EQUILIBRIUM REINSURANCE-INVESTMENT STRATEGY WITH A COMMON SHOCK UNDER TWO KINDS OF PREMIUM PRINCIPLES

This paper investigates the optimal mean-variance reinsurance-investment problem for an insurer with a common shock dependence under two kinds of popular premium principles: the variance premium principle and the expected value premium principle. We formulate the optimization problem within a game theoretic framework and derive the closed-form expressions of the equilibrium reinsurance-investment strategy and equilibrium value function under the two different premium principles by solving the extended Hamilton–Jacobi–Bellman system of equations. We find that under the variance premium principle, the proportional reinsurance is the optimal reinsurance strategy for the optimal reinsurance-investment problem with a common shock, while under the expected value premium principle, the excess-of-loss reinsurance is the optimal reinsurance strategy. In addition, we illustrate the equilibrium reinsurance-investment strategy by numerical examples and discuss the impacts of model parameters on the equilibrium strategy. Mathematics Subject Classification. 90C39, 91B30, 91G80. Received June 15, 2021. Accepted December 14, 2021.


Introduction
The study of an insurer's optimal reinsurance-investment problem has attracted a lot of attention in the literature of actuarial science in the past few years. The optimization criteria which are commonly used in these optimal reinsurance-investment problems include maximizing the expected utility of the terminal wealth of an insurer (see e.g., [10,12,18,20,21,26,27]), minimizing the ruin probability of an insurer (see e.g., [10,22,23]), and the mean-variance criterion (see e.g., [3][4][5]16]). In this paper, we take the mean-variance criterion as the optimization criterion.
For the optimal reinsurance-investment problem of an insurer with more lines of insurance business, we have to consider the dependence among the lines of the insurance business. For example, a hurricane can lead to different kinds of insurance claims such as death claims, household claims, and so on. The common shock risk model is often used to describe the dependence between different classes of an insurance business. In this model, there is a common shock affecting the claim numbers of all classes in addition to their underlying risks. Yuen et al. [30] and Liang and Yuen [17] assumed that the claim number processes were correlated through a common shock, and studied the optimal proportional reinsurance strategy. More researches about dependent insurance risks can be found in Yuen et al. [29], Bai et al. [2], Bi et al. [5], Zhang and Liang [34], Liang et al. [18], and references therein.
It is well-known that the optimal reinsurance-investment problems under the mean-variance criterion in a continuous time framework are time inconsistent due to the fact that variance lacks of the iterated-expectation property. So the dynamic programming cannot be directly applied to such problems. Actually, the optimal strategies in most of the mean-variance works mentioned above are called pre-committed strategies. A precommitted strategy means that if the decision makers can commit themselves at the initial time, they can choose a strategy that is optimal from the point of view at the initial time, and then constrain themselves to abide by it in the future, although the strategy is no longer optimal for the future time. Considering a real-world example, if an investor adopts the pre-committed strategy, it is optimal only when sitting at the initial time but no longer optimal in the remaining time interval, which is time-inconsistent, while this investor hopes to choose a strategy which is optimal at every remaining time, i.e., she/he prefers the time-consistent strategy. For this reason, more and more scholars have become to study the time-consistent strategies for the dynamic mean-variance problems. Björk and Murgoci [6] developed a general theory for time inconsistent stochastic control problems, and derived an extended Hamilton-Jacobi-Bellman (HJB) system of equations that can be used to solve for the equilibrium strategy, which is time-consistent. From then on, many works have addressed the equilibrium strategies of dynamic mean-variance problems within the game theoretic framework, including Zeng and Li [31], Li and Li [15], Björk et al. [8], Björk and Murgoci [7], Wu and Zeng [25], Björk et al. [9], Wei and Wang [24], and Liu and Chen [19].
A reinsurance policy consists of one risk sharing function which determines how the insurer and the reinsurer share risk, and one premium share function which indicates how premium is diverted between the insurer and the reinsurer. The optimal risk sharing functions have different forms under different reinsurance premium principles. Kaluszka [14] studied the optimal reinsurance under premium principles based on the mean and variance of the reinsurer's share of the total claim amount for both global reinsurance and local reinsurance. Hipp and Taksar [13] solved the problem of ruin probability minimization and they found that the proportional reinsurance was optimal under the variance premium principle. Bai et al. [2] showed that an excess-of-loss reinsurance policy was the optimal form that minimized the ruin probability under the expected value premium principle. Zeng and Luo [32] modeled reinsurance as a cooperation game, then Pareto-optimal policies were studied and classified as either excess-of-loss or proportional reinsurance based on choices of premium share functions. Zhang et al. [33] analyzed the optimal reinsurance strategy for insurers with a generalized mean-variance premium principle. They derived the form of optimal reinsurance under the criteria of maximizing the expected utility function of terminal wealth and minimizing the probability of ruin. Li et al. [16] studied an insurer's reinsurance problem under a mean-variance criterion. They showed that excess-of-loss was the equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium was computed according to the expected value premium principle. These papers motivate us to consider the optimal reinsurance forms in an optimal reinsurance-investment problem with a common shock under different premium principles.
In this paper, we investigate the equilibrium reinsurance-investment strategy with common shock dependence under two popular reinsurance premium principles in practice: the variance premium principle and the expected value premium principle. The expected value premium principle is commonly used in life insurance which has stable and smooth claim frequency and claim sizes. The variance premium principle is extensively used in property insurance. The variance principle permits the company to take the fluctuations (variance) of claims into consideration when pricing insurance contracts. We assume that the insurer has two dependent classes of insurance businesses, which are subject to a common shock. The common shock can be interpreted as a claim that affects two lines of businesses at the same time, such as motor and life insurances, or as a natural disaster that causes different kinds of claims. The insurer can invest in a financial market consisting of a risk-free asset and a risky asset whose price process follows a geometric Brownian motion. Under the mean-variance criterion, we formulate the optimal reinsurance-investment problem within a game theoretic framework. We firstly prove that the optimal reinsurance contract is a proportional reinsurance under the variance premium principle, and the optimal reinsurance contract is an excess-of-loss reinsurance under the expected value premium principle.
Then by using the technique of stochastic control theory and solving the corresponding extended Hamilton-Jacobi-Bellman (HJB) system of equations, we derive the closed-form expressions of the equilibrium reinsuranceinvestment strategy and the equilibrium value function. We further use numerical examples to show the impacts of model parameters on the equilibrium reinsurance-investment strategies under two kinds of premium principles.
The main contributions of this paper are as follows: First, we incorporate a common shock into the insurer's reserve model and then consider the time-inconsistent reinsurance-investment problem under two reinsurance premium principles. Our model extends the results in Zeng and Luo [32] and Zhang et al. [33], where they did not consider common shock dependence and time-consistent strategies. Second, for mathematical simplicity, many papers use the proportional reinsurance directly, but we use the general reinsurance form and prove that the optimal reinsurance strategies have different forms under different reinsurance premium principles. We find that the optimal reinsurance contract is a proportional reinsurance under the variance premium principle, and the optimal reinsurance contract is an excess-of-loss reinsurance under the expected value premium principle. Third, we find that under the variance premium principle, the optimal proportions for the first class and the second class of claims are the same. However, under the expected value premium principle, the optimal reinsurance strategies for the first class and the second class of claims are different.
This paper is organized as follows. Section 2 presents the assumptions and problem formulation. Section 3 studies a mean-variance reinsurance-investment problem under variance premium principle, and derives the explicit solution to this problem by solving the extended HJB system of equations. Section 4 solves the meanvariance reinsurance-investment problem under the expected value premium principle. Section 5 uses numerical examples to illustrate the derived equilibrium reinsurance-investment strategies under two kinds of premium principles. Section 6 concludes this paper.

The model
Let (Ω, ℱ, {ℱ } ≥0 , ) be a filtered probability space satisfying the usual conditions, i.e., ℱ = {ℱ } ∈[0, ] is right continuous and complete with respect to . ℱ stands for the information of the market available until time , which is generated by the standard Brownian motions 1 ( ), 2 ( ). That is, ℱ is the accumulated information up to time . > 0 is a fixed time horizon. In addition, we assume that there is no transaction cost and tax in the financial market or the insurance market, and trading takes place continuously.
The reserve process {︀¯( ) }︀ ≥0 of the insurer is modeled bȳ Here¯0 is the deterministic initial reserve of the insurer and the constant 0 is the premium rate. { 1 ( )} ≥0 , { 2 ( )} ≥0 and { ( )} ≥0 are three independent Poisson processes with intensity parameters 1 > 0, 2 > 0 and > 0, respectively. The counting processes 1 ( )+ ( ) and 2 ( )+ ( ) represent the numbers of claims during the time interval [0, ] for the first class and second class, respectively. 1 is the size of the th claim for the first class and { 1 , ≥ 1} is assumed to be an i.i.d. sequence with common distribution 2 is the size of the th claim for the second class and { 2 , ≥ 1} is assumed to be an i.i.d. sequence with common distribution 2 (·), E( 2 ) = 1 2 > 0 and E( , and { 2 , ≥ 1} are mutually independent. It is obvious that the dependence of the two classes of businesses is due to a common shock governed by the counting process { ( )} ≥0 . This model has been studied extensively in the literature; see e.g., Yuen et al. [28], Yuen et al. [29], Yuen et al. [30] and Liang and Yuen [17].
Moreover, we allow the insurance company applies reinsurance strategies 1 ( , 1 ) and 2 ( , 2 ) for the first and second classes of claims. 1 ( , 1 ) and 2 ( , 2 ) are increasing risk share functions for the two variables 1 and 2 , with 0 ≤ 1 ( , 1 ) ≤ 1 and 0 ≤ 2 ( , 2 ) ≤ 2 . 1 ( , 1 ) and 2 ( , 2 ) are the parts of each random claim retained by the insurance company while the rest 1 − 1 ( , 1 ) and 2 − 2 ( , 2 ) are ceded to the reinsurer. Let the reinsurance premium rate at time be ( 1 , 2 ). In Sections 3 and 4, we consider the variance premium principle and the expected value premium principle respectively. Let denote the associated surplus process, i.e.,̃︀ 1, 2 ( ) is the wealth of the insurer at time under the strategy ( 1 (·, ·), 2 (·, ·)). This process then evolves as︀ Due to the jumps in the reserve process , it is not feasible to derive the optimal investmentreinsurance strategy in this paper explicitly. As most studies on the optimization problem (see e.g., [1,2,10,17,28], and so on), we can consider the problem under the diffusion approximation of the reserve process According to Grandell [11] (pages 15-17), we consider the diffusion approximations, i.e., approximating the reserve process (2.2) by a Brownian motion with drift. Mathematically, such approximations are based on the theory of weak convergence of probability measures. One way to express this diffusion approximation is, if the classical risk model is regarded to be "large deviation", the diffusion model is related to "the central limit theorem". By the similar calculations as those in Bai et al. [2], we have the diffusion approximation for our two-dimensional reserve process (2.2) in the following form︀ where 1 ( ) is a standard Brownian motion. Suppose that the insurer is allowed to invest all of his/her wealth in a financial market consisting of one risk-free asset (the money market instrument or the bond) and one risky asset (stock). We consider the financial market where the two assets are traded continuously on a finite time horizon [0, ].
The price process of the risk-free asset is given by where the deterministic function 0 ( )(> 0) is the interest rate of the risk-free asset. The price of the risky asset is modeled by the following stochastic differential equation where 1 ( )(> 0 ( )) is the appreciation rate and ( ) is the volatility coefficient of the risky asset. 2 ( ) is a standard {ℱ } ≥0 -adapted Brownian motion which is independent with 1 ( ). We assume that 0 ( ), 1 ( ) and ( ) are deterministic, Borel-measurable and bounded on [0, ]. Let ( ) denote the insurer's wealth at time and ( ) denote the total market value of the insurer's wealth in the risky asset. Then ( ) − ( ) is the value of the insurer's wealth in the risk-free asset. A restriction we will consider in this paper is the prohibition of short-selling of the stock, i.e., ( ) ≥ 0. But the market value of the insurer's wealth in the risk-free asset is not constrained.
We call ( ) := ( 1 ( , ·), 2 ( , ·), ( )) ′ an admissible strategy if ( ) is ℱ -predictable and satisfies 0 ≤ We denote the set of all admissible strategies by Π. Let ( ) denote the wealth process when the strategy (·) is applied. Then the reserve process ( ) is given by We will formulate the problem within a game theoretic framework, which is developed by Björk and Murgoci [6]. The optimization problem for the insurer with the objective function which we want to maximize is given by are the conditional expectation and variance given filtration ℱ with ( ) = for ( , ) ∈ [0, ] × R, respectively, and is the risk aversion coefficient of the insurer.
Since the objective function ( , , ) in (2.5) involves with V , [·], that is, ( , , ) is a nonlinear function of the expected value E , [·], so the optimization problem is time-inconsistent. We solve this time-inconsistent problem within a game theoretic framework and look for Nash subgame perfect equilibrium solutions. Now we recall the following definition of an equilibrium control and equilibrium value function, which is taken from Björk and Murgoci [6].
Definition 2.1. Given a control law * , which can be informally viewed as a candidate equilibrium law, choose a fixed ∈ Π, a fixed real number > 0 and a fixed arbitrarily chosen initial point ( , ) ∈ [0, ] × R, and construct a control law by for all ∈ Π and ( , ) ∈ [0, ] × R, we say that * is an equilibrium control law. The equilibrium value function is defined by ( , ) = ( , , * ).
Based on the definition above, the equilibrium strategy is time-consistent, the equilibrium strategy is thus the optimal time-consistent strategy. Our objective is to find an equilibrium strategy * and the corresponding equilibrium value function.
Next we give the extended HJB system of equations and the corresponding Verification Theorem in the following theorem.

7)
and is the equilibrium value function, * is the equilibrium reinsuranceinvestment strategy, and ( , ) has the following probabilistic interpretation: where ◇ and ℋ are defined as follows Equation (2.7) is also called the extended HJB system of equations.
Proof. The derivation of the extended HJB system of equations (2.7) and the proof of the Verification Theorem can be obtained by using the standard arguments similar to those used in Section 4 of Björk and Murgoci [6]. So we omit the details here.

Equilibrium strategy under variance premium principle
In this section, we assume that the reinsurance premium is calculated according to the variance premium principle. That is, here Λ is the reinsurer's safety loading. Then the reserve process ( ) given by (2.4) becomes and the usual infinitesimal generator for process (3.2) becomes After a number of elementary calculations, we obtain the following result about the extended HJB system of equations.
In the following, we will focus on finding the equilibrium reinsurance-investment strategy and the corresponding value function.
Proof. Let 1 (·, ·) be an arbitrary risk sharing function and 1 (·, ·) be a proportional risk sharing function Then with Cauchy-Schwartz inequality, we have The inequality above becomes equality when the relationship between 1 ( , 1 ) and 1 are proportional. That is Then we have Similarly, we have The inequality becomes equality when the relationship between 1 ( , 1 ) and 1 are proportional. That is Then we have By similar calculations, we have for an arbitrary risk sharing function 2 (·) and a proportional risk sharing function Note that Λ > 0, ≥ 0, ≤ 0, = > 0, 2 ≥ 0, then we get In other words, the optimal risk sharing functions are proportional.
Next we just consider the proportional reinsurance strategy, that is 1 ( , 1 ) = 1 ( ) 1 and 2 ( , 2 ) = 2 ( ) 2 , and give the equilibrium reinsurance-investment strategy and the corresponding equilibrium value function under the variance premium principle in the following theorem. (3.14) Proof. Set 1 := ( 1 + ) 1 1 , 2 := ( 2 + ) 1 2 , 1 := √︀ ( 1 + ) 2 1 and 2 := √︀ ( 2 + ) 2 2 . For the proportional reinsurance strategy, i.e., 1 ( , 1 ) = 1 ( ) 1 and 2 ( , 2 ) = 2 ( ) 2 , (3.4) becomes The maximizer of (3.16) is in which we use the fact ( ) = 2 2 . Next we look for the equilibrium reinsurance strategy and the corresponding equilibrium value function. Let The Hessian matrix is Because of Solving the above equation, we have Since the reserve process has the linear structure, and in accordance with the forms of the boundary conditions, we conjecture that Then (3.15) becomes That is, By separating variables, we obtain the following ordinary differential equations Moreover, the second equation in (2.7) becomes By separating variables again, we obtain Then we have (3.12)-(3.14). This completes the proof. Remark 3.5. Under the variance premium principle, the proportional reinsurance is optimal. And the optimal proportions for the first class and the second class of claims are the same, which just depend on the reinsurer's safety loading Λ of the reinsurance business, the interest rate 0 of the risk-free asset, and the risk aversion coefficient of the insurer. The equilibrium investment strategy depends on the interest rate 0 of the risk-free asset, the appreciation rate 1 , the volatility coefficient of the risky asset, and the risk aversion coefficient of the insurer. Moreover, owing to the fact that the risk aversion coefficient of the insurer is a constant, the equilibrium reinsurance-investment strategy is independent of the state variable .

Equilibrium strategy under expected value premium principle
In this section, we assume that the reinsurance premium is calculated according to the expected value principle. That is, where 1 and 2 are the reinsurer's safety loadings of the insurance businesses. The usual infinitesimal generator becomes By the similar calculations as those in Section 3, we obtain the following result about the extended HJB system of equations under the expected value premium principle. Proposition 4.1. The first and second equations of the extended HJB system of equations (2.7) can be simplified as follows: and ( , ) + ]︂ ( , ) Proof. The proof of this proposition is similar to that of Proposition 3.1, so we omit it here.
Next we will find the equilibrium reinsurance-investment strategy and the corresponding value function under the expected value premium principle.
Theorem 4.2. Assume that the value function satisfies ≥ 0, ≤ 0. Under the expected value premium principle, the optimal reinsurance-investment strategy * = ( * 1 , * 2 , * ) for the mean-variance criterion is in the form of In other words, the optimal risk sharing function is excess-of-loss reinsurance.
Theorem 4.3. The equilibrium reinsurance-investment strategy * = ( * 1 , * 2 , * ) for problem (2.5) with (2.4) under expected value premium principle (4.1) is The corresponding equilibrium value function of the extended HJB system of equations (2.7) is Proof. Since the wealth process has the linear structure, and in accordance with the forms of the boundary conditions, we conjecture that By separating variables, we obtain the following ordinary differential equations Inserting (4.17), (4.18) and (4.5)-(4.7) into (4.4), we have ]︂ = 0.
Remark 4.5. Under the expected value premium principle, the excess-of-loss reinsurance is optimal. The equilibrium excess-of-loss retention limits depend on the intensity parameters 1 , 2 , of the Poisson processes, the reinsurer's safety loadings 1 , 2 , the risk aversion coefficient of the insurer, and the interest rate 0 of the risk-free asset.
Remark 4.6. The equilibrium investment strategies under the two different reinsurance premium principles are the same, depending on the interest rate 0 of the risk-free asset, the appreciation rate 1 , the volatility coefficient of the risky asset, and the risk aversion coefficient of the insurer. The equilibrium investment strategy is independent of the parameters 1 , 2 , , 1 1 , 1 2 , 2 1 , 2 2 in the reserve process, and the safety loadings 1 and 2 of the reinsurance business. This shows that the equilibrium investment strategy is unaffected by the price of reinsurance and the price of the primary insurance.

Numerical illustrations
In this section, we present some numerical examples to illustrate our results obtained in Sections 3 and 4. Without loss of generality, we assume that all the parameters of the financial market and the insurer are constants.
In Sections 3 and 4, we have derived the explicit expressions of the equilibrium reinsurance-investment strategies under both variance premium principle and expected value premium principle given in Theorems 3.3 and 4.3, respectively. In the following, we provide the numerical illustrations of the equilibrium reinsuranceinvestment strategies under the two premium principles according to Theorems 3.3 and 4.3. We use MATLAB to obtain the following numerical results.

The equilibrium strategy in Section 3
The conclusion in Theorem 3.3 shows that the optimal proportions are increasing with respect to the safety loading Λ of the reinsurance company, and decreasing with respect to the interest rate 0 of the risk-free asset and the risk aversion coefficient of the insurer. Moreover, the equilibrium investment strategy is increasing with respect to the appreciation rate 1 of the risky asset, and decreasing with respect to the interest rate 0 of the risk-free asset, the volatility coefficient of the risky asset and the risk aversion coefficient of the insurer. Set = 4, 0 = 0.02, 1 = 0.05, = 0.2, Λ = 2, = 0.5. The equilibrium reinsurance-investment strategy is shown in Figure 1.

Concluding remarks
This paper studies the equilibrium reinsurance-investment strategies for a mean-variance insurer with common shock dependence under two kinds of premium principles: the variance premium principle and the expected value premium principle. Using the technique of stochastic control theory and the corresponding extended HJB system of equations, within a game theoretic framework, we derive the closed-form expressions of the equilibrium reinsurance-investment strategies and the equilibrium value functions. We find that: under the variance premium principle, the optimal reinsurance contract is a proportional reinsurance; on the other hand, under the expected value premium principle, the optimal reinsurance contract is not a proportional reinsurance but an excess-of-loss reinsurance.
It would be interesting to extend our analysis to some other situations. In recent years, socially responsible investing has become a popular subject with both private and institutional investors. We can propose a modification allowing to incorporate not only the risk-free and risky assets but also a social responsibility measure into the investment decision making process, and reconsider the mean-variance reinsurance-investment problem. Of course, these problems are more complicated. To solve such problems, we need to adopt much more sophisticated techniques. We will explore these problems in the future research.