2. Problem formulation

Consider an insurance company under contagious financial and insurance markets. The insurance company has two business lines, i.e., ordinary insurance products and catastrophic insurance products, as

(2.1) \begin{equation} C(t)=C_{1}(t)+C_{2}(t)=\overset {N_{1}(t)}{\sum \limits _{\kappa =1}}J_{1,\kappa }+\sum \limits _{j=1}^{N_2(t)}J_{2,j}. \end{equation}

The claim process of ordinary claims $C_{1}(t)$ follows the standard Poisson process $N_{1}(t)$ with a deterministic parameter $\eta$ $\gt 0$ , and claim sizes are denoted by $J_{1,\kappa }$ with the cumulative distribution function $F_{1}(\cdot )$ .

The catastrophic claims $C_2(t)$ have a loss model as follows:

(2.2) \begin{equation} C_2(t)=\sum \limits _{j=1}^{N_2(t)}J_{2,j}, \end{equation}

where $J_{2,j}$ are catastrophic claim sizes that occur jointly with $Y_j$ in the intensity process with the joint distribution function $\Psi (y,j_2)$ , defined in (2.7). $N_{2}(t)$ follows the dynamic contagion process with a stochastic intensity process $\lambda (t)$ as follows:

(2.3) \begin{equation} N_{2}(t)=\sum \limits _{j\geq 1}\mathbb{I}\left ( T_{2,j}\leq t\right )_{j=1,2,\ldots }, \end{equation}

(2.4) \begin{equation} \lambda (t)=a+\left ( \lambda _{0}-a\right ) e^{-\delta t}+\sum \limits _{i\geq 1}K_{i}e^{-\delta \left ( t-T_{1,i}\right ) }\mathbb{I}\left ( T_{1,i}\leq t\right ) +\sum \limits _{j\geq 1}Y_{j}e^{-\delta \left ( t-T_{2,j}\right ) }\mathbb{I}\left ( T_{2,j}\leq t\right ), \end{equation}

where

1. i. $\lambda _{0}$ $\gt 0$ is the initial intensity at time $t=0$ ;

2. ii. $a$ $\geq 0$ is the constant mean-reverting level;

3. iii. $\delta$ $\gt 0$ is the rate of exponential decay;

4. iv. $\left \{ K_{i}\right \} _{i=1,2,\ldots }$ is a sequence of i.i.d. positive externally excited jumps that occur jointly with $L_i$ in the risky asset process with the joint distribution $\Phi (k,l),$ $k\gt 0,l\gt 0$ , where $L_i$ , defined in (2.9), denotes the negative effect of the $i$ th external event on the financial risky assets, at the corresponding random times $\left \{ T_{1,i}\right \} _{i=1,2,\ldots }$ following a Poisson process $M(t)$ with constant rate $\rho \gt 0$ , and $\mathbb{I}$ is the indicator function. The corresponding margins are $H(k)$ and $D(l)$ , respectively. $\Phi (k,l)$ measures the dependency between the positive effect on the intensity and the negative effect on the risky financial asset. $M(t)$ denotes the number of external events such as natural catastrophes, cyberattacks, and pandemics.

5. v. $\left \{ Y_{j}\right \} _{j=1,2,\ldots }$ is a sequence of i.i.d. positive self-excited jumps with distribution function $\Psi (y,j_{2}),$ $y\gt 0,$ $j_{2}\gt 0$ , where margins are $G(y)$ and $F_2(\cdot )$ , at the corresponding random times $\left \{ T_{2,j}\right \} _{j=1,2,\ldots }$ generated by the intensity process $\lambda (t)$ . $\Psi (y,j_{2})$ measures the dependency between the magnitude of contribution to the intensity after each claim for catastrophic insurance products and their loss size.

6. vi. $\left \{ T_{1,i}\right \} _{i=1,2,\ldots }$ , $\left \{ J_{1,\kappa }\right \} _{\kappa =1,2,\ldots }$ , $\left \{ \begin{matrix} K_{i}\\ L_{i} \end{matrix} \right \} _{i=1,2,\ldots }$ , and $\left \{ \begin{matrix} Y_{j}\\ J_{2,j} \end{matrix} \right \} _{j=1,2,\ldots }$ are assumed to be independent of each other.

In this paper, we implement the theory of copulas to consider two types of dependency structures from joint distributions $\Psi (y,j_2)$ and $\Phi (k,l).$ Copulas were first introduced by Sklar (1959) to deal with multivariate distribution functions. The formal definition of copulas is as follows.

Thanks to Sklar’s Theorem, which links copulas to joint probability distributions. We first state Sklar’s Theorem and then illustrate the setting for $\Psi (y,j_2)$ and $\Phi (k,l).$

Theorem 2.2. Let $\mathcal{F}$ be an $n$ -dimensional joint distribution with margins $\mathcal{F}_1,\ldots , \mathcal{F}_n$ . Then, there exists an $n$ -dimensional copula $\mathcal{C}$ such that, for any random vector $\textbf {x}\in \mathbb{R}^n$ ,

(2.5) \begin{equation} \mathcal{F}(x_1,\ldots ,x_n)=\mathcal{C}(\mathcal{F}_1(x_1),\ldots , \mathcal{F}_1(x_1)). \end{equation}

Furthermore, if $\mathcal{F}_1,\ldots , \mathcal{F}_n$ are continuous, then $\mathcal{C}$ is unique. Otherwise, $\mathcal{C}$ is unique on $\textbf{Ran}\mathcal{F}_1\times \ldots \times \textbf{Ran}\mathcal{F}_n$ ( $\textbf{Ran}$ denotes Range). More importantly, the copulas density function $c:[0,1]^n \rightarrow [0,\infty )$ follows

(2.6) \begin{equation} c(u_1,\ldots ,u_n)=\frac {\partial ^n\mathcal{C}(u_1,\ldots ,u_n)}{\partial u_1\ldots \partial u_n}=\frac {f(x_1,\ldots ,x_n)}{\Pi ^n_{i=1}f_i(x_i)}, \end{equation}

where $f(x_1,\ldots ,x_n)$ denotes the multivariate probability density function and $f_i(x_i)$ denotes the univariate probability density function of the margin $\mathcal{F}_i$ , for $i=1,\ldots ,n$ .

The proof of Sklar’s theorem can be found in Sklar (1959) and Schweizer & Sklar (1983). Now, Let $(u_1,u_2)\in [0,1]^2$ and $(\widehat {u}_1,\widehat {u}_2)\in [0,1]^2$ such that $u_1=G(y)$ , $u_2=F_2(j_2)$ , $\widehat {u}_1=H(k)$ , and $\widehat {u}_2=D(l)$ . Then, we introduce two bivariate copulas $\mathcal{C}_1(u_1,u_2)$ and $\mathcal{C}_2(\widehat {u}_1,\widehat {u}_2)$ with corresponding copula density functions $p_1(u_1,u_2)$ and $p_2(\widehat {u}_1,\widehat {u}_2)$ such that

(2.7) \begin{align} \mathcal{C}_1(u_1,u_2)&=\Psi (y,j_2),\nonumber\\[2pt] \mathcal{C}_2(\widehat {u}_1,\widehat {u}_2)&=\Phi (k,l),\nonumber\\[2pt] \widehat {\psi }(y,j_2)&=p_1(u_1,u_2)g_y(y)f_2(j_2),\nonumber\\[2pt] \widehat {\phi }(k,l)&=p_2(\widehat {u}_1,\widehat {u}_2)h(k)d(l), \end{align}

where $\widehat {\psi }(y,j_2)$ and $\widehat {\phi }(k,l)$ denote the bivariate joint density function of $\Psi (y,j_2)$ and $\Phi (k,l)$ , $g_y(y)$ , $f_2(j_2)$ , $h(k)$ , and $d(l)$ denote the univariate density function of margins $G(y)$ , $F_2(j_2)$ , $H(k)$ , and $D(l)$ , respectively. Different types of copulas can be considered, and we will use Gaussian copulas for numerical works in Section 5. The role of copula functions is to capture two dependency structures: one is for the dependency structure between the magnitude of the contribution of external events to the intensity and financial risky assets, and the other is the dependency structure between the magnitude of contribution to intensity after each catastrophic claim and the corresponding claim size. For notation simplification, the Laplace transform of all distribution functions is given below, where it is assumed that they are finite, $i.e.$ ,

Assumption 2.3.

\begin{align*} \overset {\wedge }{g}\left ( \zeta \right ) =\int \limits _{0}^{\infty }e^{-\zeta y}dG(y)\lt \infty \mathrm{, }\overset {\wedge }{f_{2}}\left ( \kappa \right ) =\int \limits _{0}^{\infty }e^{-\kappa j_{2}}dF_{2}(j_{2})\lt \infty \mathrm{, } \end{align*}

\begin{align*} \overset {\wedge }{h}\left ( \varpi \right ) =\int \limits _{0}^{\infty }e^{-\varpi k}\mathrm{ }dH(k)\lt \infty \mathrm{, } \overset {\wedge }{d}\left ( \varphi \right ) =\int \limits _{0}^{\infty }e^{-\varphi l}dD(l)\lt \infty \mathrm{, } \overset {\wedge }{f_{1}}\left (\varepsilon \right ) =\int \limits _{0}^{\infty }e^{-\varepsilon j_{1}}dF_{1}(j_{1})\lt \infty , \end{align*}

\begin{align*} \overset {\wedge }{\psi }\left ( \zeta ,\kappa \right ) =\int \limits _{0}^{\infty }\int \limits _{0}^{\infty }e^{-\zeta y}e^{-\kappa j_{2}}d\Psi ( y,j_{2}) \lt \infty \,\,\mathrm{ and }\, \, \overset {\wedge }{\pi }\left ( \varpi ,\varphi \right ) =\int \limits _{0}^{\infty }\int \limits _{0}^{\infty }e^{-\varpi k}e^{-\varphi l}d\Phi ( k,l) \lt \infty . \end{align*}

where $\zeta \gt 0,$ $\kappa \gt 0,$ $\varpi \gt 0,$ $\varphi \gt 0,$ and $\varepsilon \gt 0.$

The negative sign of first-order (partial) derivatives of these Laplace transforms are also given below, where it is assumed that they are finite, $i.e.$ ,

Assumption 2.4.

\begin{align*} -\frac {d\overset {\wedge }{g}\left ( \zeta \right ) }{d\zeta } =\int \limits _{0}^{\infty }ye^{-\zeta y}dG(y)\lt \infty \mathrm{, }-\frac {d\overset { \wedge }{f_2}\left ( \kappa \right ) }{d\kappa }=\int \limits _{0}^{\infty }j_2e^{-\kappa j_2}\mathrm{ }dF_2(j_2)\lt \infty , \end{align*}

\begin{align*} -\frac {d\overset { \wedge }{h}\left ( \varpi \right ) }{d\varpi }=\int \limits _{0}^{\infty }ke^{-\varpi k}\mathrm{ }dH(k)\lt \infty , \mathrm{, }-\frac {d\overset { \wedge }{d}\left ( \varphi \right ) }{d\varphi }=\int \limits _{0}^{\infty }le^{-\varphi l}\mathrm{ }dD(l)\lt \infty , \end{align*}

\begin{align*} -\frac {\partial \overset {\wedge }{\psi }\left ( \zeta ,\kappa \right ) }{ \partial \zeta }=\int \limits _{0}^{\infty }\!\int \limits _{0}^{\infty }\!ye^{-\zeta y}e^{-\kappa j_{2}}d\Psi ( y,j_{2}) \lt \infty \,\,\mathrm{and}\,\,-\frac {\partial \overset {\wedge }{\pi }\left ( \varpi ,\varphi \right ) }{\partial \varpi }=\int \limits _{0}^{\infty }\!\int \limits _{0}^{\infty }\!k e^{-\varpi k}e^{-\varphi l}d\Phi ( k,l) \lt \infty . \end{align*}

where $\zeta \gt 0,$ $\kappa \gt 0,$ $\varpi \gt 0,$ and $\varphi \gt 0.$

Based on Dassios & Zhao (Reference Dassios and Zhao2012), to have a stationary intensity process $\lambda (t)$ , we need $\delta \gt m_{1y}$ , where $m_{1y}=\int \limits _0^{\infty }ydG(y)\lt \infty .$ Furthermore, we assume that there are two available assets in the financial market.

The risk-free asset $B(t)$ follows

(2.8) \begin{equation} dB(t)=rB(t)dt, \end{equation}

where $r\gt 0$ denotes the risk-free rate. The risky asset $S(t)$ follows

(2.9) \begin{equation} dS(t)=\mu S(t)dt+\sigma S(t)dW(t)-LS(t)dM(t), \end{equation}

where $\mu \gt r$ denotes the drift, $\sigma \gt 0$ denotes the volatility, and $L\gt 0$ denotes the negative effect of external events on the risky financial asset, with a joint distribution $\Phi (k,l)$ mentioned before.

Formally, we consider a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ that is generated by the above point processes and the Brownian motion and satisfies usual conditions. A filtration $\{\mathcal{F}_t\}$ represents the information up to time t. Let $\pi (t)$ denotes the dollar amount allocated into the risky asset, and let $q_1(t)$ and $q_2(t)$ denote the proportion of ordinary and catastrophic claims covered by the insurer, respectively. Let $\nu (t)=(\pi (t),q_1(t),q_2(t))$ denote a trading strategy. The controlled surplus process $X^{\nu }(t)$ becomes

(2.10) \begin{align} dX^{\nu }(t)&=\frac {X^{\nu }(t)-\pi (t)}{B(t)}dB(t)+\frac {\pi (t)}{S(t)}dS(t)+(c_1-\tilde {c}_1(1-q_1(t))+c_2-\tilde {c}_{2}(1-q_2(t)))dt\nonumber\\[2pt] &\quad -q_1(t)dC_1(t)-q_2(t)dC_2(t)\nonumber\\[2pt] &=\left[(\mu -r)\pi (t)+rX^{\nu }(t)+c_1-\tilde {c}_{1}(1-q_1(t))+c_2-\tilde {c}_2(1-q_2(t))\right]dt+\sigma \pi (t)dW(t)\nonumber\\[2pt] &\quad -q_1(t)J_{1}dN_1(t)-q_2(t)J_2dN_2(t)-\pi (t)LdM(t). \end{align}

In (2.10), $c_1$ and $\tilde {c}_1$ denote the insurance and reinsurance premium of ordinary claims. $c_2$ and $\tilde {c}_2$ denote the insurance and reinsurance premium of catastrophic claims. $J_{1,\kappa }$ and $J_{2,j}$ are assumed to be $i.i.d.$ random variables with magnitude $J_1$ and $J_2$ , respectively. For the insurance premium rate $c_i$ , we consider two popular premium principles in this paper, $i.e.$ , the net-profit principle and the deviation-premium principle. Under the net-profit principle, we have $c_it=(1+\theta _i)\mathbb{E}(C_i(t))$ with $\theta _i\gt 0$ , for $i=1,2.$ Under the deviation-premium principle, we assume that $c_it=\mathbb{E}(C_i(t))+\theta _i\sqrt {Var(C_i(t))}$ . For ordinary claims, under the net-profit principle, we have $c_1=(1+\theta _1)\eta m_{1j_1}$ , where $m_{1j_1}=\int \limits _0^{\infty }j_1dF_1(j_1)$ , since the claim amount is independent of the claim process. Under the deviation-premium principle, we have $c_1t=\eta m_{1j_1}t+\theta _1\sqrt {\eta m_{2j_1}t}$ , where $m_{2j_1}=\int \limits _0^{\infty }j^2_1dF_1(j_1)$ . For catastrophic claims, we follow the work of Dassios & Zhao (Reference Dassios and Zhao2012) for the premium of catastrophic claims under enhanced dynamic contagion processes, which are captured in Theorem2.6.

Now, we define the admissible set of trading strategies for the insurer by the following definition:

Theorem 2.6. Under the assumption $\delta \gt m_{1y}$ , the premium of catastrophic claims under the net-profit principle follows

(2.11) \begin{equation} c_2=(1+\theta _2)\frac {m_{1k}\rho +a\delta }{\delta -m_{1y}}m_{1j_2}, \end{equation}

where $m_{1k}=\int \limits _0^{\infty }kdH(k)$ and $m_{1j_2}=\int \limits _0^{\infty }j_2dF_2(j_2).$

Under the deviation-premium principle, the premium of catastrophic claims follows

(2.12) \begin{equation} c_2t=\mathbb{E}(C_2(t))+\theta _2\sqrt {Var(C_2(t))}, \end{equation}

where

(2.13) \begin{equation} \mathbb{E}(C_2(t))=\frac {a\delta +\rho m_{1k}}{\delta -m_{1y}}m_{1j_2}t. \end{equation}

(2.14) \begin{align} Var(C_2(t))& = 2m_{1j_{2}}\left [ \begin{array}{c} m_{1j_{2}}\mathrm{ }\left [ \frac {\left \{ 2\left ( m_{1k}\mathrm{ }\rho +a\delta \right ) +m_{2y}\right \} \left ( m_{1k}\mathrm{ }\rho +a\delta \right ) }{2\left ( \delta -m_{1y}\right ) ^{3}}+\frac {m_{2k}\mathrm{ }\rho }{2\left ( \delta -m_{1y}\right ) ^{2}}\right ] \left \{ t-\left ( \frac { 1-e^{-\left ( \delta -m_{1y}\right ) t}}{\delta -m_{1y}}\right ) \right \} \\[15pt] +m_{1\Psi _{y,j_{2}}}\mathrm{ }\left \{ \frac {m_{1k}\mathrm{ }\rho +a\delta }{\left ( \delta -m_{1y}\right ) ^{2}}\right \} \left \{ t-\left ( \frac { 1-e^{-\left ( \delta -m_{1y}\right ) t}}{\delta -m_{1y}}\right ) \right \} \\[15pt] -\left ( m_{1k}\mathrm{ }\rho +a\delta \right ) m_{1j_{2}}\left \{ \frac { m_{1k}\mathrm{ }\rho +a\delta }{\left ( \delta -m_{1y}\right ) ^{3}} \right \} \left \{ t-\left ( \frac {1-e^{-\left ( \delta -m_{1y}\right ) t}}{ \delta -m_{1y}}\right ) \right \} \end{array} \right ] \nonumber \\[15pt] & \quad +m_{2j_{2}}\left (\frac {a\delta +\rho m_{1k}}{\delta -m_{1y}}\right ) t, \end{align}

where $m_{2y}=\int \limits _0^{\infty }y^2dG(y)$ , $m_{2k}=\int \limits _0^{\infty }k^2dH(k)$ , $m_{2j_2}=\int \limits _0^{\infty }j_2^2dF_2(j_2)$ , and $m_{1\Psi _{y,j_{2}}}=\int \limits _{0}^{\infty }\int \limits _{0}^{\infty }yj_{2}d\Psi ( y,j_{2})$ .Footnote ¹

The proof of Theorem2.6 is provided in Appendix A.

For the reinsurance premium rate $\tilde {c}_{i}$ , $i=1,2,$ we assume that the reinsurer applies the same premium principle as the insurer does but with a different risk loading factor $\theta _i$ , $i=1,2.$ Furthermore, $\theta _i$ , $i=1,2,$ satisfy the following conditions

Assumption 2.7.

(2.15) \begin{equation} \begin{aligned} &\tilde {\theta }_i\geq \theta _i, \quad i, j\in \{1,2\}\\ &c_i-\tilde {c}_{i}+c_j\geq 0, \quad i, j\in \{1,2\}, i\neq j, \end{aligned} \end{equation}

which can eliminate arbitrage opportunities and sets an upper bound for the reinsurance premium for the two business lines.

Given the initial surplus $X(t)=x\gt 0$ and the intensity of catastrophic claims $\lambda (t)=\lambda \gt 0$ at time $t\geq 0$ , the insurance company’s management wants to maximize the expected utility of the surplus at the terminal date $T\geq t$ .

(2.16) \begin{equation} V(t,x,\lambda )=\sup _{\nu (t)\in \mathcal{V}}\mathbb{E}_{t,x,\lambda }\left [U(X^{\nu }(T))\right ], \end{equation}

where $U(\cdot )$ denotes a suitable utility function and $\mathbb{E}_{t,x,\lambda }[\cdot ]$ denotes the conditional expectation given that $X(t)=x$ and $\lambda (t)=\lambda$ . Note that we omit the variable $t$ in $\pi (t), q_1(t),$ and $q_2(t)$ to simplify notations thereafter.

Define an operator $\mathcal{K}$ on a function $f(t,x,\lambda )$ within its domain $[0,T]\times (0,\infty )\times (0,\infty )$ as follows:

(2.17) \begin{align} \mathcal{K}f(t,x,\lambda )=&\left .(rx+(\mu -r)\pi +c_1-(1-q_1)\tilde {c}_{1}+c_2-(1-q_2)\tilde {c}_{2}\right .)\frac {\partial f(t,x,\lambda )}{\partial x}\nonumber\\[3pt] &+\delta (\alpha -\lambda )\frac {\partial f(t,x,\lambda )}{\partial \lambda }+0.5\sigma ^2\pi ^2\frac {\partial ^2f(t,x,\lambda )}{\partial x^2}\nonumber\\[3pt] & +\eta \int \limits _0^{\infty } (f(t,x-q_1j_1,\lambda )-f(t,x,\lambda ))dF_1(j_1)\nonumber\\[3pt] &+\lambda \int \limits _0^{\infty }\int \limits _0^{\infty } (f(t,x-q_2j_2,\lambda +y)-f(t,x,\lambda ))d\Psi (y,j_2)\nonumber\\[3pt] &+\rho \int \limits _0^{\infty }\int \limits _0^{\infty } (f(t,x-\pi l,\lambda +k)-f(t,x,\lambda ))d\Phi (k,l), \end{align}

where the domain of the operator $\mathcal{K}$ contains the class of functions $f(t,x,\lambda )\in \mathcal{C}^{1,2,1}([0,T]\times (0,\infty )\times (0,\infty ))$ such that for every $(t,x,\lambda )\in ([0,T]\times (0,\infty )\times (0,\infty ))$ , we have

(2.18) \begin{align} &\mathbb{E}\left [\int \limits _0^{\infty } \left(f(t,x-q_1j_1,\lambda )-f(t,x,\lambda )\right)dF_1(j_1)\right ]\lt \infty ,\nonumber\\[3pt] &\mathbb{E}\left [\int \limits _0^{\infty }\int \limits _0^{\infty } \left(f(t,x-q_2j_2,\lambda +y)-f(t,x,\lambda )\right)d\Psi (y,j_2)\right ]\lt \infty ,\nonumber\\[3pt] &\mathbb{E}\left [\int \limits _0^{\infty }\int \limits _0^{\infty } \left(f(t,x-\pi l,\lambda +k)-f(t,x,\lambda )\right)d\Phi (k,l)\right ]\lt \infty . \end{align}

By standard techniques, we can obtain the HJB equation and the terminal condition for the value function on its domain $[0,T]\times (-\infty ,\infty )\times (0,\infty )$ as follows:

(2.19) \begin{align} \frac {\partial V}{\partial t}+\sup _{\nu (t)\in \mathcal{V}}\mathcal{K}V(t,x,\lambda ) & =0,\nonumber\\ V(T,x,\lambda ) &= U(x). \end{align}

3. Main results

In this paper, we consider exponential utilities $U(x)=\frac {-1}{\gamma }e^{-\gamma x}$ with $\gamma \gt 0$ . Exponential utilities play a significant role in the insurance sector and financial mathematics since it is the only utility function where the principle of “zero utility” gives a fair premium that is independent of the insurer’s surplus level. This important feature is also reflected in this paper, which will be demonstrated later. Motivated by Duffie et al. (Reference Duffie, Pan and Singleton2000), Aït-Sahalia & Hurd (Reference Aït-Sahalia and Hurd2015), and Liu et al. (2021), if the claim intensity $\lambda (t)$ follows (2.4), we conjecture that the value function $V(t,x,\lambda )$ satisfies an exponential affine structure, which can be separated as follows:

(3.1) \begin{align} V(t,x,\lambda )&=\frac {-1}{\gamma }e^{A(t,T)x}g(t,\lambda ),\nonumber\\[4pt] g(t,\lambda )&=e^{B(t,T)+C(t,T)\lambda }, \end{align}

where $A(t,T)$ , $B(t,T)$ , and $C(t,T)$ are functions to be determined. The explicit solution for $A(t,T)$ can be obtained, while $B(t,T)$ and $C(t,T)$ are too complex to be solved. To describe the value function, $g(t,\lambda )$ is applied to represent the implicit version of the affine structure. Hence, the main task in the following sections is to solve $A(t,T)$ and $g(t,\lambda )$ . Note that we will denote $A(t, T)$ by $A$ thereafter to simplify notations unless it is necessary to be specified.

Replacing $V(t,x,\lambda )$ by (3.1), we can simplify the HJB equation as follows:

(3.2) \begin{align} 0 & =\inf _{\pi ,q_1,q_2}\frac {\partial g}{\partial t}+\frac {\partial A}{\partial t}xg+\left [rx+(\mu -r)\pi +c_1-(1-q_1)\tilde {c}_1+c_2-(1-q_2)\tilde {c}_2\right ]Ag\nonumber\\ &\quad +\delta (\alpha -\lambda )\frac {\partial g}{\partial \lambda }+0.5\sigma ^2\pi ^2A^2g+\eta g\int \limits _0^{\infty } \left(e^{-Aq_1j_1}-1\right)dF_1(j_1)\nonumber\\ &\quad +\lambda \int \limits _0^{\infty }\!\int \limits _0^{\infty }\!\left(e^{-Aq_2j_2}g(t,\lambda +y)-g(t,\lambda )\right)d\Psi (y,j_2)+\rho \int \limits _0^{\infty }\!\int \limits _0^{\infty } \!\left(e^{-A\pi l}g(t,\lambda +k)-g(t,\lambda )\right)d\Phi (k,l), \end{align}

with the terminal condition

(3.3) \begin{equation} g(T,\lambda )=1. \end{equation}

According to the first-order condition, optimal controls are as follows, which are independent of the surplus level

(3.4) \begin{align} \pi ^*&=\arg \inf _{\pi }(\mu -r)\pi Ag+0.5\sigma ^2\pi ^2A^2g+\rho \int \limits _0^{\infty }\int \limits _0^{\infty } \left (e^{-A\pi l}g(t,\lambda +k)-g(t,\lambda )\right )d\Phi (k,l),\nonumber\\ q_1^*&=\arg \inf _{q_1\in [0,1]}Ag\tilde {c}_1q_1+\eta g\int \limits _0^{\infty }\left (e^{-Aq_1j_1}-1\right )dF_1(j_1),\nonumber\\ q_2^*&=\arg \inf _{q_2\in [0,1]}Ag\tilde {c}_2q_2+\lambda \int \limits _0^{\infty }\int \limits _0^{\infty }\left (e^{-Aq_2j_2}g(t,\lambda +y)-g(t,\lambda )\right )d\Psi (y,j_2), \end{align}

which admits the analytic solution for the optimal reinsurance policy with respect to ordinary claims. Now, we provide a verification result in terms of the simplified HJB equation (3.2).

The proof of Theorem3.1 is provided in Appendix B.

For the unknown function $A(t,T)$ and the optimal reinsurance policy with respect to ordinary claims, we have

Lemma 3.2. Given any initial conditions $X(t)=x\gt 0$ and $\lambda (t)=\lambda \gt 0$ at any time $t\geq 0$ , the unknown $A(t,T)$ follows $A(t,T)=-\gamma e^{-r(t-T)}$ , and the analytic solution of the optimal reinsurance policy with respect to ordinary claims under a general ordinary claim distribution $F_1(j_1)$ is provided as follows:

(3.5) \begin{equation} \begin{aligned} q_1^*=\min \left (1,\tilde {q}_1\right ), \end{aligned} \end{equation}

where $\tilde {q}_1$ is the unique positive root of the following equation

(3.6) \begin{align} \mathrm{Under the net-profit principle:} \quad \quad (1+\tilde {\theta }_1)m_{j_1}&=\int \limits _0^{\infty } j_1e^{-A(t,T)\tilde {q}_1j_1}dF_1(j_1).\nonumber\\ \mathrm{Under the deviation-premium principle:} \quad \quad m_{1j_1}+\tilde {\theta }_1\sqrt {\frac {m_{2j_1}}{\eta t}}&=\int \limits _0^{\infty } j_1e^{-A(t,T)\tilde {q}_1j_1}dF_1(j_1). \end{align}

Furthermore, suppose $J_1\sim Exp(\epsilon _1), \epsilon _1\gt 0$ , we have

(3.7) \begin{equation} \begin{aligned} \mathrm{Under the net-profit principle:}\quad \quad q_1^*&=\min \left (1,\frac {\epsilon _1}{A(t,T)}\left (\frac {1}{\sqrt {1+\tilde {\theta }_1}}-1\right )\right ).\\[5pt] \mathrm{Under the deviation-premium principle:} \quad \quad q_1^*&=\min \left (1,\frac {\epsilon _1}{A(t,T)}\left (\sqrt {\frac {1}{1+\tilde {\theta }_1\sqrt {\frac {2}{\eta t}}}}-1\right )\right ). \end{aligned} \end{equation}

The proof of Lemma 3.2 is provided in Appendix C.

Given the closed-form expression in Lemma 3.2, we can further simplify the HJB equation as follows:

\begin{align*} 0 & = \inf _{\pi ,q_2}\frac {\partial g}{\partial t}+\delta (\alpha -\lambda )\frac {\partial g}{\partial \lambda }+(\mu -r)\pi Ag+0.5\sigma ^2\pi ^2A^2g\nonumber\\& \quad +\rho \int \limits _0^{\infty }\int \limits _0^{\infty } \left(e^{-A\pi l}g(t,\lambda +k)-g(t,\lambda )\right)d\Phi (k,l)\end{align*}

(3.8) \begin{align} &\quad +\left [c_1-\tilde {c}_1+c_2-\tilde {c}_2\right ]Ag+Ag\tilde {c}_1q^*_1+\eta g\int \limits _0^{\infty } \left(e^{-Aq^*_1j_1}-1\right)dF_1(j_1)\nonumber\\ &\quad +Ag\tilde {c}_2q_2+\lambda \int \limits _0^{\infty }\int \limits _0^{\infty } \left(e^{-Aq_2j_2}g(t,\lambda +y)-g(t,\lambda )\right)d\Psi (y,j_2). \end{align}

Note that the infinitesimal generator $\mathcal{A}$ for a function $f(t,\lambda )$ in its domain $[0,T]\times (0,\infty )$ is given by

(3.9) \begin{equation} \begin{aligned} \{\mathcal{A}f\}(t,\lambda )&=\frac {\partial f(t,\lambda )}{\partial t}+\delta (\alpha -\lambda )\frac {\partial f(t,\lambda )}{\partial \lambda }\\ &\quad +\lambda \int \limits _0^{\infty }\int \limits _0^{\infty } \Big (f(t,\lambda +y)-f(t,\lambda )\Big )d\Psi (y,j_2)+\rho \int \limits _0^{\infty }\int \limits _0^{\infty } \Big (f(t,\lambda +k)-f(t,\lambda )\Big )d\Phi (k,l). \end{aligned} \end{equation}

Then, the HJB equation (3.8) can be further simplified as follows:

(3.10) \begin{equation} \{\mathcal{A}g\}(t,\lambda )+\left [c_1-\tilde {c}_1+c_2-\tilde {c}_2\right ]Ag+\inf _{\pi }\chi _1(\pi ,g(t,\lambda ))+\chi _2(q_1^*,g(t,\lambda ))+\!\!\inf _{q_2\in [0,1]}\chi _3(q_2,g(t,\lambda ))=0, \end{equation}

where

(3.11) \begin{align} \chi _1(\pi , g(t,\lambda ))&=(\mu -r)\pi Ag+0.5\sigma ^2\pi ^2A^2g+\rho \int \limits _0^{\infty }\int \limits _0^{\infty } \left(e^{-A\pi l}g(t,\lambda +k)-g(t,\lambda +k)\right)d\Phi (k,l),\nonumber\\ \chi _2(q_1^*, g(t,\lambda ))&=q_1^*\tilde {c}_1Ag+\eta g\int \limits _0^{\infty } (e^{-Aq_1^*j_1}-1)dF_1(j_1),\nonumber\\ \chi _3(q_2, g(t,\lambda ))&=q_2\tilde {c}_2Ag+\lambda \int \limits _0^{\infty }\int \limits _0^{\infty } \left(e^{-Aq_2j_2}g(t,\lambda +y)-g(t,\lambda +y)\right)d\Psi (y,j_2) ,\nonumber\\ \pi ^* & =\arg \inf _{\pi }\chi _1(\pi , g(t,\lambda )),\mathrm{ and } q_2^*=\arg \inf _{q_2\in [0,1]}\chi _3(q_2,g(t,\lambda )). \end{align}

Given Lemma 3.2 and (3.10), we know that $g(t,\lambda )$ can be expressed as the following conditional expectation based on the Feynman–Kac formula,

(3.12) \begin{equation} \begin{aligned} g(t,\lambda )&=\mathbb{E}\left [\int \limits _t^Te^{\int \limits _t^s\widehat {A}(\tau ,T)d\tau }[\chi _1(\pi ^*(s),g(s,\lambda (s)))+\chi _2(q_1^*(s),g(s,\lambda (s)))\right .\\ &\quad \left .+\chi _3(q_2^*(s),g(s,\lambda (s)))]ds+e^{\int \limits _t^T\widehat {A}(\tau ,T)d\tau }|\mathcal{F}_t \right ], \end{aligned} \end{equation}

where $\widehat {A}(\tau ,T)=(c_1-\tilde {c}_1+c_2-\tilde {c}_2)A(\tau ,T)$ . Then, we can develop an implicit formula of $g(t,\lambda )$ based on (3.12). Given the initial guess of $g(t,\lambda )$ , denoted by $g^{0}(t,\lambda )$ , the iterative scheme is as follows:

(3.13) \begin{equation} \begin{aligned} g^{n+1}(t,\lambda )&=\mathbb{E}\left [\int \limits _t^Te^{\int \limits _t^s\widehat {A}(\tau ,T)d\tau }[\chi _1(\pi _n^*(s),g^n(s,\lambda (s)))+\chi _2(q_{1n}^*(s),g^n(s,\lambda (s)))\right .\\[5pt] &\quad \left .+\chi _3(q_{2n}^*(s),g^n(s,\lambda (s)))]ds+e^{\int \limits _t^T\widehat {A}(\tau ,T)d\tau }|\mathcal{F}_t \right ], \end{aligned} \end{equation}

where $\pi ^*_n(t)$ and $q_{in}^*(t)$ denote the optimal allocation policy and reinsurance policy in (3.11) with a given $g^n(t,\lambda )$ for $i=1,2,$ and $n=0,1,2,\ldots$ .

4. Proof of convergence

To prove the convergence of the iterative method in (3.13) is equivalent to proving the existence and uniqueness of the solution to (3.12). Define an operator $\mathcal{L}$ acting on the function $g(t,\lambda )$ as follows:

(4.1) \begin{equation} \begin{aligned} (\mathcal{L}g)(t,\lambda )&=\mathbb{E}\left [\int \limits _t^Te^{\int \limits _t^s\widehat {A}(\tau ,T)d\tau }[\chi _1(\pi ^*(s),g(s,\lambda (s)))+\chi _2(q_1^*(s),g(s,\lambda (s)))\right .\\[5pt] &\quad \left .+\chi _3(q_2^*(s),g(s,\lambda (s)))]ds+e^{\int \limits _t^T\widehat {A}(\tau ,T)d\tau }|\mathcal{F}_t \right ]. \end{aligned} \end{equation}

Hence, the function $g(t,\lambda )$ in (3.12) satisfies a fixed point equation $(\mathcal{L}g)(t,\lambda )=g(t,\lambda ).$ To prove the existence and uniqueness of the solution to (3.12), we only need to prove that the operator $\mathcal{L}$ is a contraction mapping. Note that, to enhance the readability and organization of the manuscript, we consolidate all technical proofs related to the results of Section 4 in Appendix D.

Lemma 4.1. The function $g(t,\lambda )$ is bounded for any $t\in [0,T]$ and $\lambda \in (0,\infty )$ and increasing in $t\in [0,T]$ , and the operator $\mathcal{L}$ on $g(t,\lambda )$ is also bounded for any $t\in [0,T]$ and $\lambda \in (0,\infty )$ , $i.e.$ ,

(4.2) \begin{equation} \begin{aligned} 0\lt (\mathcal{L}g)(t,\lambda )\lt \beta , \end{aligned} \end{equation}

where $\beta$ is a positive constant depending on the initial surplus $x$ .

The proof of Lemma 4.1 is provided in Appendix D.1.

The proof of Lemma 4.2 is provided in Appendix D.2.

Based on Lemmas 4.1 and 4.2, we define a space of continuous functions $\zeta$ on $[0,T]\times (0,\infty )$ by $\mathfrak{C}_e([0,T]\times (0,\infty ))$ , where $0\lt \zeta (t,\lambda )\lt \beta$ and

(4.3) \begin{equation} \zeta (t,\lambda )=e^{\tilde {\psi }(t,T)+\tilde {\psi }_1(t,T)\lambda }, \end{equation}

where $\tilde {\psi }(t,T)$ and $\tilde {\psi }_1(t,T)$ are bounded and Lipschitz continuous for $t\in [0,T]$ and $\lambda \in (0,\infty )$ . In this section, we will prove that the mapping $\mathcal{L}$ : $\mathfrak{C}_e([0,T]\times (0,\infty ))$ $\to$ $\mathfrak{C}_e([0,T]\times (0,\infty ))$ is a contraction mapping with some suitable metric. Then, (4.1) admits a unique solution, which is $g(t,\lambda )\in \mathfrak{C}_e([0,T]\times (0,\infty )).$

Note that a metric is defined on $\mathfrak{C}_e([0,T]\times (0,\infty ))$ as follows: for any two functions $v_1(t,\lambda )\in \mathfrak{C}_e([0,T]\times (0,\infty ))$ and $v_2(t,\lambda )\in \mathfrak{C}_e([0,T]\times (0,\infty ))$ , let

(4.4) \begin{equation} d(v_1,v_2)=\sup _{(t,\lambda )\in [0,T]\times (0,\infty )}\left |e^{-\alpha (T-t)}\left (v_1(t,\lambda )-v_2(t,\lambda )\right )\right |, \end{equation}

where $\alpha$ denotes a positive constant to be determined.

The proof of Lemma 4.3 is provided in Appendix D.3.

Now, given Lemma 4.3, to show that the operator $\mathcal{L}$ is a contraction mapping, we only need to prove $d(\mathcal{L}v_1,\mathcal{L}v_2)\leq \phi d(v_1,v_2)$ for some $\phi \in [0,1)$ .

(4.5) \begin{align} d(\mathcal{L}v_1,\mathcal{L}v_2) & = \sup _{t,\lambda }\left |e^{-\alpha (T-t)}\left (\mathcal{L}v_1-\mathcal{L}v_2\right )\right |\nonumber\\[4pt] & =\sup _{t,\lambda }\Bigg|e^{-\alpha (T-t)}\mathbb{E}\Bigg[\int \limits _t^Te^{\int \limits _t^s\widehat {A}(\tau ,T)d\tau }\left(\chi _1(\pi _{v_1}^*(s),v_1(s,\lambda (s)))-\chi _1(\pi _{v_2}^*(s),v_2(s,\lambda (s)))\right.\nonumber\\[4pt] &\quad +\chi _2(q_{1v_1}^*(s),v_1(s,\lambda (s)))-\chi _2(q_{1v_2}^*(s),v_2(s,\lambda (s)))\nonumber\\[4pt] &\quad \left. +\chi _3\left(q_{2v_1}^*(s),v_1(s,\lambda (s))\right)-\chi _3\left(q_{2v_2}^*(s),v_2(s,\lambda (s))\right)\right)ds|\mathcal{F}_t\Bigg]\Bigg|, \end{align}

where $\chi _1(\pi _{v_i}^*(s),v_i(s,\lambda (s)))$ , $\chi _2(q_{1v_i}^*(s),v_i(s,\lambda (s)))$ , and $\chi _3(q_{2v_i}^*(s),v_i(s,\lambda (s)))$ , $i=1,2,$ follow (3.11). $\pi ^*_{v_i}(s)$ , $q_{1v_i}^*(s)$ , and $q_{2v_i}^*(s)$ denote the optimal allocation and reinsurance policies of ordinary claims and catastrophic claims with respect to $v_1(t,\lambda )$ and $v_2(t,\lambda )$ .

Lemma 4.4. For any two continuous functions $v_1,v_2\in \mathfrak{C}_e([0, T]\times (0,\infty ))$ , let $\pi ^*_{v_i}(s)$ , $q_{1v_i}^*(s)$ , and $q_{2v_i}^*(s)$ denote the optimal allocation and reinsurance policies of ordinary claims and catastrophic claims with respect to $v_1(t,\lambda )$ and $v_2(t,\lambda )$ . Suppose that joint distributions $\Phi (k,l)$ and $\Psi (y,j_2)$ as well as marginal distributions $H(k), D(l), F_2(j_2)$ , and $G(y)$ follow Assumptions 2.3 and 2.4 and the distribution $F_1(j_1)$ follows Assumption 2.3 , we have the following inequality

(4.6) \begin{equation} \begin{aligned} & |\chi _1(\pi _{v_1}^*(s),v_1(s,\lambda (s)))-\chi _1(\pi _{v_2}^*(s),v_2(s,\lambda (s)))\\ &\quad +\chi _2(q_{1v_1}^*(s),v_1(s,\lambda (s)))-\chi _2\left(q_{1v_2}^*(s),v_2(s,\lambda (s))\right)\\ &\quad +\chi _3(q_{2v_1}^*(s),v_1(s,\lambda (s)))-\chi _3\left(q_{2v_2}^*(s),v_2(s,\lambda (s))\right)|\lt \Delta D_{\lambda }(v_1,v_2), \end{aligned} \end{equation}

where $\Delta$ denotes some positive constant and $D_{\lambda }(v_1,v_2)=\sup _{\lambda }|v_1(s,\lambda )-v_2(s,\lambda )|$ .

The proof of Lemma 4.4 is provided in Appendix D.4.

Lemma 4.5. Given any two continuous functions $v_1,v_2\in \mathfrak{C}_e([0,T]\times (0,\infty ))$ and Lemma 4.4 , $d(\mathcal{L}v_1,\mathcal{L}v_2)$ in (4.5) satisfies the following inequality:

(4.7) \begin{equation} \begin{aligned} d(\mathcal{L}v_1,\mathcal{L}v_2)\leq \Delta \left (\int _t^Te^{\int \limits _t^s\widehat {A}(\tau ,T)d\tau -\alpha (s-t)}ds\right )d(v_1,v_2), \end{aligned} \end{equation}

where $\widehat {A}(\tau ,T)=(c_1-\tilde {c}_1+c_2-\tilde {c}_2)A(\tau ,T)\gt 0.$

The proof of Lemma 4.5 is provided in Appendix D.5.

The proof of Theorem4.6 is provided in Appendix D.6.

5. Numerical results

In this section, we provide several numerical examples to demonstrate the optimal asset allocation policy, optimal reinsurance policies for ordinary claims and catastrophic claims, and the value function under a copula. Gaussian copulasFootnote ² is assumed to dominate the dependence structure of $\Psi (y,j_2)$ and $\Phi (k,l)$ with default linear correlation parameters $\rho _1=\rho _{(y,j_2)}=0$ and $\rho _2=\rho _{(k,l)}=0$ , respectively.

Furthermore, we assume that ordinary claims ( $J_1$ ) follow exponential random variables with the parameter $\epsilon _1\gt 0$ and marginal distributions of catastrophic claims ( $J_2$ ), self-excited jumps ( $Y$ ), externally excited jumps ( $K$ ), and risky-asset crashes ( $L$ ) also follow exponential random variables with parameters $\epsilon _2\gt 0$ , $\epsilon _3\gt 0$ , $\epsilon _4\gt 0$ , and $\epsilon _5\gt 0$ , respectively. The values for exponential random variable parameters of $\epsilon _2\gt 0$ , $\epsilon _3\gt 0$ , and $\epsilon _4\gt 0$ are chosen for their simplicity, where their means are the same. The value for the average risky-asset crash is chosen to be $20\%$ .

The parameters of the financial and insurance market mainly follow the works of Meng et al. (2015), Hainaut (Reference Hainaut2017), and Liu et al. (2021). Unless otherwise stated, the value of parameters can be referred to in Table 1. To describe the insurance market, it is assumed that $\eta =1$ , which implies that, on average, an ordinary claim occurs once per time unit. This setting ensures a reasonable claim frequency that aligns with empirical observations in general insurance markets. For the catastrophic claim process, we assume that $a=2$ and $\delta =4$ , meaning that the mean-reverting level for catastrophic claims is twice per time unit, and the excitation effects decay by around $80\%$ per time unit. These choices reflect the characteristics of catastrophic events, which typically exhibit temporary clustering due to external shocks but eventually revert to a stable level over time. The fast decay rate of excitation captures the reality that while major disasters can trigger secondary events, their influence diminishes relatively quickly.

Table 1. Parameters and values

The claim severity assumptions are also well-founded. The average ordinary claim size is set to $0.25$ units, which is significantly smaller than the average catastrophic claim size of $0.5$ units. This distinction is consistent with empirical data, as catastrophic claims tend to have much larger impacts than ordinary claims. Moreover, the externally exciting and self-exciting effects are assumed to be $40\%$ of the mean-reverting level. This reflects a high-risk insurance market where disasters and systemic risks have a substantial impact, potentially leading to claim clustering and financial stress for insurers. Such a setting is particularly relevant in regions prone to natural disasters or extreme events. The assumption that $\tilde {\theta }_i\gt \theta _i$ , $i=1,2,$ ensures that reinsurance is not overly cheap for both ordinary and catastrophic claims. This prevents moral hazard and ensures that insurers are incentivized to manage risks properly rather than relying excessively on reinsurance. It also aligns with real-world market conditions, where reinsurers charge premiums that reflect the risk exposure they assume.

For the financial market, the risk-free rate is set at $0.02$ , with the risky asset having a drift of $0.05$ and volatility of $0.2$ . These parameters are broadly in line with historical financial market trends, where the excess return of risky assets over the risk-free rate is moderate, and volatility remains within a reasonable range. The relatively low risk-free rate reflects modern economic conditions, where safe investments yield limited returns. Overall, these parameter settings provide a realistic and well-calibrated representation of both the insurance and financial markets, capturing key risk characteristics while ensuring model robustness.

5.1 Optimal reinsurance policy

5.1.1 Ordinary claims

Figure 1 illustrates the behavior of the optimal reinsurance policy with respect to ordinary claims, where the closed-form expression under exponential marginal distributions is obtained in Lemma 3.2. First, the insurer transfers a little bit more risk on ordinary claims to the reinsurer when the operating time increases (as $t$ decreases). Second, note that there is no linear relationship between the optimal reinsurance policy with respect to ordinary claims and the initial time point, shown by Lemma 3.2. Figure 1 looks like linear functions because the scope of the time interval is small. Third, $q_1^*(t)$ increases as the average claim size decreases (as $\epsilon _1$ increases), which means more risks are retained by the insurer as the average ordinary claim size decreases. Finally, comparing Figure 1a and b, we can see that $q_1^*(t)$ under the net-profit principle is smaller than that under the deviation-premium principle. Since ordinary claims are assumed to be exponentially distributed, the reinsurance premium under the deviation-premium principle is larger than that under the net-profit principle with the same risk loadings. To maximize the expected terminal surplus, the insurer reduces the reinsurance cost by retaining a larger proportion of ordinary claim risks.

Figure 1 Optimal ordinary reinsurance policy under dynamic contagion processes.

5.1.2 Catastrophic claims

Figure 2 shows the behavior of the optimal reinsurance strategy with respect to catastrophic claims under different parameter settings and two premium principles. First, the optimal reinsurance policy $q^*_2(t,\lambda )$ decreases as the claim intensity increases in general. When the claim intensity is small, the insurer rejects the proportional reinsurance products ( $q_2^*(t,\lambda )=1$ ). According to (2.4), externally exciting and self-exciting factors can only have positive effects on claim intensities. Hence, when the claim intensity is low, the insurer recognizes that the intensity will eventually revert to at least the constant mean-reverting level. This predictability allows the insurer to assess its baseline risk exposure with confidence. Since claim occurrences are expected to remain manageable at such levels, the insurer can rely on its own capital reserves and risk management strategies rather than purchasing costly reinsurance. Moreover, in a low-intensity regime, the probability of sudden claim clustering remains relatively low, reducing the insurer’s concern about extreme losses. As a result, retaining more risk is financially optimal, as the insurer avoids paying high reinsurance premiums for coverage that may not be immediately necessary. This aligns with standard industry practice, where insurers typically increase retention when claim risks are stable and foreseeable. However, when the claim intensity is large, the insurer is unsure about the behavior of claim intensities. The mean-reverting feature will move the claim intensity back to the constant mean-reverting level, while the externally exciting and self-exciting effects will further increase the claim intensity. To mitigate the potential contagious claim risk, the insurer transfers those risks to the reinsurer by purchasing the proportional reinsurance contract.

Second, when catastrophic claims and self-exciting effects are negatively correlated (i.e., $\rho _1\lt 0$ ), catastrophic claim risks decrease, and the insurer can bear more catastrophic risks by itself. Otherwise, more catastrophic risks are transferred to the reinsurer when $\rho _1\gt 0$ , as shown by Figure 2a, c, and e under the net-profit principle and Figure 2b, d, and f under the deviation-premium principle.

Third, the effect of $\rho _2$ on $q^*_2(t,\lambda )$ is negligible. This is because $\rho _2$ measures the dependence structure between asset crashes and the external factor on the claim intensity, which only has an indirect effect on the insurance market. The insurer can adjust the asset allocation policy accordingly and maintain the optimal reinsurance policy for catastrophic claims.

Finally, comparing the left column and the right column in Figure 2, we can see that $q^*_2(t,\lambda )$ under the deviation-premium principle is slightly larger than that under the net-profit principle. According to Theorem2.6, we know that the reinsurance premium of catastrophic claims under the deviation-premium principle is larger than that under the net-profit principle. To maximize the expected terminal surplus, the insurer also reduces the reinsurance cost by retaining a larger proportion of catastrophic claim risks.

Figure 2 Optimal catastrophic reinsurance policy under dynamic contagion processes.

5.2 Optimal asset allocation strategy

Figure 3 demonstrates the behavior of the optimal asset allocation strategy under different parameter settings and two premium principles. First, the optimal asset allocation policy decreases as the increase of $t$ , which implies that the insurer decreases the holdings of the risky asset under the contagious market when the operating time decreases (the initial time $t$ increases to the terminal date $T$ ). When $t$ is small, claim risks dominate. The insurer invests more in risky assets to utilize the high growth rate and mitigate the insurance risk, even though negative jumps could appear in the risky asset. As $t$ goes to the $T$ , claims risks decrease. The insurer decreases risky asset holdings to reduce total risks from the contagious financial and insurance markets.

Second, $\pi ^*(t,\lambda )$ decreases as $\lambda$ increases. The insurer decreases risky asset holdings to reduce total risks from the contagious financial and insurance markets when claim risks increase. Third, comparing Figure 3a, c, and e under the net-profit principle and Figure 3b, d, and f under the deviation-premium principle, we observe that $\pi ^*(t,\lambda )$ is affected by the correlation between catastrophic claims and self-exciting effects. When catastrophic claims and self-exciting effects are positively correlated (i.e. $\rho _1\gt 0$ ), catastrophic claim risks are magnified. The insurer decreases the risky asset holdings to reduce the total risk level. Otherwise, when $\rho _1\lt 0,$ a larger amount of resources can be allocated to the risky assets to take advantage of the higher growth rate.

Fourth, comparing Figure 3g and i under the net-profit principle and Figure 3h and j under the deviation-premium principle, we observe that $\pi ^*(t,\lambda )$ can also be affected by the correlation between asset crashes and externally exciting effects. When asset crashes and externally exciting effects are positively correlated (i.e. $\rho _2\gt 0$ ), the financial risk increases. The insurer decreases the risky asset holdings to reduce the total risk level. Otherwise, when $\rho _2\lt 0$ , more resources can be allocated to risky assets. Finally, $\pi ^*(t,\lambda )$ under the deviation-premium principle is larger than that under the net-profit principle. This is because the more insurance risks are retained by the insurer under the deviation-premium principle, the more premiums are collected by the insurer. Hence, the insurer can allocate more resources to the risky asset to have larger investment returns and cover future potential obligations.

Figure 3 Optimal asset allocation policy under dynamic contagion processes.

5.3 Value function

Figure 4 shows the value function $V(t,1,\lambda )$ with a fixed initial surplus $x=1$ under different parameter settings and two premium principles. First, it is shown that the value function $V(t,1,\lambda )$ decreases as the increase of $t$ , which implies that the insurer can effectively manage the contagious claim risks to obtain a larger expected utility of the terminal surplus as the operating time increases. Second, the value function decreases with the increase of initial claim intensity. This is because claim risks have negative effects on the terminal surplus. As time moves to the terminal time point, the negative effect of the initial claim intensity decreases gradually.

Third, comparing Figure 4a, c, and e under the net-profit principle and Figure 4b, d, and f under the deviation-premium principle, we observe that the positive correlation between catastrophic claims and self-exciting effects $(\rho _1\gt 0)$ has a negative impact on the insurer’s value function, while a negative correlation $(\rho _1\lt 0)$ enhances it. This result reflects the economic intuition that when larger catastrophic claims trigger larger self-exciting effect, the overall risk exposure of the insurer increases, leading to a lower value function. In contrast, when $\rho _1\lt 0$ , catastrophic claims are less likely to generate additional shocks, reducing long-term risk accumulation and improving the insurer’s financial position. This suggests that in markets where catastrophic claims exhibit strong self-excitation, insurers face higher systemic risk and may require more robust capital reserves or reinsurance strategies. Furthermore, comparing Figure 4g and i under the net-profit principle and Figure 4h and j under the deviation-premium principle, we observe that the correlation between asset crashes and externally exciting effects $\rho _2$ also influences the value function. Specifically, a positive correlation $(\rho _2\gt 0)$ moderately reduces the value function, while a negative correlation $(\rho _2\lt 0)$ enhances it. This effect is more pronounced when both $\lambda$ and $t$ are small.

The economic interpretation of this result is that when asset crashes and externally exciting effects are positively correlated, financial market downturns coincide with heightened claim risks, exacerbating the insurer’s financial distress. This increases the likelihood of capital depletion and reduces the insurer’s overall value. Conversely, when $\rho _2\lt 0$ , financial downturns are less likely to be accompanied by increased claim risk, allowing the insurer to better manage its capital and risk exposure, ultimately leading to a higher value function. This finding highlights the importance of diversification in risk management – insurers operating in environments where financial and insurance risks are highly correlated may need stronger capital buffers or adaptive reinsurance strategies to mitigate the compounded risks.

Finally, the value function under the deviation-premium principle is slightly larger than that under the net-profit principle. This is just because the insurance premium under the deviation-premium principle is larger than that under the net-profit principle. The insurer has more resources to deal with the contagious financial and insurance markets.

Figure 4 Value function under dynamic contagion processes.