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We mainly study a general risk model and investigate the precommitted strategy and the time-consistent strategy under mean-variance criterion, respectively. A lagrange method is proposed to derive the precommitted investment strategy. Meanwhile from the game theoretical perspective, we find the time-consistent investment strategy by solving the extended Hamilton-Jacobi-Bellman equations. By comparing the precommitted strategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Furthermore, we theoretically and numerically provide the effect of the parameters on these two optimal strategies and the corresponding value functions.

There exist two important risk models: the Cramér-Lundberg model (the C-L risk model) and the dual risk model. The C-L risk model describes the surplus process of an insurance company. The insurer has two opposing cash flows: incoming cash premiums and outgoing claims. It can get the premium at a rate

By incorporating the C-L risk model and the dual risk model, a general risk model can be given by the following equation:

The literature for mean-variance (MV) analysis of the general risk model has not appeared. Mean-variance analysis for optimal asset allocation is an important result of financial economics. Markowitz [

Nowadays two basic ways are used to deal with time-inconsistency in optimal control problems in the literature. One way is to study the precommitted problem where “optimal” is interpreted as “optimal from the point of view of time zero” and the decision makers themselves follow the policies chosen at the initial time in the future. Zhou and Li [

Recently many scholars have paid more attention to mean-variance analysis for the risk model. Bäuerle [

In this paper, we are concerned with the optimal investment problem for the general risk model under mean-variance criterion. Our study contributes to the literature in three ways. Firstly, we study the general risk model under mean-variance criterion and the precommitted strategy and the time-consistent strategy are derived. Secondly, we propose a simple technique (lagrange technique) to deal with the precommitted investment problem. Our method is different from the lagrange technique proposed by Zhou and Li [

The rest of this paper is organized as follows. Section

In this section, we start with a filtered complete probability space (

The financial market consists of a risk-free asset and

For

For

A strategy

SDE (

In addition, let

This section will provide the precommitted investment strategy for problem (MV). We firstly state the main idea of solving problem (MV).

Let

We will calculate the optimal investment strategy and the value function by solving the related Hamilton-Jacobi-Bellman (HJB) equation for problem (

If there exist a real function

Now, we will solve the HJB equation in Lemma

For problem (

By virtue of

For problem (

Problem (MV) can be finally solved by virtue of the relationship of

For problem

The efficient frontier at initial state

The precommitted investment strategy is stochastically dependent on the current wealth which means that

When all the parameters are all constants and

In this section, we will provide optimal time-consistent investment strategy and the equilibrium value function for problem (MV) by solving the extended HJB equations.

Firstly define problem (

For any fixed chosen initial state

It is easy to see that the equilibrium strategy is time-consistent. So the equilibrium strategy

If there exist two real functions

Next, we will find the solution to the extended HJB equations. By using the infinitesimal generator (

On one hand, differentiating the function in the left bracket of (

For problem (MV), optimal time-consistent strategy

By virtue of (

Equation (

This time-consistent investment strategy is independent on the current wealth which means

When all the parameters are all constants and

In the next two subsections, we study the effect of parameters on the optimal strategies (precommitted strategy and time-consistent strategy) and the corresponding value functions and provide some numerical examples to illustrate the effects. Finally, compare the precommitment results with the time-consistent ones by some numerical analysis. For convenience but without loss of generality, all the parameters involved are constants and

In this subsection, we will work on numerical analysis of the precommitted strategy and the value function.

Firstly, we will show how the coefficients involved impact on the precommitted strategy. Since the precommitted investment strategy is stochastically dependent on the current wealth, we explore the effect of parameters of the financial market and the risk aversion by stochastic simulation. Because the precommitted strategy is indeed a stochastic process, we investigate the effect of different parameters in a same sample trajectory. In order to model the trajectory, we assume that

The effect of parameters on optimal precommitted strategy for the general risk model.

The effect of

The effect of

The effect of

The effect of

The effect of

The effect of

The effect of parameters on optimal precommitted strategy for the C-L risk model.

The effect of

The effect of

The effect of

The effect of

The effect of

The effect of

Secondly, we will show how the coefficients involved impact the value function. For convenience, introduce the notation

The effect of parameters on the value function.

The effect of

The effect of

The effect of

The effect of

In this subsection, we will work on numerical analysis of the time-consistent strategy and the equilibrium value function.

Firstly, we work on how the coefficients involved impact optimal time-consistent investment strategy. From (

The effect of parameters on optimal time-consistent strategy.

The effect of

The effect of

The effect of

The effect of

Secondly, we will show how the coefficients involved impact the value function. From (

The effect of parameters on the equilibrium value function.

The effect of

The effect of

The effect of

The effect of

In this subsection, we compare the optimal investment strategy, the corresponding value function, and the efficient frontier under the precommitted framework with the ones under the time-consistent framework.

Firstly, we compare the precommitted strategy with the time-consistent strategy. The time-consistent strategy is time deterministic but the precommitted strategy depends on the current wealth. We also assume that

The comparisons between the precommitted strategy and the time-consistent strategy.

The comparison of optimal strategies

The comparison of the value functions

The comparison of different efficient frontiers

Secondly, we compare the optimal value function

Thirdly, we compare the efficient frontiers derived from two different perspectives. From (

It seems to be true that the precommitted strategy is prior to the time-consistent strategy from the comparison of the value functions and the efficient frontiers, but we cannot conclude that the precommitted strategy is better than that from the game theoretical framework. Because the latter strategy is time-consistent and it can make the company ensure a consistent return for the whole time horizon. Meanwhile, the precommitted strategy is a global optimal strategy which only can make the company’s mean-variance utilities maximized at

In this paper, we have investigated the optimal investment strategy for a general risk model under mean-variance criterion. The precommitted strategy is derived by the lagrange method and the time-consistent strategy is also calculated via the approach based on the time-consistent equilibrium controls. In the end, we theoretically and numerically provide the effect of the parameters on the optimal investment strategies and the corresponding value functions. The value function and the efficient frontier under the precommitted strategy are prior to the ones under the time-consistent strategy, we cannot conclude that the precommitted strategy is better than that from game theoretical framework, because the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Meanwhile, the precommitted strategy is a global optimal strategy and it only can make the company’s mean-variance utilities maximized at initial time

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the National Natural Science Foundation of China (Grant nos. 11201335 and 71071111) and the Research Project of the Social Science and Humanity on Young Fund of the Ministry of Education (Grant no. 11YJC910007). The authors would like to thank anonymous reviewers for very helpful suggestions which improved this paper greatly.