{\pi_{i}, (\pi_{m}^{*})_{m\neq i}}V^i(t, \hat x_i, \bar s, z)\}=0, \; \; \; \; 0\leq t<T \end{eqnarray} $

(3.3)

with the boundary condition $ V^i(T, \hat x_i, \bar s, z)=U_i(\hat x_i) $.

In the post-default case, we have $ p(t, T_1)=0 $, $ \tau\leq t\leq T $. Thus $ \pi_{ip}^{*}(t)=0 $ for $ i\in{1, 2, \cdots, n} $. Form the definition of $ \mathcal{A}^{\pi_{i}, (\pi_{m})_{m\neq i}}V^i $, Eq. (3.3) is expressed as

$ \begin{eqnarray} \begin{split} &\sup\limits_{\pi_{i}\in\Pi_i}\Big\{V_t^i+[r\hat x_i+a_i(\theta_i+\eta_{i_2}q_i)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_{m}^{*})\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj}^{*})]V_{\hat x_i}^i+\sum \limits_{j=1}^{n}\mu_js_jV_{s_j}^i+\frac{1}{2}[\sum \limits_{j=1}^{n}f^2(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta}+\sigma_{i_0}^2q_i^2\\ &+\frac{\tau_i^2}{(n-1)^2}\sum\limits_{m\neq i}\sigma_{m_0}^2q_{m}^{*2}]V_{\hat x_i\hat x_i}^i+\sum\limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta+1}V_{\hat x_is_j}^i+\frac{1}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}V_{s_js_j}^i\Big\}=0. \end{split} \end{eqnarray} $

(3.4)

We try to infer that the solution of Eq. (3.4) satisfies the following functional form

$ \begin{eqnarray} V^i(t, \hat x_i, \bar s, 1)=V^i(t, \hat x_i, s_1, s_2, \cdots, s_n, 1)=-\frac{1}{\gamma_i}e^{-\gamma _i[e^{r(T-t)}\hat x_i+\sum\limits_{j=1}^{n}G_{ij}(t, s_j)-F_{i1}(t)]}, \end{eqnarray} $

(3.5)

with the boundary conditions $ F_{i1}(T)=0 $ and $ G_{ij}(T, s_j)=0 $, $ j=1, 2, \cdots, n $. Substituting (3.5) into Eq. (3.4), we can obtain

$ \begin{eqnarray} \begin{split} &\sup\limits_{\pi_{i}\in\Pi_i}\Big\{-\sum\limits_{j=1}^{n}\frac{\partial G_{ij}}{\partial t}+F'_{i1}(t)-[a_i(\theta_i+\eta_{i_2}q_i)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_{m}^{*})\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj}^{*})]e^{r(T-t)}-\sum \limits_{j=1}^{n}\mu_js_j\frac{\partial G_{ij}}{\partial s_j}+\frac{\gamma_i}{2}[\sum \limits_{j=1}^{n}f^2(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta}\\ &+\sigma_{i_0}^2q_i^2+\frac{\tau_i^2}{(n-1)^2}\sum\limits_{m\neq i}\sigma_{m_0}^2q_{m}^{*2}]e^{2r(T-t)}+\gamma_ie^{r(T-t)}\sum\limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta+1}\frac{\partial G_{ij}}{\partial s_j}\\ &+\frac{\gamma_i}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}(\frac{\partial G_{ij}}{\partial s_j})^2-\frac{1}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}\frac{\partial^2 G_{ij}}{\partial s_{j}^{2}}\Big\}=0. \end{split} \end{eqnarray} $

(3.6)

By the first order condition of Eq.(3.6), we have:

$ \begin{eqnarray} &&q_{i}^{*}(t)=\frac{a_i\eta_{i_2}}{\gamma_i\sigma_{i_0}^{2}e^{r(T-t)}}, \end{eqnarray} $

(3.7)

$ \begin{eqnarray} &&\pi_{ij}^{*}(t)=\frac{\mu_j-r}{\gamma_i\sigma_{j}^{2}s_{j}^{2\beta}e^{r(T-t)}}-\frac{s_j}{e^{r(T-t)}}\frac{\partial G_{ij}}{\partial s_j}+\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\pi_{mj}^{*}(t). \end{eqnarray} $

(3.8)

Inserting Eqs.(3.7) and (3.8) into Eq.(3.6), we have:

$ \begin{eqnarray} \begin{split} &-\sum\limits_{j=1}^{n}[\frac{\partial G_{ij}}{\partial t}+rs_j\frac{\partial G_{ij}}{\partial s_j}+\frac{1}{2}\sigma_j^2s_j^{2\beta+2}\frac{\partial^2 G_{ij}}{\partial s_{j}^{2}}+\frac{(\mu_j-r)^2}{2\gamma_i\sigma_j^2s_j^{2\beta}}-\frac{\tau_ia_j^2\eta_{j_2}^{2}}{(n-1)\gamma_j\sigma_{j_0}^{2}} -\frac{\gamma_i\tau_i^2a_j^2\eta_{j_2}^2}{2(n-1)^2\gamma_j^2\sigma_{j_0}^{2}}]\\&+F'_{i1}(t)-[a_i\theta_i-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m\theta_m]e^{r(T-t)}-(1+\frac{\tau_i}{n-1})^2\frac{a_i^2\eta_{i_2}^{2}}{2\gamma_i\sigma_{i_0}^{2}}=0. \end{split} \end{eqnarray} $

(3.9)

Eq.(3.9) can be decomposed into two equations

$ \begin{eqnarray} &&\frac{\partial G_{ij}}{\partial t}+rs_j\frac{\partial G_{ij}}{\partial s_j}+\frac{1}{2}\sigma_j^2s_j^{2\beta+2}\frac{\partial^2 G_{ij}}{\partial s_{j}^{2}} +\frac{(\mu_j-r)^2}{2\gamma_i\sigma_j^2s_j^{2\beta}} -\frac{\tau_ia_j^2\eta_{j_2}^{2}}{(n-1)\gamma_j\sigma_{j_0}^{2}} -\frac{\gamma_i\tau_i^2a_j^2\eta_{j_2}^2}{2(n-1)^2\gamma_j^2\sigma_{j_0}^{2}}=0, \\ \end{eqnarray} $

(3.10)

$ \begin{eqnarray} &&F'_{i1}(t)-[a_i(\theta_i-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m\theta_m]e^{r(T-t)}-(1+\frac{\tau_i}{n-1})^2\frac{a_i^2\eta_{i_2}^{2}}{2\gamma_i\sigma_{i_0}^{2}}=0. \end{eqnarray} $

(3.11)

In order to solve Eq.(3.10), we define

$ \begin{eqnarray} G_{ij}(t, s_j)=A_{ij}(t)+B_{ij}(t)s_{j}^{-2\beta}, \end{eqnarray} $

(3.12)

and the boundary conditions are $ A_{ij}(T)=0 $ and $ B_{ij}(T)=0 $. Plugging Eq.(3.12) into (3.10), we derive

$ \begin{eqnarray} A'_{ij}(t)&+&\beta(2\beta+1)\sigma_{j}^{2}B_{ij}(t)-\frac{\tau_ia_j^2\eta_{j_2}^{2}}{(n-1)\gamma_j\sigma_{j_0}^{2}} -\frac{\gamma_i\tau_i^2a_j^2\eta_{j_2}^2}{2(n-1)^2\gamma_j^2\sigma_{j_0}^{2}}\\ &+&[B'_{ij}(t)-2\beta rB_{ij}(t)+\frac{(\mu_j-r)^2}{2\gamma_i\sigma_j^2}]s_{j}^{-2\beta}=0. \end{eqnarray} $

(3.13)

By matching coefficients, we obtain two differential equations. Considering the boundary conditions $ A_{ij}(T)=0 $ and $ B_{ij}(T)=0 $, we have

$ \begin{eqnarray} \begin{split} &A_{ij}(t)=\beta(2\beta+1)\sigma_{j}^{2}\int_{t}^{T}B_{ij}(u)du-\frac{\tau_i}{n-1}[1-\frac{\tau_i\gamma_i}{2(n-1)\gamma_j}]\frac{a_j^2\eta_{j_2}^2}{\gamma_j\sigma_{j_0}^{2}}(T-t), \\ &B_{ij}(t)=\frac{(\mu_j-r)^2}{4\beta r\gamma_i\sigma_j^2}[1-e^{-2\beta r(T-t)}]. \end{split} \end{eqnarray} $

(3.14)

Finally, the solution of Eq. (3.11) is

$ \begin{eqnarray} F_{i1}(t)=-[a_i\theta_i-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m\theta_m]\int_{t}^{T}e^{r(T-u)}du-(1+\frac{\tau_i}{n-1})^2\frac{a_i^2\eta_{i_2}^{2}}{2\gamma_i\sigma_{i_0}^{2}}(T-t). \end{eqnarray} $

(3.15)

Lemma 3.1  Eqs.(3.8) has a unique root $ \pi_{ij}^{*}(t) $ is given by

$ \begin{eqnarray*} \pi_{ij}^{*}(t)=\frac{\phi_{ij}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} +\frac{\frac{\tau_i}{n-1}}{1+\frac{\tau_i}{n-1}}\frac{1}{1-\psi_n}\sum\limits_{i=1}^{n}\frac{\phi_{ij}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} \; \; \; i, j=1, 2, \cdots, n, \end{eqnarray*} $

where

$ \begin{eqnarray} \phi_{ij}(t)=\frac{\mu_j-r}{\gamma_j\sigma_j^2s_{j}^{2\beta}}+2\beta B_{ij}(t)s_{j}^{-2\beta}, \; \; \; \; \; \; \; \psi_n=\sum\limits_{i=1}^{n}\frac{\frac{\tau_i}{n-1}}{1+\frac{\tau_i}{n-1}}. \end{eqnarray} $

(3.16)

Proof  Similar to the one of Lemma 4.2 in ^[17], and we omit it.

In the pre-default case, Eq.(3.3) can be rewritten as follows

$ \begin{eqnarray} \begin{split} &\sup\limits_{\pi_{i}\in\Pi_i}\Big\{V_t^i+[r\hat x_i+a_i(\theta_i+\eta_{i_2}q_i)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_{m}^{*})+\delta f(\pi_{ip}, \pi_{mp}^{*})(t)\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj}^{*})]V_{\hat x_i}^i+\sum \limits_{j=1}^{n}\mu_js_jV_{s_j}^i+\frac{1}{2}[\sum \limits_{j=1}^{n}f^2(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta}+\sigma_{i_0}^2q_i^2\\ &+\frac{\tau_i^2}{(n-1)^2}\sum\limits_{m\neq i}\sigma_{m_0}^2q_{m}^{*2}]V_{\hat x_i\hat x_i}^i+\sum\limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta+1}V_{\hat x_is_j}^i+\frac{1}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}V_{s_js_j}^i\\ &+h^p[V^i(t, \bar s, \hat x_i-\zeta f(\pi_{ip}, \pi_{mp}^{*}), 1)-V^i(t, \bar s, \hat x_i, 0)]\Big\}=0. \end{split} \end{eqnarray} $

(3.17)

Similarly, we guess that $ V^i(t, \hat x_i, \bar s, 0) $ is of the following form

$ \begin{eqnarray} V^i(t, \hat x_i, \bar s, 0)=V^i(t, \hat x_i, s_1, s_2, \cdots, s_n, 0)=-\frac{1}{\gamma_i}e^{-\gamma _i[e^{r(T-t)}\hat x_i+\sum\limits_{j=1}^{n}G_{ij}(t, s_j)-F_{i0}(t)]} \end{eqnarray} $

(3.18)

with the boundary condition $ F_{i0}(T)=0 $. Inserting (3.5) and (3.18) into HJB equation (3.17), we derive

$ \begin{eqnarray} \begin{split} \sup\limits_{\pi_{i}\in\Pi_i}\Big\{&-\sum\limits_{j=1}^{n}\frac{\partial G_{ij}}{\partial t}+F'_1(t)-[a_i(\theta_i+\eta_{i_2}q_i)-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m(\theta_m+\eta_{m_2}q_{m}^{*})\\ &+\sum \limits_{j=1}^{n}(\mu_j-r)f(\pi_{ij}, \pi_{mj}^{*})+\delta f(\pi_{ip}, \pi_{mp}^{*})]e^{r(T-t)}-\sum \limits_{j=1}^{n}\mu_js_j\frac{\partial G_{ij}}{\partial s_j}\\ &+\frac{\gamma_i}{2}[\sum \limits_{j=1}^{n}f^2(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta}+\sigma_{i_0}^2q_i^2+\frac{\tau_i^2}{(n-1)^2}\sum\limits_{m\neq i}\sigma_{m_0}^2q_{m}^{*2}]e^{2r(T-t)}\\ &+\gamma_ie^{r(T-t)}\sum\limits_{j=1}^{n}f(\pi_{ij}, \pi_{mj}^{*})\sigma_j^2s_j^{2\beta+1}\frac{\partial G_{ij}}{\partial s_j}+\frac{\gamma_i}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}(\frac{\partial G_{ij}}{\partial s_j})^2\\ &-\frac{1}{2}\sum\limits_{j=1}^{n}\sigma_j^2s_j^{2\beta+2}\frac{\partial^2 G_{ij}}{\partial s_{j}^{2}}+\frac{h^p}{\gamma_i}e^{\gamma_i[\zeta e^{r(T-t)}f(\pi_{ip}, \pi_{mp}^{*})+F_{i1}(t)-F_{i0}(t)]}-\frac{h^p}{\gamma_i}\Big\}=0. \end{split} \end{eqnarray} $

(3.19)

The first order maximizing condition for the optimal investment strategy is

$ \begin{eqnarray} &&q_{i}^{*}(t)=\frac{a_i\eta_{i_2}}{\gamma_i\sigma_{i_0}^{2}e^{r(T-t)}}, \end{eqnarray} $

(3.20)

$ \begin{eqnarray} &&\pi_{ij}^{*}(t)=\frac{\mu_j-r}{\gamma_i\sigma_{j}^{2}s_{j}^{2\beta}e^{r(T-t)}}-\frac{s_j}{e^{r(T-t)}}\frac{\partial G_{ij}}{\partial s_j}+\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\pi_{mj}^{*}(t), \end{eqnarray} $

(3.21)

$ \begin{eqnarray} &&\pi_{ip}^{*}(t)=\frac{ln\frac{\delta}{\zeta h^p}}{\zeta\gamma_ie^{r(T-t)}}-\frac{F_{i1}(t)-F_{i0}(t)}{\zeta e^{r(T-t)}}+\frac{\tau_i}{n-1}\sum \limits_{m\neq i}\pi_{mp}^{*}(t). \end{eqnarray} $

(3.22)

Inputting Eqs.(3.20), (3.21) and (3.22) into Eq.(3.19), we get

$ \begin{eqnarray} \begin{split} F'_{i0}(t)&-\frac{\delta}{\zeta}F_{i0}(t)+\frac{\delta}{\zeta}F_{i1}(t) -[a_i\theta_i-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m\theta_m]e^{r(T-t)}\\ &+\frac{\delta}{\zeta\gamma_i}(1-ln\frac{\delta}{\zeta h^p})-\frac{h^p}{\gamma_i}-(1+\frac{\tau_i}{n-1})^2\frac{a_i^2\eta_{i_2}^{2}}{2\gamma_i\sigma_{i_0}^{2}}=0, \; \; \; \; \; \; F_{i0}(T)=0. \end{split} \end{eqnarray} $

(3.23)

By solving Eq.(3.23), we obtain

$ \begin{eqnarray} F_{i0}(t)=\frac{\delta}{\zeta}e^{\frac{\delta}{\zeta}t}\int_{t}^{T}F_{i1}(u)e^{-\frac{\delta}{\zeta}u}du -f_{i1}[e^{-\frac{\delta}{\zeta}(T-t)}-e^{-r(T-t)}]-\frac{\zeta}{\delta}f_{i2}[e^{-\frac{\delta}{\zeta}(T-t)}-1], \end{eqnarray} $

(3.24)

here $ f_{i1}=\frac{1}{r-\frac{\delta}{\zeta}}[a_i\theta_i-\frac{\tau_i}{n-1}\sum \limits_{m\neq i}a_m\theta_m] $ and $ f_{i2}=\frac{\delta}{\zeta\gamma_i}(1-ln\frac{\delta}{\zeta h^p})-\frac{h^p}{\gamma_i}-(1+\frac{\tau_i}{n-1})^2\frac{a_i^2\eta_{i_2}^{2}}{2\gamma_i\sigma_{i_0}^{2}} $.

Lemma 3.2  Eqs.(3.22) has a unique root $ \pi_{ip}^{*}(t) $ which is given by

$ \begin{eqnarray*} \pi_{ip}^{*}(t)=\frac{\varphi_{i}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} +\frac{\frac{\tau_i}{n-1}}{1+\frac{\tau_i}{n-1}}\frac{1}{1-\psi_n}\sum\limits_{i=1}^{n}\frac{\varphi_{i}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} , \; \; i=1, 2, \cdots, n, \end{eqnarray*} $

here,

$ \begin{eqnarray} \varphi_i(t)=\frac{ln\frac{\delta}{\zeta h^p}}{\zeta\gamma_i}-\frac{F_{i1}(t)-F_{i0}(t)}{\zeta}. \end{eqnarray} $

(3.25)

Proof  Similar to Lemma 3.1, thus we omit it.

Theorem 3.1  For the wealth process (2.3) with $ n $ insurers and Problem 1, the $ i $th $ (i=1, 2, \cdots, n) $ insurer's equilibrium reinsurance and investment strategy $ \pi_i^*(t) $ is given by

$ \begin{align*} q_{i}^{*}(t)=&\frac{a_i\eta_{i_2}}{\gamma_i\sigma_{i_0}^{2}}e^{-r(T-t)}, \\ \pi_{ij}^{*}(t)=&\frac{\phi_{ij}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} +\frac{\frac{\tau_i}{n-1}}{1+\frac{\tau_i}{n-1}}\frac{1}{1-\psi_n}\sum\limits_{i=1}^{n}\frac{\phi_{ij}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)}, \; \; j=1, 2, \cdots, n, \\ \pi_{ip}^{*}(t)=&\frac{\varphi_{i}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)} +\frac{\frac{\tau_i}{n-1}}{1+\frac{\tau_i}{n-1}}\frac{1}{1-\psi_n}\sum\limits_{i=1}^{n}\frac{\varphi_{i}(t)}{1+\frac{\tau_i}{n-1}}e^{-r(T-t)}, \; \; t\in[0, \tau\wedge T], \end{align*} $

where $ \psi_n $ and $ \phi_{ij}(t) $ are shown in Eq. (3.16), $ \varphi_{i}(t) $ is given by Eq. (3.25).

Besides, the equilibrium value functions are

$ \begin{align*} V^i(t, \hat x_i, s_1, s_2, \cdots, s_n, 1)=&-\frac{1}{\gamma_i}e^{-\gamma _i[e^{r(T-t)}\hat x_i+\sum\limits_{j=1}^{n}G_{ij}(t, s_j)-F_{i1}(t)]}, \\ V^i(t, \hat x_i, s_1, s_2, \cdots, s_n, 0)=&-\frac{1}{\gamma_i}e^{-\gamma _i[e^{r(T-t)}\hat x_i+\sum\limits_{j=1}^{n}G_{ij}(t, s_j)-F_{i0}(t)]}, \end{align*} $

where $ G_{ij}(t, s_j) $ are given by (3.12), $ F_{i1}(t) $ and $ F_{i0}(t) $ are given by (3.15) and (3.24).

The following theorem verifies that the solution to HJB equation (3.3) given in Theorem 3.1 is indeed the solution to Problem 1.

Theorem 3.2  (Verification theorem) If $ W^i(t, \hat x_i, \bar s, z) $ be the solution of HJB equation (3.3) with the boundary condition $ V^i(T, \hat x_i, \bar s, z)=U_i(\hat x_i) $, and for $ \forall j=1, 2, \cdots, n $, the parameters satisfy one of the following conditions

$ \begin{eqnarray*} &&(a)\; r>(1-\frac{1}{\sqrt{6}})\mu_j;\\ &&(b)\; r<(1-\frac{1}{\sqrt{6}})\mu_j\; {\rm and}\; T <\frac{1}{\beta\sqrt{6(\mu_j-r)^2-r^2}}\arctan(-\frac{\sqrt{6(\mu_j-r)^2-r^2}}{r}), \end{eqnarray*} $

then the optimal value function is $ V^i(t, \hat x_i, \bar s, z)=W^i(t, \hat x_i, \bar s, z) $, and the optimal strategy is $ \pi_i^*(t) $, which is given in Theorem 3.1.

The proof of the verification theorem can be adapted from Theorem 2 of Gu et.al [7]. We hence omit it here.

In this section, we present a series of numerical simulations to analyze the effects of different model parameters on the equilibrium reinsurance-investment strategies. In the following examples, unless otherwise specified, we select the basic parameters as $ t=0 $, $ T=10 $, $ r=0.03 $, $ \beta=1 $, $ \zeta=0.4 $, $ h^p=0.005 $, $ \delta=0.02 $, and other model parameters are given in the Table 1.

Form the expression of $ q_{i}^{*}(t) $, we derive $ \frac{\partial q_{i}^{*}(t)}{a_i}>0 $, $ \frac{\partial q_{i}^{*}(t)}{\eta_{i_2}}>0 $, $ \frac{\partial q_{i}^{*}(t)}{\gamma_i}<0 $, $ \frac{\partial q_{i}^{*}(t)}{\sigma_{i_0}}<0 $ and $ \frac{\partial q_{i}^{*}(t)}{r}<0 $, which mean that as the expectation of claim $ a_i $ or the safety loading of the reinsurer $ \eta_{i_2} $ increases, the cost of reinsurance becomes higher, and thus the $ i $th insurer will take more risks by buying less reinsurance or acquiring more new businesses. When the $ i $th insurer's risk aversion coefficient $ \gamma_i $ is higher, or the volatility $ \sigma_{i_0} $ is larger, the $ i $th insurer prefers to transfer more risks into reinsurance, so as to purchase more reinsurance or acquire less new business. If the interest rate $ r $ is larger, the insurer will invest more funds in risk-free asset. As a result, the insurer will also buy more reinsurance or acquire less new business.

In this subsection, without loss of generality, we assume that the financial market consists of one risk-free bond, one defaultable bond and three risky assets, and that there are three insurers investing in the financial market.

Fig. 1 and Fig. 2 show the influences of $ \gamma_1 $, $ \gamma_2 $ and $ \gamma_3 $ on $ \pi_{11}^*(0) $, $ \pi_{12}^*(0) $, $ \pi_{13}^*(0) $, $ \pi_{21}^*(0) $ and $ \pi_{31}^*(0) $. From Fig. 1 and Fig. 2 we can see that $ \pi_{11}^*(0) $, $ \pi_{12}^*(0) $, $ \pi_{13}^*(0) $, $ \pi_{21}^*(0) $ and $ \pi_{31}^*(0) $ decrease with regard to $ \gamma_1, \gamma_2 $ and $ \gamma_3 $, respectively. It is observed that with the increase of $ \gamma_i(1\leq i\leq3) $ the insurers tend to present stronger risk-aversion. Hence, a larger $ \gamma_i(1\leq i\leq3) $ would cause less investment in risk assets. Because we consider the mutual influence among insurers, the increase in one insurer's risk-aversion coefficient would cause that the other two insurers also decrease their investment. However, from Fig. 1, we notice that each insurer is more sensitive to his own risk-aversion coefficient.

Fig. 3 displays the effects of $ \tau_1 $, $ \tau_2 $ and $ \tau_3 $ on $ \pi_{11}^*(0) $, $ \pi_{12}^*(0) $ and $ \pi_{13}^*(0) $, respectively. We can observe from Fig. 3 (a) that, with the increase of $ \tau_1 $, the first insurer increases his investments in three risky assets. Here, a larger $ \tau_1 $ means that the first insurer would put more weight on the relative average performance, and that the competition becomes more intense. To beat competitors, the first insurer would invest more of his surplus not only in the first risky asset, but also in the second and third risky asset. The similar phenomenon also appears in Fig. 3 (b) and (c). As $ \tau_2 $ or $ \tau_3 $ gradually increases, the first insurer would invest more of its wealth in risky assets in order to gain an advantage over the competition.

Fig. 4 further captures how $ \pi_{11}^*(0) $, $ \pi_{21}^*(0) $ and $ \pi_{31}^*(0) $ vary with respect to the competition parameters $ \tau_1 $, $ \tau_2 $ and $ \tau_3 $. Similar to the result for Fig. 3, the growth of $ \tau_i(1\leq i\leq3) $ indicates that the $ i $th insurer cares more about his competitor's performance at terminal time, and hence he tends to increase exposure on risky assets, i.e., investing more amount of surplus in the risky assets. It can also be seen that each insurer's relative concern parameter has a more significant influence on their own investment.

Fig. 5 illustrates the effects of $ \gamma_1, \gamma_2 $ and $ \gamma_3 $ on $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $, $ \pi_{3p}^*(0) $. Here $ \gamma_i(1\leq i\leq3) $ is the $ i $th insurer's risk-aversion coefficient. Similar to that in Fig. 1, $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $, $ \pi_{3p}^*(0) $ decrease with regard to $ \gamma_1 $, $ \gamma_2 $ and $ \gamma_3 $.

To demonstrate the impacts of $ \tau_i(1\leq i\leq3) $ on the equilibrium pre-default bond investment strategies $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $, we refer to Fig. 6. From these subfigures, we find that $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $ are increasing functions of $ \tau_i(1\leq i\leq3) $ in all three cases. In other words, if an insurer is more concerned with his relative terminal wealth, he would retain more risks, so more money would invest in the defaultable bond. In addition, from Fig. 6 (a), for fixed $ \tau_2 $ and $ \tau_3 $, a greater $ \tau_1 $ produces a larger $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $, which implies that the second and third insurer would invest more surplus in the corporate bond when his competitor is aggressive. Fig. 6 (b) and (c) reveal a similar phenomenon.

Fig. 7 depicts the effects of loss rate $ \zeta $, credit spread $ \delta $ and default intensity $ h^p $ on the equilibrium investment strategies $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $. As is well illustrated in Fig. 7 (a) and (b), $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $ decrease w.r.t. $ \zeta $ while increase w.r.t. $ \delta $. So the insurer will invest less in the defaultable bond when he has higher loss rate or lower credit spread. From Fig. 7 (c), we find that $ \pi_{1p}^*(0) $, $ \pi_{2p}^*(0) $ and $ \pi_{3p}^*(0) $ have a positive relationship with $ h^p $. Indeed, a larger $ h^p $ means a stronger default intensity, and therefore, the less money is invested in the defaultable bond.

In this subsection, we take the first insurer as an example to illustrate the influences of the number of insurers on insurer's investment strategies. Fig. 8 and Fig. 9 give the sensitivity of $ \pi_{11}^*(0) $, $ \pi_{12}^*(0) $, $ \pi_{13}^*(0) $ and $ \pi_{1p}^*(0) $ to the number of insurers. Here we respectively assume that there are one insurer, two insurers and three insurers investing in the above financial market. We can see from Fig. 8 and Fig. 9 that the investment strategies $ \pi_{11}^*(0) $, $ \pi_{12}^*(0) $, $ \pi_{13}^*(0) $ and $ \pi_{1p}^*(0) $ increase with the rising number of insurers. This is because, with the increase of the number of insurers, the insurer will have access to more market information. That would boost his confidence and make him invest more of his surplus in risky assets and the defaultable bond.

In this paper, we investigate a non-zero-sum stochastic differential investment and reinsurance game involving a defaultable security for $ n $ competitive insurers. The insurer's claim process is assumed to follow a drifted Brownian motion. We allow each insurer to dynamically purchase proportional reinsurance and allocate his surplus to a risk-free asset, a defaultable bond and n risky assets. Moreover, we adopt the CEV model to describe each risky asset's price process. By adopting a stochastic control approach, we derive the HJB equations under both pre-default case and post-default case. Explicit expressions of equilibrium strategy that maximize the expected exponential utility of the terminal wealth relative to that of his competitors and corresponding equilibrium value functions are obtained. At last, numerical examples and sensitivity analyses are presented to illustrate the impacts of parameters on the equilibrium strategy. Results indicate that the risk-aversion coefficient of both the insurer and his competitors have an impact on the insurer's investment strategy. Furthermore, we find that competition will increase each insurer's investment in risky assets and defaultable bond, and the number of insurers will also influence each insurer's investment strategy.