Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a Dual Risk Model

We are concerned with optimal investment strategy for a dual risk model. We assume that the company can invest into a risk-free asset and a risky asset. Short-selling and borrowing money are allowed. Due to lack of iterated-expectation property, the Bellman Optimization Principle does not hold. Thus we investigate the precommitted strategy and time-consistent strategy, respectively. We take three steps to derive the precommitted investment strategy. Furthermore, the time-consistent investment strategy is also obtained by solving the extendedHamilton-Jacobi-Bellman equations.We compare the precommitted strategywith time-consistent strategy and find that these different strategies have different advantages: the former can make value function maximized at the original time t = 0 and the latter strategy is time-consistent for the whole time horizon. Finally, numerical analysis is presented for our results.


Introduction
We start from a dual risk model; namely, the surplus process of a company is given by where  is the initial capital and  is the rate of expenses; the positive incomes or profits   ( = 1, 2, 3, . ..) arrive as a Poisson process {()} ≥0 with intensity ;   ( = 1, 2, 3, . ..) are independent and identically distributed (i.i.d) nonnegative random variables with the first and second moments   and  2  .The expected increase of the surplus per unit time satisfies the positive loading condition:  =   −  > 0. The model is called dual as opposed to the Cramer-Lundberg risk model with applications to insurance.
The dual risk model can be described as the amount of capital of a company engaging in research and development.The company pays expenses and gets occasional revenues.
Revenues are interpreted as the values of future gains from an invention or discovery, while the decrease of surplus can represent costs of production, payments to employees, maintenance of equipment, and so forth.This is also the case for a portfolio of life annuities, where the risk consists of survival and the event death leads to gains.Furthermore, many scholars investigated the dual risk model under different criterions.Stochastic control theory and the related methodologies are the main tools for finding the related minimum probability of ruin or the maximum value for the expected utility of terminal wealth or the maximum value for the expected present value of the dividends minus capital injections.For related works see, for example, Avanzi et al. [1,2], Zhu and Yang [3], Dai et al. [4], and Yao et al. [5] and references therein.
In this paper, we are concerned with optimal investment strategy for the dual risk model under mean-variance criterion.Our objective is to find the optimal investment strategy such that the expected terminal wealth is maximized and the variance of the terminal wealth is minimized.According to the biobjective optimization theory, the alternative objective is to find a strategy which maximizes the expected terminal wealth minus the variance of the terminal wealth; namely, (MV) { { {  0, ( ()) −  2 Var 0, ( ()) → max where  is a prespecified risk aversion coefficient,  0, [ It is well known that this criterion lacks the iteratedexpectation property, so problem (MV) is time-inconsistent which means that Bellman Optimality Principle does not hold.The lack of time consistency leads to conceptual as well as computational problems.Recently there are two approaches to the conceptual problem.
We then simply disregard the fact that at later points in time the strategy will not be optimal.Such a strategy is known as a precommitted strategy.Zhou and Li [6] and Li and Ng [7] made big breakthrough works and proposed an embedding method to derive the precommitted strategy for problem (MV).
(ii) We take the time-inconsistency seriously and formulate the problem within a game theoretic framework.Problem (MV) is viewed as a game, where the players are the future incarnations of our own preferences and Nash equilibrium points can be derived in order to address the general time-inconsistency.See more references in Björk and Murgoci [8], Ekeland and Lazrak [9], and Kryger and Steffensen [10].
Thus, problem (MV) can be reduced to a resolvable problem by virtue of some techniques including the embedding technique [6,7] and the game theoretical technique [8].Our contributions are as follows.Firstly, we take three steps to deal with the precommitted investment problem.This method is different from the embedding technique in Zhou and Li [6] and Li and Ng [7].Secondly, the time-consistent strategy from the game theoretical perspective is also derived and we also present economics implications and sensitivity analysis for our results.
This paper is organized as follows.Section 2 formulates problem (MV) and gives the main results.Section 3 provides numerical analysis for our results.

Model and Main Results
This section starts with a filtered complete probability space (Ω, F, {F  } 0≤≤ , P), where  represents the fixed time horizon and F  stands for the information available at time .We recall that the dual risk model and the surplus process of the company are subject to the following stochastic process: The company invests its wealth into a Black-Scholes market consisting of a risk-free asset and a risky asset.Denote the amount of money invested in the risky asset at time  by (), and the allowance of short-selling and borrowing money implies () ∈ R. The price of the risk-free asset  0 () is modeled by and the price of the risky asset  1 () evolves according to the dynamics where () is a standard Brownian motion which is independent of ∑ () =1   , the appreciation rate  1 () and risk-free rate  0 () satisfy  1 () >  0 (), () is the volatility coefficient, and they are all bounded continuous functions.Denote the resulting surplus process after incorporating strategy  into (3) by   (), and   () is subject to the following stochastic equation: where () =  1 () −  0 ().Denote that  1,2 () = {(, ) | (, ⋅) is once continuously differentiable on [0, ] and (⋅, ) is twice continuously differentiable on R}.
In order to solve problem (MV), we firstly give the definition of admissible strategy for the dual risk process   ().
Denote the set of all admissible strategies over the time horizon [0, ] by .Our aim is to find a strategy which maximizes the expected terminal wealth minus the variance of the terminal wealth; in other words, we aim to solve problem (MV).In the next subsection we will give the optimal precommitted investment strategy and the time-consistent strategy.

The Optimal Precommitted Investment Strategy for Problem (MV).
This subsection puts forward an idea to solve problem (MV).
Firstly, problem (MV) can be transformed into a constrained problem.For  ∈ R, define the following problem with a constrained expectation of terminal wealth: Considering the terminal condition  0, (  ()) =  sufficiently, problem (CMV) equals the following problem: (, ) and () represent the value functions for problem (CMV) and problem (MV), respectively, and they have the following relationship () = sup ∈ (, ).
Secondly, problem (CMV) is solved by Lagrange technique.By introducing a Lagrange multiplier  := (, ) ∈ R, define a quadratic problem without the constrained expectation of the terminal wealth:

𝑉(𝑥, 𝑑, 𝑦) represents the value function for problem (BM).
The duality theory implies that the value function for problem (CMV) satisfies that (, ) = inf ∈R (, , ); then we have We will take three steps to derive the precommitted investment strategy for problem (MV).Based on the discussion in Appendix A, we can derive the precommitted investment strategy and the value function.
Theorem 2. For problem (MV), the optimal precommitted investment strategy is given by

) and the value function is given by
2 .
(13) Furthermore, the efficient frontier of problem (MV) at initial state (0, ) is given by Remark 3. Our method is different from the embedding technique proposed by Zhou and Li [6].Problem (MV) can be embedded into a class of auxiliary stochastic linearquadratic (LQ) problems, and the precommitted strategy for problem (MV) was derived by solving the LQ problems.Correspondingly, we take three steps to derive the precommitted strategy.Compared with the embedding method, our method is simple but easy to implement for solving problem (MV).

The Time-Consistent Investment Strategy for Problem (MV).
This subsection provides the time-consistent investment strategy for problem (MV).Firstly, define a dynamic problem According to the theory stated in Björk and Murgoci [8], we can convert this time-inconsistent problem into a timeconsistent problem.The equilibrium strategy is defined by the same way in Björk and Murgoci [8] or Zeng et al. [11].
Definition 4 (equilibrium strategy).For any fixed chosen initial state (, ) ∈  := [0, ] × , consider an admissible strategy  * (, ).Choose two fixed real numbers l > 0 and  > 0 and define the following strategy: then  * (, ) is called an equilibrium strategy and the corresponding equilibrium value function (, ) for problem (MV) is given by By Definition 4, we know that the equilibrium strategy is time-consistent.So the equilibrium strategy  * () and the equilibrium value function (, ) are called optimal timeconsistent strategy and optimal equilibrium value function for problem (MV), respectively.It is easy to see that the equilibrium value function  * () for problem (MV) satisfies  * () = (0, ).Based on the discussion in Appendix B, the time-consistent investment strategy and the equilibrium value function for problem (MV) are given by the following theorem.
Theorem 5.For the dual model, the optimal time-consistent strategy  * () is given by and the optimal equilibrium value function is given by Furthermore, the efficient frontier under time-consistent strategy is given by the following equation: Remark 6.The time-consistent strategy is not affected by the initial information and the current wealth which is different from the precommitted strategy.While the timeconsistent strategy is time deterministic, the precommitted strategy is stochastically dependent on the current wealth.We exploit Monte Carlo methods to simulate the precommitted investment strategy.We compare the average of 1000 tracks of the precommmitted investment strategy with the timeconsistent strategy.Figure 1(a) shows that the time-consistent investment strategy is smaller than the average of the precommitted investment strategy; that is to say, the company should invest more money into the risky asset under the precommitted investment strategy.
Remark 7. The optimal value function () and the equilibrium value function  * () are not the same for problem (MV).From ( 13) and (20), it is not hard to calculate that () >  * () which is illustrated by Figure 1(b).It follows from ( 14) and (21) that the efficient frontiers for problem (MV) are not straight lines but hyperbolas in mean-standard variance plane and that the expectation of terminal wealth under the time-consistent strategy is never bigger than the one under the precommitted strategy for the same variance of terminal wealth which is illustrated by Figure 1(c).
Remark 8. From Remark 7, we can see that the value function () is larger than the equilibrium value function  * () and that the efficient frontier under the time-consistent equilibrium strategy is never above the efficient frontier under the precommitted strategy.But the conclusion that the precommitted strategy is prior to the time-consistent equilibrium strategy is not right, for the precommitted strategy is a global optimal strategy only at  = 0 and the timeconsistent strategy is a suboptimal strategy for all  ≥ 0.

Numerical Analysis
This section provides some numerical examples to illustrate the effect of parameters on the optimal strategies and the corresponding value functions.For convenience but without loss of generality, all the parameters involved are constants.

Analysis of the Precommitted Strategy and the Corresponding Value
Function.This subsection will provide numerical  examples to show how the parameters effect the precommitted strategy and the value function.
Because the precommitted investment strategy depends on the current wealth, we will investigate the effect of all the parameters for the precommitted strategy in the same sample trajectory by stochastic simulations.In order to model the trajectory, we assume that {()} ≥0 is a Poisson process with intensity  = 1 and the profit or the income   ( = 1, 2, 3, . ..) is exponentially distributed with mean 1.From (12), we can see that the precommitted investment strategy increases when the current wealth decreases; namely, if the current wealth is smaller, the company should invest more money in the risky asset.Figure 1 shows how the coefficients involved impact on the optimal precommitted investment strategy.From Figure 2, we can conclude the following: (1) the precommitted investment strategy is decreasing with respect to  and  which shows that the more the company dislikes risk or the larger the market's risk is, the less amount the company invests in the risky asset; (2) the precommitted investment strategy is also decreasing with respect to  which shows that the more the company's expenditure is, the less amount the company invests in the risky asset; (3) the precommitted investment strategy has a more complex relation with  1 and  0 because the increase of  1 or  0  can increase the deterministic part and the stochastic part (current wealth) of the precommited strategy which results in the uncertainty of their difference.
Secondly, Figure 3 shows how the coefficients involved impact on the value function.Figure 3(a) illustrates that the value function is decreasing with respect to , namely, the larger risk aversion the company has, the smaller the value function is; Figure 3(b) reveals when the risk-free rate is small enough, the value function decreases and when the risk-free rate is close to the the appreciation rate, the value function increases; Figure 3(c) illustrates that the value function is increasing with respect to  1 , namely, the bigger the appreciation rate is, the bigger the value function is; Figure 3(d   with respect to , namely, the bigger the volatility of the market's risky asset is, the smaller the value function is.Recalling  =  − , Figure 3(e) illustrates that the value function is increasing with respect to , namely, the larger the expectation of the positive income is or the smaller the expense rate of the company is or the bigger the intensity of the jumps of the profit is, the bigger the value function becomes; Figure 3(f) shows the value function is increasing with respect to   , namely, the smaller the second moment of the positive income is, the bigger the value function becomes.

Analysis of the Time-Consistent Strategy and the
Equilibrium Value Function.This subsection will work on numerical analysis of the time-consistent strategy and the equilibrium value function.
Firstly, we will show how the coefficients involved impact on the time-consistent investment strategy.From (19) we can see that the time-consistent investment strategy is independent of the current wealth.Figure 4(a) illustrates that the time-consistent investment strategy is decreasing with respect to , namely, the more the company dislikes risk, the less amount the company invests in the risky asset; Figure 4(b) reveals that the time-consistent investment strategy is decreasing with respect to  0 , namely, the smaller the risk-free rate is, the more amount the company invests in the risky asset; Figure 4(c) reveals that the time-consistent investment strategy is increasing with respect to  1 , namely,  when the appreciation rate  1 increases, the company should invest more money in the risky asset; Figure 4(d) tells that the time-consistent investment strategy is decreasing with respect to , namely, when volatility of the risky asset increases, the company should invest more money in the riskfree asset.
Secondly, Figure 5 shows how the coefficients involved impact on the equilibrium value function.The parameters ,  1 , , , and   have the similar effect on the equilibrium value function as their effect on the value function with precommitment discussed in Section 3.1.Furthermore, the equilibrium value function is increasing with respect to the risk-free rate.

Conclusion
In this paper, optimal investment strategies for a dual risk model are explored under mean-variance criterion.We assume that a company can invest into a finance market which consists of a risk-free asset and a risky asset.We have derived the optimal precommitted strategy and the time-consistent equilibrium strategy for problem (MV).In the end, numerical analysis is given for optimal investment strategies and the value functions.From the comparisons on the value functions and efficient frontiers, it seems to be that the precommitted strategy is better than the timeconsistent strategy.Unfortunately, the precommitted strategy is not time-consistent.Thus these investment strategies have different advantages.These strategies are all important for the company.
Step 3. Finally problem (MV) is solved by virtue of () = sup ∈ (, ).Differentiating (, ) at , we can conclude that From the extreme value theory, the optimal expected terminal wealth  * does exist. * can be derived by (, )/ = 0 and is given by the following equation .

B.
This section will prove Theorem 5. Due to the relationship of the equilibrium value function for problem (MV) and problem (MV), we firstly derive the time-consistent investment strategy for problem (MV).From standard arguments as in Björk and Murgoci [8], we obtain the extended HJB equations for problem (MV).
strategy (c) The comparison of different efficient frontiers
The effect of  0 on the optimal precommitted strategy The effect of  1 on the optimal precommitted strategy The effect of  on the optimal precommitted strategy The effect of  on the optimal precommitted strategy

Figure 2 :
Figure 2: The effect of parameters on the optimal precommitted strategy.
) reveals that the value function is decreasing The effect of  on the optimal value function The effect of  0 on the optimal value function 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0The effect of  1 on the optimal value function The effect of  on the optimal value function The effect of  on the optimal value function The effect of   on the optimal value function

Figure 3 :
Figure3: The effect of parameters on the optimal value function.

Figure 4 :
Figure4: The effect of parameters on the optimal time-consistent strategy.
The effect of  on the optimal equilibrium value function The effect of  0 on the optimal equilibrium value function The effect of  1 on the optimal equilibrium value function The effect of  on the optimal equilibrium value function The effect of  on the optimal equilibrium value function The effect of   on the optimal equilibrium value function

Figure 5 :
Figure5: The effect of parameters on the optimal equilibrium value function.