Multi-time state mean-variance model in continuous time

In the continuous time mean-variance model, we want to minimize the variance (risk) of the investment portfolio with a given mean at terminal time. However, the investor can stop the investment plan at any time before the terminal time. To solve this kind of problem, we consider to minimize the variances of the investment portfolio at multi-time state. The advantage of this multi-time state mean-variance model is that we can minimize the risk of the investment portfolio along the investment period. To obtain the optimal strategy of the multi-time state mean-variance model, we introduce a sequence of Riccati equations which are connected by a jump boundary condition. Based on this sequence Riccati equations, we establish the relationship between the means and variances of this multi-time state mean-variance model. Furthermore, we use an example to verify that minimizing the variances of the multi-time state can affect the average of Maximum-Drawdown of the investment portfolio.


Introduction
In the financial market, standard deviation or variance is always used to measure the risk of a given risky asset. Variance is an important metric in the investment, which can help an investor determine whether it will be profitable. Investors can achieve the best asset allocation in a portfolio by analyzing the variance of the returns among the assets in the portfolio. Semi-deviation is proposed to capture the downside risk of an investment which focuses only on negative price fluctuations. Conversely, variance treats all deviations from the mean as the same regardless of their direction. Thus, in this paper, we use variance to measure the risk of an investment.
To balance the return (mean) and risk (variance) in a single-period portfolio selection model, references [19,20] proposed the mean-variance model. Since then, many related works focused on these topics. Under some mild assumptions, [22] solved the single-period problem analytically. Reference [24] studied a mean-variance model in which a single stock with a constant risk-free rate was introduced. Dynamic asset allocation in a meanvariance framework was studied by [1]. Reference [16] embedded the discrete-time multi-period mean-variance problem within a multi-objective optimization framework and obtained an optimal strategy. By extending the embedding technique introduced in [16] and applying the results from the stochastic linear quadratic control in the continuous time case, [33] investigated an optimal pair for the continuous-time mean-variance problem. Further results in the mean-variance problem included those with bankruptcy prohibition, transaction costs, and random parameters in complete and incomplete markets, see [5,9,17,18,27]. The aforementioned precommitted strategies in the continuous time multi-period, differed from those of the single-period case. For further details, see [15].
The optimal strategy in the multi-period mean-variance framework is a pre-committed strategy that strengths the premise that the investor needs to follow the strategy given at the initial time. When the optimal strategy is not time-consistent, the investor may not obey this strategy in the following investment period. A game theoretic approach to study the time-inconsistency of the mean-variance model was developed in [2]. Reference [11] considered a general time-inconsistent stochastic linear-quadratic control problem with an equilibrium instead of optimal control. Based on a local mean-variance efficiency, [8] developed a time-consistent formulation of the mean-variance portfolio selection problem in a general semi-martingale setting. Subsequently, the meanvariance portfolio selection problem in a complete market with unbounded random coefficients was studied in [25]. The large population stochastic dynamic games and the Nash certainty equivalence based on control laws were established in [12]. Based on the adjoint equation approach, [4] studied the linear-quadratic mean field games (see [3]). The mean-variance problem with state dependent risk aversion was considered in [6]. A general framework to study the time-inconsistent stochastic control in the continuous time framework was established in [7]. In addition, [23] investigated a robust time-inconsistent stochastic control problem, and applied it to the robust dynamic mean-variance portfolio selection problem. In the incomplete markets with stochastic volatility, [28] considered the open-loop control framework for time-consistent mean-variance portfolio problems. A dynamic mean-variance analysis for log returns within the game-theoretic approach was proposed in [10]. Further, [13] introduced a "dynamic utility" under which the original time-inconsistent problem became a time-consistent one, and investigated possible approaches to study a general time-inconsistent optimization problem. Reference [14] used a time-consistent dynamic convex risk measure to evaluate the risk of a portfolio and showed that the dynamic mean-risk problem satisfies a set-valued Bellman's principle.
In the financial market, for a given terminal time T , Y π (T ) represents a portfolio asset with strategy π(·), while E[Y π (T )] and Var(Y π (T )) = E Y π (T ) − E[Y π (T )] 2 represent the mean and variance of Y π (T ), respectively.
In the classical mean-variance model, we want to minimize the variance of the portfolio asset Var(Y π (T )) for a given mean E[Y π (T )] = L, where L is a constant. The investor can stop the investment plan at an uncertain horizon time τ before the terminal time T , where τ ≤ T . Therefore, there are many related works on the meanvariance portfolio model with an uncertain horizon time. Reference [21] considered static mean-variance analysis with an uncertain time horizon. Moreover, [31] studied the mean-variance model of a multi-period asset-liability management problem under uncertain exit time; see [26,30,32] for additional studies in this vein. However, in the literature of mean-variance model under uncertain or random exit time, authors always suppose that the uncertain horizon time τ satisfies a distribution (or a conditional distribution) and investigate the related mean-variance model at time τ . However, in general, we do not know the information of τ at initial time t 0 = 0. Given a probability space (Ω, F, P ), notice that for a given partition 0 = t 0 < t 1 < · · · < t N = T of interval [0, T ] and ω ∈ Ω, there exists To reduce the variance of the portfolio asset Y π (·) at τ ∈ (0, T ], we consider minimizing the variances of the portfolio asset at multi-time state (Y π (t 1 ), Y π (t 2 ), . . . , Y π (t N )) with constraint on means of multi-time state (Y π (t 1 ), Y π (t 2 ), . . . , Y π (t N )). Therefore, we introduce the following multi-time state mean-variance model: where α i ∈ [0, 1] and N i=1 α i = 1, with constraints on the multi-time state means, In this multi-time state mean-variance model, we can minimize the risk of the investment portfolio within the multi-time (t 1 , t 2 , . . . , t N ). Note that the multi-time state (Y π (t 1 ), Y π (t 2 ), . . . , Y π (t N )) of the investment portfolio can affect the value of each other, and we cannot solve the multi-time state mean-variance model via one classical Riccati equation directly. To obtain the optimal strategy of the multi-time state mean-variance model, we introduce a sequence of Riccati equations, which are connected by jump boundary conditions (see Eqs. (3.5) and (3.6)). Based on this sequence of Riccati equations, we investigate an optimal strategy (see Thm. 3.2) and establish the relationships between the means and variances of this multi-time state mean-variance model (see Lem. 3.4).
The Maximum-Drawdown of the asset Y π (·) is an important index to evaluate a strategy in the investment portfolio model, where the Maximum-Drawdown of the asset Y π (·) is defined in the interval [0, h], h ≤ T , by Based on the simulation results of the multi-time state mean-variance model (see subsection 4.2), we can see that the constrained condition (1.2) can affect the average of MD h Y π of the portfolio asset Y π (·) (see Fig. 3). This study is most closely related to the study of [29], in which the author established the necessary and sufficient conditions for stochastic differential systems with multi-time state cost functional.
The remainder of this paper is organized as follows. In Section 2, we formulate the multi-time state meanvariance model. Subsequently, in Section 3, we investigate an optimal strategy and establish the relationships between multi-time state means and variances for the proposed model. In Section 4, based on the main results of Section 3, we compare the multi-time state mean-variance model with the classical mean-variance model. Finally, we conclude the paper in Section 5.
The set of admissible strategies π(·) is defined as: is the set of all square integrable measurable R n valued {F t } t≥0 adaptive processes. If there exists a strategy π * (·) ∈ A that yields the minimum value of the cost functional (2.2), then we say that the multi-time state mean-variance model (2.2) is solved.

Optimal strategy
In this section, we investigate an optimal strategy π(·) for the problem defined in (2.2), with a constraint on the multi-time state mean (2.3). Here, we describe how to construct an optimal strategy for (2.2) with constrained condition (2.3).
Similar to [33], we introduce the following multi-time state mean-variance problem: minimizing the cost functional, where µ i = κ i α i , 1 ≤ i ≤ N , and κ i = 0 is used to combine the mean and variance at time t i . To solve the cost functional (3.1), we employ the following model: Note that, when α i = 0 at some 1 ≤ i ≤ N , we need to delete the i-th term. For notation simplicity, we consider µ i = 0 and λ i ∈ R, i = 1, 2, . . . , N . For the given µ i , i = 1, 2, . . . , N , we suppose π * (·) is an optimal strategy of cost functional (3.1). Based on Theorem 3.1 of [33], taking . . , N , we can show that π * (·) is an optimal strategy of cost functional (3.2). Conversely, let π * (·) be an optimal strategy of cost functional (3.2). When . . , N , we can show that π * (·) is an optimal strategy of cost functional (3.1).
Then, we embed cost functional (3.1) into (3.2). Furthermore, (3.1) and (3.2) admit the same optimal strategy π * (·), when . It should be noted that, we cannot solve the cost functional (3.2) by applying the embedding technique of [33] for the multi-time state mean-variance model via the classical Riccati equation directly, as the value Y π (t i ) can affect Y π (t i+1 ), for i = 0, 1, . . . , N − 1. Denoting Thus, the cost functional (3.2) is equivalent to Now, we construct a sequence of Riccati equations that are connected by jump boundary conditions, in which the jump boundary condition can offset the interaction of Y π (t i+1 ) and Y π (t i ), for i = 0, 1, . . . , N − 1. We first introduce a sequence of deterministic Riccati equations: and related equations, which is used to obtain the following results.
Theorem 3.2. Let Assumptions H 1 and H 2 hold, there exists an optimal strategy π * (·) for cost functional (3.3), where the optimal strategy π * (·) is given as follows: Thus, (3.7) Applying Itô formula to z π i (t)g i (t), it follows that (3.8) We add the equations (3.7) and (3.8) together and integrate from t i−1 to t i . The left terms of equations (3.7) and (3.8) show that where the second equality is derived by the following results, where z π 0 (t 0 ) = y, ρ 0 = 0. The right terms of equations (3.7) and (3.8) show that Thus, we have (3.9) Adding i on both sides of equation (3.9) from 1 to N , it follows that 10) and thus Based on the representation of E N i=1 µ i 2 z π i (t i ) 2 , we can obtain an optimal strategy π * (·) for J 4 (π(·)), for ).
E[Y * (·)] and E[Y * (·) 2 ] satisfy the following linear ordinary differential equations: and (3.14) In the following, we investigate the efficient frontier of the multi-time state mean-variance Var(Y * (t i )) and E[Y * (t i )], i = 1, 2, . . . , N . Lemma 3.4. Let Assumptions H 1 and H 2 hold, the relationship of Var(Y * (t i )) and E[Y * (t i )] is given as follows:
Remark 3.5. Specially, for i = 1, one obtains which is the same as the efficient frontier in [33].

19)
such that the optimal strategy π * (·) of cost functional (3.3) is an optimal strategy of cost functional (3.2).
Proof. By Theorem 3.2, an optimal strategy of model (3.2) can be solved by (3.3), let Note that E[Y * (t i )] depends on λ * i . To solve the parameters λ * i , i = 1, 2, . . . , N , by equation (3.16), we first consider the case i = N , and Thus, we have Based on the representation of λ * N , by equation (3.21), we have (3.24) It follows that Note that the coefficient of λ * N −1 is which indicates that there exists a unique solution for λ * N −1 ,  Similar to the case i = N − 1, we can solve λ * i , µ i , i = 1, 2, . . . , N − 1 step by step from N − 1 to 1, and (3.28) Therefore, the optimal strategy π * (·) of cost functional (3.3) is an optimal strategy of cost functional (3.2). This completes the proof.
By equality (4.14), we have Example 4.3. Let b, r, σ be constants, and T = 2, y = 1, b = 2r, β = r, L 1 = 2e r , L 2 = 3e 2r . Now, we consider two special cases of the model in this part: N=1,2. For the case N = 2, t 1 = 1, and t 2 = 2, by (4.10), (4.11), and (4.12). We have and (4.16) and the variances of Y * (·) at t 1 = 1, t 2 = 2 are given as follows: Var(Y * (1)) = e 2r e r − 1 ; Var(Y * (2)) = e 3r + e 4r e r − 1 . (4.17) For the case N = 1, which is the classical mean-variance model, the explicit formulas of the optimal asset and related parameters are and the related optimal strategy is The mean E[Y # (·)] and variance Var(Y # (·)) satisfy .  As we have shown in this example, we can use the multi-time state mean-variance model to control the average return of the asset before terminal time T . In particular, when N = 2, we want to minimize the variances of the asset at time t 1 and t 2 with the given returns L 1 , L 2 at time t 1 and t 2 . Thus, the advantages of the multi-time state mean-variance model are that we can adjust the return at t 1 and minimize the variance at time t 1 .

Simulation analysis
Let r = 0.04, b = 0.12, σ = 0.2, β = 0.16, we show the simulation results of the case N = 2, and case N = 1, where case N = 1 is same with the classical continuous time mean-variance model.
In Figure 1, we take L 2 = e 5r , L 1 = e 2.1r , which satisfies conditions (4.9). The expectations of Y * (·) and Y # (·) are given as follows, respectively,   These results show that if we want to minimize the variances of the wealth at times t 1 = 1, t 2 = 2 together, the means of the investment portfolio may be smaller than that of the classical mean-variance model in continuous time.
In Figure 2, we plot the values Y * (·) and Y # (·) in pathwise. The left shows the pathwise of the function Y * (·) along with E[Y * (·)], while the right shows that of Y # (·). We can see that the variance of Y * (1) is bigger than that of Y * (1), and the variance of Y * (2) is almost the same as that of Y * (2). These phenomena verify the results of Corollary 4.2. In addition, in Figure 1, we can see that E[Y * (t)] < E[Y # (t)], 0 < t ≤ 1, while Figure 2 shows that the variance of Y * (·) is smaller than that of Y # (·) before time 1.
In Figure 3

Conclusion
For given 0 = t 0 < t 1 < · · · < t N = T , to reduce the variance of the mean-variance model at the multitime state (Y π (t 1 ), . . . , Y π (t N )), we propose a multi-time state mean-variance model with a constraint on the multi-time state mean value. In the proposed model, we solve the multi-time state mean-variance model by introducing a sequence of Riccati equations.
The main results of this study are as follows: -We can use the multi-time state mean-variance model to manage the risk of the investment portfolio along the multi-time 0 = t 0 < t 1 < · · · < t N = T . -A sequence of Riccati equations that are connected by jump boundary conditions are introduced, on which we obtain an optimal strategy for the multi-time state mean-variance model.