Abstract
In this paper, we consider the problem of investment and reinsurance with time delay under the compound Poisson model of two-dimensional dependent claims. Suppose an insurance company controls the claim risk of two kinds of dependent insurance businesses by purchasing proportional reinsurance and invests its wealth in a financial market composed of a risk-free asset and a risk asset. The risk asset price process obeys the geometric Brownian motion. By introducing the capital flow related to the historical performance of the insurer, the wealth process described by stochastic delay differential equation (SDDE) is obtained. The extended HJB equation is obtained by using the stochastic control theory under the framework of game theory. Under the reinsurance expected premium principle, optimal time-consistent investment and reinsurance strategy and the corresponding value function are obtained. Finally, the influence of model parameters on the optimal strategy is explained by numerical analysis.
1. Introduction
Since insurance companies have been allowed to enter the financial market for investing risk assets, the optimal investment strategy has become an important research topic in recent years. Many literature have studied the maximization of the utility of the terminal wealth or the minimization of the ruin probability of the insurer. Browne [1] uses the surplus process given by the diffusion risk model to study the investment problem of maximizing the utility of the terminal wealth and minimizing the ruin probability of an enterprise and obtains the explicit optimal solution. Hipp and Plum [2] apply the Cramer–Lundberg model to describe the insurance surplus process, based on the assumption that there is only one risky asset in the financial market and the time is discrete; the investment problem is studied. Wang et al. [3] use martingale approach to study the optimal portfolio selection of insurers under the criteria of mean-variance and constant absolute risk aversion utility maximization. For more similar literature, see Liu and Yang [4], Yang and Zhang [5], Wang [3], and Bai and Guo [6].
In addition to market risk, the insurer will also consider insurance risk. It is impossible to avoid insurance risk by investing in bonds and other assets in the market alone. However, reinsurance business provides a way for the insurer to avoid this risk. In recent years, this approach has been widely concerned. Reinsurance business mainly adopts two different forms of insurance: excess-of-loss reinsurance and proportional reinsurance. Promislow and Young [7] first investigate the proportional reinsurance and investment. Bauerle [8] considers proportional reinsurance and investment also, and the optimal explicit solution of the investment-reinsurance problem is obtained under the mean-variance criterion. Zeng and Li [9] also study proportional reinsurance and obtain the efficient frontier of the mean-variance under the multidimensional risky asset model. The stock price in the above model generally follows the geometric Brownian motion; the market price of stock-related risk is constant, but in the real market, stock price may have other characteristics, such as stochastic volatility. Liang et al. [10] used the Ornstein–Uhlenbeck process to characterize the instantaneous return of stocks and obtained the optimal reinsurance and investment strategy. Gu et al. [11] investigate the excess-of-loss reinsurance-investment problem under the constant elasticity variance (CEV) model.
There are two deficiencies in the above literature that deserve further discussion. On the one hand, these literature implicitly assume that all insurance businesses of insurers are independent of each other, so they only study the investment and reinsurance of a single insurance business. However, in the real insurance market, there are often interdependencies between insurance businesses. For example, during the 2019-nCoV, medical claims and death claims often occur together. In order to depict this kind of dependency between different insurance businesses, the risk dependent model is proposed. The main works in this area are as follows. Yuen et al. [12], taking the expected utility maximization of the terminal wealth as the criterion, considered the optimal proportional reinsurance problem with multidimensional risk dependence by using the diffusion approach method. For the detailed process of diffusion approximate to the compound Poisson process, see Gandell [13]. Liang and Yuen [14], under the principle of variance premium, investigated the optimal proportional reinsurance of the Poisson model and diffusion approximation model. Ming et al. [15] derive the explicit expression of the optimal proportional reinsurance under the mean-variance criterion by using stochastic linear quadratic control. Considering the combination of investment and reinsurance, Bi et al. [16] obtains the optimal investment-reinsurance strategy for mean-variance under the diffusion approximation model. Bi and Chen [17], under the criterion of maximizing the expected utility of terminal wealth, arrived at the optimal investment and reinsurance strategies. On the other hand, most of the literature on optimal investment-reinsurance and other optimal control problems focus on time-delay free controlled systems. In fact, financial markets tend to rely on the past, Chang [18] considers the investment and consumption problems related to the return on risk assets and the historical performance. Federico [19] introduces the time-delay state process by considering the capital inflow/outflow related to performance. Peng et al. [20] and Yu et al. [21] study the optimal dividend policy based on observing the information of past time points to determine the behavior of the next moment. In fact, this is a discrete case of time delay. However, the stochastic control problems of systems with time-delay state may be infinite-dimensional in continuous cases; hence, it is difficult to find the analytical solutions. Only in some special cases, it is finite-dimensional and the problem has explicit solution. Elsanosi et al. [22], ksendal and Sulem [23], and David [24] provide a theoretical basis for solving such problems. Shen and Zeng [25] first introduced the time delay in the investment and insurance problem. They introduced the inflow/outflow of capital in the wealth process of insurer and then depicted the wealth process of insurance companies through the stochastic delay differential equation (SDDE). After that Li [26] and Lai [27] studied the optimal investment-reinsurance problem with time delay under Heston and CEV models, respectively.
Inspired by the above research, this paper combines risk dependence with time delay to consider investment-reinsurance problem. The structure of the rest of this paper is as follows. In Section 2, the financial model framework of this paper is given, assuming that an insurer can invest in a risk-free asset and a risky asset, and in the case of two-dimensional dependent claim compound Poisson model and the introduction of the historical performance of the insurance company, the company’s wealth process with time delay is obtained. In Section 3, considering the mean-variance preference criterion, the time-inconsistent optimization problem is defined, and the extended HJB equation is obtained by using the stochastic control theory in the framework of game theory. In Section 4, under the principle of reinsurance expected premium, the explicit solutions of optimal investment and reinsurance strategies and their corresponding value functions are derived. In Section 5, the numerical calculation process of optimal investment and reinsurance strategies are introduced through numerical examples, and the influence of important model parameters on optimal strategy is analyzed. Section 6 concludes this paper.
2. The Model
Suppose that model is based on the probability space of information flow which satisfies the general assumptions of right continuity and completeness, where is a finite constant, representing the operation cycle of an insurance company, and is the sum of information available up to time . All stochastic processes involved in this paper are assumed to adapt to .
Suppose an insurer has an insurance portfolio business, which is composed of two different insurance businesses, such as medical insurance and death insurance. Suppose that the random variable represents the claim amount of the first type of insurance business; they are independent and have the same distribution function . represents the claim amount of the second type of insurance business; they are independent and have the same distribution function . We assume that if , then . Otherwise, . And also assume that if , then . Otherwise, . In addition, their moment generating functions and exist. The cumulative claim process of the two insurance businesses are as follows:where and represent the number of claims for the first and second categories of insurance business up to time , respectively. And suppose , , , and are independent of each other.
For different insurance businesses, it is assumed that they are interdependent as follows:where , , and are three independent Poisson processes and the corresponding intensities are , , and , respectively. Therefore, the total claim amount of these two types of the insurance business is
Suppose for arbitrary , and exist. And, for some , there are and .
For convenience of writing, we definewhere , , , and .
Considering the financial market, it is assumed that assets are traded continuously in time interval , and tax and transaction costs are not considered. Suppose the insurer can invest its wealth in the financial market composed of a risk-free asset and a risky asset. The risk-free asset price process is
The risky asset price process is as follows:where , , and are constants, representing risk-free interest rate, drift rate, and volatility, respectively. Define .
As usual, the surplus process from the insurer up to time is defined as follows:where is the initial surplus and is the premium rate. In addition, it is assumed that insurance companies can continuously reinsurance insurance business in a certain proportion to control business risk. We denote the retention ratio of categories 1 and 2 insurance business by and . When the claim occurs, the insurance company pays or , while the reinsurance company pays or . Let the reinsurance rate be at time t.
Let denote the wealth process of insurance companies at time , denote the amount of capital invested in the risky asset, and then denote the amount of wealth invested in the risk-free asset. The investment-reinsurance strategy will be applied by the insurer. Given an investment-reinsurance strategy , the wealth process of an insurer satisfies the following stochastic differential equation:
Next, we consider the influence of historical performance on the wealth process. Suppose that represents the inflow/outflow of capital, then the wealth process of insurers with time delay is given by the following stochastic delay differential equation (SDDE):
To make the problem easier to deal with, consider a linear capital inflow/outflow function, that is,where and are constants, . , , and represent the integrated, average, and point by point delay information of wealth process in time interval . and are given average parameters and delay parameters, respectively. Note that is defined as the weighted average value of wealth process in time interval , and the exponential decay factor represents the weight. When , and represent the average gain or loss and absolute gain or loss of wealth of insurers in the last operating cycle. Because the inflow/outflow of capital is closely related to the past performance of the wealth process. If the past performance is good, the company will give part of its earnings to shareholders or give bonuses to the management, which shows the outflow of capital, i.e., . At this time, and . On the contrary, if the past performance of the insurance company is not good, the company needs additional financing to achieve the predetermined goal. This shows capital inflow, i.e., when and . Therefore, the function considers the average and absolute performance of the wealth process in .
Substituting (10) into (9), the following stochastic delay differential equation (SDDE) is obtained:
Furthermore, suppose , which can be interpreted as that the insurance company has the initial wealth of at . There is no business operation during , and the wealth has no change. The integrated delay wealth initial value can be calculated to get .
Definition 1. (admissible strategy). For any fixed , an investment-reinsurance strategy is said to admissible if (i) is progressively measurable, (ii) for , , , and , and (iii) SDDE (11) has a unique strong solution such that . Let be the set of all admissible investment-reinsurance strategy.
3. Optimization Problem
To take historical operating performance into account, the insurer will focus on both terminal wealth and historical average operating performance ; thus, the following objective function is defined:where risk aversion coefficient and delay parameter are constants. and represent conditional expectation and conditional variance based on , , and , respectively. is the weight of , indicating the degree of terminal wealth affected by historical average performance. If we write , then . In addition, according to Chang [18], delay optimal control problem is generally an infinite-dimensional problem. In order to obtain the optimal solution, some additional conditions will be attached. We assume that the value function is only related to and , but is related to ; in order to make only depend on , the problem can obtain the optimal solution, and we assume the following conditions hold:
Therefore, this paper aims at the following optimization problems:where and .
Remark 1. (i)According to Shen and Zeng [25], condition (13) can be regarded as exogenous technical conditions that need to be determined in advance by the insurance company. Firstly, the average delay wealth and point by point delay wealth are determined by selecting the average parameter and delay time . Secondly, it selects the weight . Finally, it calculates the weight ratios and of historical performance and according to the two assumptions in (13) and adjusts the inflow/outflow of capital accordingly.(ii)Because there is a nonlinear function of the expectation of the terminal value wealth in the variance term, problem (14) is time inconsistent, which leads to the failure of Bellman’s optimal principle. Many works of literature deal with the mean-variance problem by setting a precommitment, so the optimal strategy obtained are time-inconsistent. However, for a rational decision maker, time consistency is often not negligible. Rational decision makers hope that the equilibrium strategy they find is not only optimal at this time but also optimal in the future with the evolution of time, that is to say, the equilibrium strategy is time consistent. Therefore, for problem (14), this paper aims to find the equilibrium strategy.
Definition 2. Consider a control law . Choose arbitrarily , , and and define the control law :We call that is an equilibrium strategy if for any and . If the equilibrium strategy exists, the equilibrium value function is defined as .
According to Definition 2, the equilibrium strategy is time consistent. For simplicity, we denote that is once continuously differentiable on , is twice continuously differentiable on , and is once continuously differentiable on . To provide verification theorem and derive conveniently extended HJB equation, for , , and given control law , we define variational operator as follows:The following theorem provides verification for the extended HJB equation in problem (14).
Theorem 1 (verification theorem). For problem (14), we assume that there exist two real-valued functions satisfying the following extended HJB equation:
Then, , and is an equilibrium investment-reinsurance strategy.
The proof process of Theorem 1 is similar to that of Björk et al. [28], so it is omitted here.
In Definition 1, the policy set is allowed to require the reinsurance policy to satisfy the constraint and . To facilitate the solution, we do not consider this constraint temporarily and record all the policy sets satisfying (i) and (iii) as . According to the variational operator (16), the extended HJB (17) can be expanded as follows:
Suppose that the solution of the above extended HJB equation has the following structure:with the boundary condition and .
Differentiating and with respect to , , and , we obtain
Through simple calculation, we can also obtain
Putting the above results back into (18), we can arrive atwhere
According to , we have
For the convenience of writing, let
4. Optimal Time-Consistent Strategy
This section assumes that the reinsurance premium rate is calculated by the expected premium principle, i.e.,where and are the reinsurer’s safety loading of the insurance business.
Substituting the above formula into (24), we have
To facilitate derivation, we rewrite (25) as
Differentiating with respect to , , and , we can derive
From (29), we obtain the following Hessian matrix:where
Lemma 1. The function in (28) is concave with respect to .
Proof. In order to prove Lemma 1, we only need to prove that the matrix is negative definite. From (43), we know , thus . According to (30), we only need to prove that matrix B is positive definite.
and . Let denote the transposition of a vector or matrix, thenSo, matrix B is positive definite.
From (29), we haveBy solving the above equations, we can obtainwhere and .
From Lemma 1, we know that is the point where function takes the maximum value. Putting into (22), we can obtainAccording towe haveBy separating variables of , we can obtainBy solving the above equations, we haveAccording to the above discussion, the following proposition can be obtained.
Proposition 1. For problem (14), the time-consistent investment-reinsurance strategy in set is as follows:
The corresponding equilibrium function iswhere H and F are given by (43).
Let for . Let for . For (), we set (). And for (), we set (). To make sure that the optimal reinsurance strategies satisfy and , we introduce the following lemma.
Lemma 2. For , , , , , , and given in (4), the following inequality holds:
Proof. Using inequality, we can easily get and and then we can obtainIn addition, for any positive number , , , and , if , then . In combination with inequality (47), inequality (46) is easily proved.
From Lemma 2, we will investigate the optimal results in the following four cases: Case 1: Case 2: Case 3: Case 4: Next, the optimal time-consistent strategy in admissible strategy set and the corresponding value function are discussed. In order to have a clear classification discussion, it is assumed that . Case 1: in this case, we have and ; thus, . Let . By substituting into (28) and maximizing function , we can get the maximum point: Let . For , it is easy to see , and then we have . Putting into (41) and (45), we obtain where where is a constant whose value will be determined in a later calculation. For , we have . Substituting it into (41) and (45), we can obtain where To make the value function continuous, let ; then, Case 2: in this case, we have , and , and it is easy to see . For , we have , , and thus . Substituting it into (41) and (45), we can derive where For , we have , and thus . Let . Putting into (28) and maximizing function , we can get the maximum point: Let and . It is easy to see that . For , we have . Inserting it into (41) and (45), we can obtain where For , we have , and thus . Putting it into (41) and (45), we can arrive at where Let then Case 3: in this case, we have , . And , so . For , we have . Substituting it into (41) and (45), we can obtain For , we have , and thus . Denote by the function in (28). By maximizing , we derive Let and . It is easy to see that . For , we have . By substituting it into (41) and (45), we can obtain where For , we have . Putting it into (41) and (45), we can obtain Let We derive Case 4: in this case, we have and , and thus . Let . By maximizing , we arrive atLet .
For , we have . Inserting it into (41) and (45), we can derivewhereFor , we have . Putting it into (41) and (45), we can obtainwhereLetWe haveFrom the above discussion, we can get the following theorem.
Theorem 2. Assuming , the optimal time-consistent investment and reinsurance strategies for problem (14) are as follows:(i)If Case 1 holds, the optimal investment-reinsurance strategies for model (14) are and the value function is given by where , , , , and are given by (50)–(55), respectively.(ii)If Case 2 holds, the optimal investment-reinsurance strategies for model (14) are and the value function is given by where , , , , , , and are given by (57)–(67), respectively.(iii)If Case 3 holds, the optimal investment-reinsurance strategies for model (14) are and the value function is given by where , , , and are given by (71)–(76), respectively.(iv)If Case 4 holds, the optimal investment-reinsurance strategies for model (14) areand the value function is given bywhere , , and are given by (79)–(83), respectively.
Remark 2. (i)Since is a continuous function for any . Furthermore, which includes that is a classical solution to the extended HJB (18).(ii)According to Theorem 2, the investment and reinsurance strategy of the insurer is not directly affected by the average parameter and the delay time , but according to (13), the average parameter and the delay time have an indirect influence on the investment and reinsurance strategy of insurance companies.(iii)Note that, in the classification discussion of Theorem 2, in order to make the classification clear, we assume that . For , we can also make a similar discussion.When , problem (14) degenerates to the case without time delay.
Corollary 1. Without time delay, the optimal time-consistent investment and reinsurance policies of problem (14) are as follows:(i)If Case 1 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by where(ii)If Case 2 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by where(iii)If Case 3 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by where(iv)If Case 4 holds, the optimal investment-reinsurance strategies for problem (14) areand the value function is given bywhere
5. Numerical Simulations
In this section, Example 1 will be used to illustrate the specific numerical calculation process of finding the optimal time-consistent strategy, and Example 2 will be used to analyze the influence of important parameters on the optimal time-consistent strategy. Assuming that the claim amount and are exponentially distributed with parameters and , respectively, then , , , and .
Example 1. Let , , , , , , , , , , , , and and according to Remark 1, we can calculate and , and thus . According to the above model parameters, Table 1 can be calculated.
From Table 1, for , we have . According to the analysis of Theorem 2, it is easy to see that and , , and hence . For , we have . So, recalculate Table 1 to obtain Table 2.
Example 2. If there is no special description in this example, the basic parameter values are as follows: , , , , , , , , , , , , and .
Figures 1 and 2 depict the influence of risk aversion parameter and delay parameter on the optimal time-consistent investment strategy. From Figure 1, we can see that the optimal time-consistent investment strategy decreases with the increase of risk aversion parameter , that is to say, the higher the risk aversion degree of the insurer is, the less the amount of risk investment will be. Because parameter includes the information of average parameter and delay , it is a comprehensive time-delay parameter, so we only analyze . Figure 2 shows that the larger the delay parameter is, the larger the number of investment in risky assets will be. Note that if , then the insurer decision-making is only based on the current information, so it may take short-term risk-taking behavior for the immediate possible high return. For , when the insurer is making decision, the comprehensive performance in the past period will be taken into account. Insurer focuses on information in a period when making decisions. According to (12), the greater the value of , the greater the proportion of average wealth in performance measurement. That is, the insurer can change the inflow/outflow of the insurer’s capital by adjusting the size of the parameter beta, thus changing the risk faced by the insurer. The bigger the beta, the smaller the risk, so the insurer will consider increasing the number of risky assets.
Figures 3–6 depict the influence of risk aversion coefficient and delay parameter on two types of insurance reinsurance. According to Figures 3 and 4, and decrease with respect to . The higher the risk aversion degree of the insurer, the more reinsurance he will buy to reduce his risk, so the retention ratio of and will be reduced. Figures 5 and 6 show that the retention ratio () increase with respect to the parameter . As the impact of on investment strategy . The larger the , the stronger the insurer’s ability to adjust capital inflow/outflow, that is, the stronger the insurer’s risk control ability. To a certain extent, the profitability of the insurer will be stronger, so the insurer will reduce the purchase of reinsurance, and the proportion of reinsurance retention () will increase. This is consistent with economic reality, which the more information investors observe, the more profit they will make.
Figures 7–9 depict the effect of the claim intensity , , and on reinsurance. In Figure 7, the larger the is, the larger the is and the smaller the is. Because the larger the is, the greater the expected claim amount of the first type of insurance business will be, so the insurer will purchase more reinsurance for the first type of insurance business and reduce the proportion of retained insurance . At this time, will remain unchanged, that is, the expected claim amount of the second type of insurance business will remain unchanged. Based on the consideration of constant total risk and more profits, the insurer will increase the retention ratio of reinsurance. A similar analysis can explain why; with the increase of , decreases and increases in Figures 8 and 9 which shows that the retention ratios and of the two types of insurance businesses decrease with the increase of lambda. Because the larger the lambda is, the greater the expected claim amount of the two types of insurance businesses will be. Therefore, in order to control the risk within a certain range, the insurer will buy more reinsurance for the two types of insurance businesses and reduce the retention ratio and .
6. Conclusion
In this paper, we study the optimal investment-reinsurance problem with delay and risk dependence under the mean-variance preference criterion. Considering the time-delay effect and risk dependence, we obtain the extended HJB equation with delay based on the time delay stochastic control framework and the equilibrium stochastic control method. The results show that the optimal time-consistent investment and reinsurance strategy will be affected by the time delay effect. The larger the capital flow related to the historical business performance, the greater the risk faced by the insurance company. In a prudent attitude, the insurer will reduce the amount invested in a risk asset and reduce the reinsurance retention ratio of all insurance businesses. In addition, risk dependence is linked by common risk shock sources. The greater the risk common shock intensity is, the smaller the reinsurance retention ratio will be. From the numerical analysis results, we can see not only the numerical calculation process of the optimal strategy but also the intuitive verification of the above conclusions.
In this paper, we study the risk assets under geometric Brownian motion. To better simulate the real financial market, the following research will consider the introduction of CEV, Heston, and other stochastic volatility models, Vasicek, CIR, and other stochastic interest rate models.
Data Availability
The data in this paper can be used publicly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the science and technology research project of Chongqing Education Commission under Grant KJQN201801529 and Doctoral Research Projects for Central Universities under Grant JBK2007190.