Mean-Variance Portfolio Selection with Margin Requirements

Abstract

We study the continuous-time mean-variance portfolio selection problem in the situation when investors must pay margin for short selling. The problem is essentially a nonlinear stochastic optimal control problem because the coefficients of positive and negative parts of control variables are different. We can not apply the results of stochastic linearquadratic (LQ) problem. Also the solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation is not smooth. Li et al. (2002) studied the case when short selling is prohibited; therefore they only need to consider the positive part of control variables, whereas we need to handle both the positive part and the negative part of control variables. The main difficulty is that the positive part and the negative part are not independent. The previous results are not directly applicable. By decomposing the problem into several subproblems we figure out the solutions of HJB equation in two disjoint regions and then prove it is the viscosity solution of HJB equation. Finally we formulate solution of optimal portfolio and the efficient frontier. We also present two examples showing how different margin rates affect the optimal solutions and the efficient frontier.

1. Introduction

Modern portfolio theory was introduced by Markowitz in 1952 [1, 2]. It is a theory of finance which attempts to minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. In the theory variance of portfolio return was chosen as measure of the risk by Markowitz. Markowitz also established concept of the efficient frontier of the optimal portfolio: it is a curve showing the relation of the best possible expected level of return with respect to its level of risk (the standard deviation of the portfolio's return). This theory has been widely accepted both in financial industry and academy. There are lots of extensions and applications in the discrete time models [3]. Relatively there are less discussions of the mean-variance portfolio problem about continuous-time model [4–6]. In 2000, Zhou and Li [5] solved the continuous time mean-variance problem without short selling constraint by using the stochastic linear quadratic optimal control theory. Li et al. [6] in 2002 considered mean-variance portfolio selection with short selling prohibiting condition by solving HJB equation.

In financial market, the potential loss on a short sale can be huge when the price of the security goes up, therefore in practice the short seller will be required to post margin or collateral to cover possible losses. In this paper, we consider the situation that short selling is allowed and deposit of certain percentage margin according to the shorting is needed to avoid loss by default of short seller. We also want to investigate how margin rate 𝜆 affect the optimal portfolio. In the extreme cases, when 𝜆=0, it is the case short selling is free of margin and in another case 𝜆=∞ means that short selling is not allowed.

There is some literature [7–9] that discussed the margin requirements for portfolio selections. For example, Cuoco and Liu [7] examined the optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements. They established existence of optimal policies by using martingale and duality techniques [10] under general assumptions on the securities price process and the investors preferences. However, the conditions for utility functions in [7] fail for mean-variance problem.

We use the model from [7], formulate a closed form of the optimal policies. The main difficulty lies when we use the approach of [6]: the positive part and the negative part of control variables 𝜋 have different coefficients; if we define new control variables (𝜋⁺,𝜋⁻) then the new volatility matrix is singular and moreover 𝜋⁺,𝜋⁻ are not independent therefore the key lemma for optimization is not applied in our case. We carefully decompose the problem into several subproblems and figure out the solutions of HJB equation in two regions and then prove it is the viscosity solution of HJB equation. Finally we formulate solution of optimal portfolio and the efficient frontier.

2. Problem Formulation

First, we prescribe a few notations. Let (Ω,ℱ,𝑃) be a complete probability space and {𝑊(𝑡),0≤𝑡≤𝑇} a standard d-dimensional Brownian motion defined on the probability space {ℱ^𝑊_𝑡}_𝑡≥0 the natural filtration generated by 𝑊(⋅). 𝒩 denotes the set composed of all the P null set and its subsets; define the augmented filtration as ℱ_𝑡∶=𝜎(ℱ^𝑊_𝑡⋃𝒩), for all𝑡∈[0,𝑇], where ℱ^𝑊_𝑡=𝜎(𝑊_𝑠∣𝑠∈[0,𝑡]), then {ℱ_𝑡}_𝑡≥0 is complete and continuous filtration.

We define 𝑎⁺∶=max{0,𝑎}, 𝑎⁻∶=max{0,−𝑎} for the positive part and negative part of a real number 𝑎. 𝐑^𝑛 represents n-dimensional real Euclidean space. 𝐑^𝑛₊∶={𝑥∈𝐑^𝑛,𝑥_𝑖≥0,𝑖=1,…,𝑛} is nonnegative subset of n-dimensional real Euclidean space. 𝐿²_ℱ(0,𝑇;𝐑^𝑛)∶={𝑓(⋅)∣𝑓∶[0,𝑇]×Ω→𝐑^𝑛,𝑓(𝑡) is ℱ_𝑡 measurable for all 𝑡∈[0,𝑇], and 𝐸∫^𝑇₀‖𝑓(𝑡)‖²𝑑𝑡<+∞}.

We consider the Merton's market model. There are 𝑑+1 securities, trading continuously in finite time horizon [0,𝑇]. Assuming the interest rate is 𝑟>0, the time 𝑡 price of the risk-free asset satisfies 𝑑𝑃₀(𝑡)=𝑟(𝑡)𝑃₀(𝑡)𝑑𝑡,𝑡∈[0,𝑇] (1) with the initial price 𝑃₀(0)=𝑝₀>0. The remaining 𝑑 risky assets prices at time 𝑡 are, respectively, 𝑃₁(𝑡),…,𝑃_𝑑(𝑡), which satisfy the following stochastic differential equations: 𝑑𝑃_𝑖(𝑡)=𝑃_𝑖(𝑡)𝑏_𝑖(𝑡)𝑑𝑡+^𝑑_𝑗=1𝜎_𝑖𝑗(𝑡)𝑑𝑊_𝑗(𝑡)[],𝑃,𝑡∈0,𝑇_𝑖(0)=𝑝_𝑖>0, (2) where 𝑏(𝑡)=(𝑏₁(𝑡),…,𝑏_𝑑(𝑡))^𝑇 is the appreciation rate of the risky assets and 𝜎(𝑡)=(𝜎_𝑖𝑗(𝑡))_𝑑×𝑑 is the volatility matrix.

Throughout the paper, we assume the following.(A1) Function 𝑟(𝑡),𝑏(𝑡),𝜎(𝑡) are deterministic and piecewise continuous on [0,𝑇]. (A2) Non-degenerate Condition: there exists a constant 𝛿>0, s.t. 𝜎(𝑡)𝜎(𝑡)^𝑇≥𝛿𝐼,∀𝑡∈[0,𝑇]. (3)

We also define the relative risk coefficient 𝜃(𝑡)∶=𝜎(𝑡)⁻¹(𝑡)[𝑏(𝑡)−𝑟(𝑡)𝟏],∀𝑡∈[0,𝑇], (4) where 𝟏=(1,…,1)^𝑇.

Let 𝑣>0 be initial wealth of the investor. Suppose 𝜋_𝑖(𝑡),𝑖=0,1,…,𝑑 is time 𝑡 total wealth of the investor on 𝑖th asset, 𝜋(𝑡)∶=(𝜋₁(𝑡),…,𝜋_𝑛(𝑡))^𝑇 is time 𝑡 wealth vector of risky assets. Then time 𝑡 total wealth of the investor is 𝑋(𝑡)=∑^𝑑_𝑖=0𝜋_𝑖(𝑡)=𝜋₀(𝑡)+𝟏^𝑇𝜋(𝑡). Suppose the investor is self-financing, then 𝑑𝑋(𝑡)=𝑟(𝑡)𝑋(𝑡)+[𝑏(𝑡)−𝑟(𝑡)𝟏]^𝑇𝜋(𝑡)−𝜆𝑟(𝑡)𝟏^𝑇𝜋⁻(𝑡)𝑑𝑡+𝜋^𝑇(𝑡)𝜎(𝑡)(𝑡),[],𝑋(0)𝑑𝑊𝑡∈0,𝑇=𝑣, (5) where 𝜆≥0 is the margin rate.

Note. When 𝜆=0, there is no margin requirement for short selling and it has been discussed in [5].

In general, the terminal wealth 𝑋(𝑇) of the investor is a random variable. In the mean-variance problem, people consider 𝜇=𝐸[𝑋(𝑇)] as his return whereas variance of 𝑋(𝑇) as his risk. Therefore mean-variance problem is to minimize the risk of the investment for fixed expected return 𝜇=𝐸[𝑋(𝑇)]: var[𝑋(𝑇)]=𝐸[𝑋(𝑇)−𝐸[𝑋(𝑇)]]²=𝐸𝑋(𝑇)−𝜇². (6)

The portfolio selection problem can be formulated as (⋅)[𝑋(𝑇𝐸[𝑋(𝑇𝜋(⋅)min𝐽𝜇,𝜋=var)]s.t.)]=𝜇,∈𝐿²_ℱ0,𝑇,𝐑^𝑑,𝑋(⋅)satisfies(5), (7) where the expected return is chosen as 𝜇≥𝜇₀∶=𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}. If the investor puts all the money in the risk-free asset, his return will be 𝜇₀ with zero risk. Since the investor will always choose high return with low risk, therefore the lowest expected return must be 𝜇₀.

For fixed 𝜇≥𝜇₀, the optimal solution 𝜋^∗(𝜇) is called efficient strategy. Corresponding minimal variance is 𝑉(𝜇)=min𝐽(𝜇,𝜋(⋅)) and (𝑉(𝜇),𝜇) is called efficient point. When 𝜇 runs all over [𝜇₀,∞), the set constructed by all the efficient points in variance-mean plane is called the efficient frontier.

3. A Class of Stochastic Control Problem

The problem mentioned in the previous section is a special class of stochastic nonlinear quadratic control problem. As the state equation is nonlinear on control variables 𝜋, the results of LQ problem can not apply. In this part, we will solve this control problem.

Considering one dimensional system 𝑑𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵₁(𝑡)𝜋(𝑡)−𝐵₀(𝑡)𝜋⁻(𝑡)+𝑓(𝑡)𝑑𝑡+𝜋^𝑇(𝑡)𝐷^𝑇₁(𝑡)(𝑡),𝑥(𝑠)𝑑𝑊=𝑦∈𝐑, (8) where 0≤𝑠≤𝑇, 𝐴(𝑡)∈𝐑₊, 𝐵^𝑇₀(𝑡), 𝐵^𝑇₁(𝑡)∈𝐑^𝑑₊, 𝐷₁(𝑡)∈𝐑^𝑑×𝑑, we assume matrix 𝐷₁(𝑡) is non-degenerate. These coefficients are all deterministic and piecewise continuous on [0,𝑇], therefore they are all bounded on [0,𝑇].

The class of admissible controls is the set 𝓊_𝑠=𝐿²_ℱ(𝑠,𝑇,𝐑^𝑑). For any given 𝜋(⋅)∈𝓊_𝑠, by the result of stochastic differential equations theory there is a unique solution 𝑥(⋅;𝑦,𝜋(⋅))∈𝐿²_ℱ(𝑠,𝑇,𝐑) of (8), and (𝑥(⋅),𝜋(⋅)) is called feasible pair. Our goal is to find admissible 𝜋(⋅)∈𝓊_𝑠 to minimize the following cost function 𝐽(⋅=1𝑠,𝑦;𝜋)2𝐸[𝑥(𝑇)]². (9)

We define the value function of (9) 𝑉𝑠,𝑦=inf_{𝜋(⋅)∈𝓊𝑠}𝐽𝑠,𝑦;𝜋(⋅). (10) Obviously, it is nonnegative.

3.1. HJB Equation

We will try to find the optimal value function 𝑉(𝑠,𝑦). According to stochastic optimal control theory [11], the value function is satisfying the following HJB equation: 𝑣_𝑡(𝑡,𝑥)+inf_{𝑢∈𝐑𝑑}𝑣_𝑥()𝑔()+1𝑡,𝑥𝑡,𝑥,𝑢2𝑣_𝑥𝑥(𝑡,𝑥)×𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢𝑣()=1=0,𝑇,𝑥2𝑥², (11) where 𝑔(𝑡,𝑥,𝑢)=𝐴(𝑡)𝑥(𝑡)+𝐵₁(𝑡)𝑢−𝐵₀(𝑡)𝑢⁻+𝑓(𝑡). Our idea is to find local solution firstly, then prove that it is the viscosity solution of HJB equation and finally construct the optimal control by using verification theorem.

In order to find solution of the HJB equation, it is clear that for fixed (𝑡,𝑥)∈[0,𝑇]×𝐑^𝑑 if 𝑣_𝑥𝑥(𝑡,𝑥)>0 (that will be proved later) we must solve min_{𝑢∈𝐑𝑑}12𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢+𝛼(𝑡,𝑥)𝐵₁(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻, (12) where 𝐵₂(𝑡)=𝐵₀(𝑡)+𝐵₁(𝑡), 𝛼(𝑡,𝑥)=𝑣_𝑥(𝑡,𝑥)/𝑣_𝑥𝑥(𝑡,𝑥). We need the following lemma [10].

Lemma 1. If 𝐷∈𝐑^𝑑×𝑑 and |𝐷|≠0 then there is a unique 𝑧∈𝐑^𝑑₊, s.t. 𝐷^𝑇−1𝑏+𝐷^𝑇−1𝑧²=min_{𝑧∈𝐑𝑑+}𝐷^𝑇−1𝑏+𝐷^𝑇−1𝑧². (13)
For nonnegative constant 𝛼, the unique minimizer of ℎ(𝑧)1∶=2𝑧^𝑇𝐷^𝑇𝐷𝑧−𝛼𝑧^𝑇𝑏, (14) over 𝑧∈𝐑^𝑑₊ is 𝛼𝜆, 𝜆=𝐷⁻¹(𝐷^𝑇)⁻¹𝑏̂, here 𝑏̂=𝑏+𝑧. Furthermore, 𝑧^𝑇𝜆=0 and ℎ𝛼𝜆1=−2𝛼²𝐷^𝑇−1𝑏̂². (15)

To solve the HJB equation (11), above lemma is not directly applicable. Now we introduce some transformation.

Note for any 𝑢∈𝐑^𝑑, there is (may be not unique) a matrix 𝐶=diag(𝑐₁,…,𝑐_𝑑) with 𝑐_𝑖=1 or −1 for 𝑖=1,…,𝑑, such that 𝑢=𝐶𝑤, 𝑤∈𝐑^𝑑₊,𝑢⁺=𝐶⁺𝑤,𝑢⁻=𝐶⁻𝑤. Then 12𝑢^𝑇𝐷^𝑇₁𝐷₁𝑢+𝛼𝐵₁𝑢⁺−𝐵₂𝑢⁻=12𝑤^𝑇𝐶^𝑇𝐷^𝑇₁𝐷₁𝐶𝑤+𝛼𝐵₁𝐶⁺−𝐵₂𝐶⁻𝑤. (16)

Define space 𝒟={𝐶=diag(𝑐₁,…,𝑐_𝑑) with 𝑐_𝑖=1 or −1 for 𝑖=1,…,𝑑}.

We have following important lemma.

Lemma 2. If 𝛼 is negative, there is at least one 𝑏₁∈𝐑^𝑑 and a minimizer 𝛼𝜆₁ of ℎ(𝑢)1∶=2𝑢^𝑇𝐷^𝑇₁𝐷₁𝑢+𝛼𝐵₁𝑢⁺−𝐵₂𝑢⁻ (17) over 𝐑^𝑑 and ℎ(𝛼𝜆₁)=−(1/2)𝛼²‖𝑏₁‖² for all 𝛼<0.
If 𝛼 is positive, there is at least one 𝑏₂∈𝐑^𝑑 and a minimizer 𝛼𝜆₂ of ℎ(𝑢) over 𝐑^𝑑 and ℎ(𝛼𝜆₂)=−(1/2)𝛼²‖𝑏₂‖² for all 𝛼>0.

Proof. In order to solve (12), we fixed a diagonal matrix 𝐶 with diagonals either 𝑐_𝑖=1 or −1 for 𝑖=1,2,…,𝑑. Let ℎ_𝐶(𝑤)1∶=2𝑤^𝑇𝐶^𝑇𝐷^𝑇₁𝐷₁𝐶𝑤+𝛼𝐵₁𝐶⁺−𝐵₂𝐶⁻𝑤. (18)
If 𝛼<0, take 𝐷=𝐷₁𝐶 and 𝑏=(𝐵₁𝐶⁺−𝐵₂𝐶⁻)^𝑇. Since 𝐷 is nonsingular, by Lemma 1, there exist vectors 𝑏₁, 𝜆₁∈𝐑^𝑑, such that 𝛼𝜆₁ is minimizer of ℎ_𝐶 and ℎ_𝐶(𝛼𝜆₁)=−(1/2)𝛼²‖𝑏₁‖².
Now because min_{𝑢∈𝐑𝑑}ℎ(𝑢)=min_𝐶∈𝒟min_{𝑤∈𝐑𝑑+}ℎ_𝐶(𝑤). (19) Note 𝒟 is a finite set, first part of lemma holds.
If 𝛼>0, take 𝐷=𝐷₁𝐶 and 𝑏=−(𝐵₁𝐶⁺−𝐵₂𝐶⁻)^𝑇. because 𝐷 is nonsingular, by Lemma 1 again, there exist vectors 𝑏₂,𝜆₂∈𝐑^𝑑, such that 𝛼𝜆₂ is minimizer of ℎ_𝐶 and ℎ_𝐶(𝛼𝜆₂)=(−1/2)𝛼²‖𝑏₂‖².
Because (19) is true for this case, we have the second part of lemma.

Note. (1) The minimizer may be not unique. In lemma 𝜆₁,𝜆₂ is dependent only on the sign of 𝛼.

(2) For 𝛼=0, it is obvious that there is unique minimizer 𝑢=0 and minℎ(𝑢)=0.

Following the approach of LQ problem, we assume the value function is quadratic: 𝑉()=1𝑡,𝑥2𝑃(𝑡)𝑥²+𝑔(𝑡)𝑥+𝑐(𝑡). (20)

Note the coefficients 𝑃(𝑡), 𝑔(𝑡), 𝑐(𝑡) may be different for 𝛼 positive and negative in Lemma 2.

Substituting (20) into HJB equation we have following ordinary differential equations (for 𝑖=1,2): 𝑃̇_𝑖(𝑡)=−2𝐴(𝑡)+𝑏_𝑖(𝑡)²𝑃_𝑖(𝑡),𝑃_𝑖(𝑇)𝑔̇=1,_𝑖(𝑡)=−𝐴(𝑡)+𝑏_𝑖(𝑡)²𝑔_𝑖(𝑡)−𝑓(𝑡)𝑃_𝑖(𝑡),𝑔_𝑖(𝑇)𝑐̇=0,_𝑖(𝑡)=−𝑓(𝑡)𝑔_𝑖(𝑡)+12𝑏_𝑖(𝑡)²𝑃_𝑖(𝑡)⁻¹𝑔_𝑖(𝑡)²,𝑐_𝑖(𝑇)=0. (21)(22)(23) Here 𝑏₁(𝑡), 𝑏₂(𝑡) are determined by Lemmas 1 and 2. If there are two solutions in some interval of [0,𝑇], we can always take same 𝐶 on that interval for it is piecewise constant. That makes our solution piecewise continuous.

Since 𝑉_𝑥𝑥(𝑡,𝑥)=𝑃_𝑖(𝑡)=𝑒^{∫𝑇𝑡(2𝐴(𝑠)−‖𝑏𝑖(𝑠)‖2)𝑑𝑠}>0, the 𝛼 in Lemma 2 equals 𝑉_𝑥(𝑡,𝑥)/𝑉_𝑥𝑥(𝑡,𝑥) having same sign as 𝑉_𝑥(𝑡,𝑥). By the lemmas, we know 𝑏₁(𝑡) and 𝑏₂(𝑡) are independent of 𝑥,𝑉_𝑥(𝑡,𝑥) and 𝑉_𝑥𝑥(𝑡,𝑥) except the sign of 𝑉_𝑥(𝑡,𝑥).

Defining 𝜂_𝑖(𝑡)∶=𝑔_𝑖(𝑡)/𝑃_𝑖(𝑡), 𝑖=1,2, from (21) and (22) we can get 𝜂̇_𝑖(𝑡)=𝑃_𝑖(𝑡)𝑔̇_𝑖(𝑡)−𝑃̇_𝑖(𝑡)𝑔_𝑖(𝑡)𝑃_𝑖(𝑡)²=𝐴(𝑡)𝑃_𝑖(𝑡)𝑔_𝑖(𝑡)−𝑓(𝑡)𝑃_𝑖(𝑡)²𝑃_𝑖(𝑡)²=𝐴(𝑡)𝜂_𝑖(𝑡)−𝑓(𝑡). (24) By the terminal constraint 𝜂_𝑖(𝑇)=0, noting the equations are same for 𝑖=1,2, we obtain 𝜂₁(𝑡)=𝜂₂(𝑡)=𝑒^{−∫𝑇𝑡𝐴(𝑠)𝑑𝑠𝑇}_𝑡𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧. (25) Therefore, for 𝑖=1,2, 𝑔_𝑖(𝑡)=𝑃_𝑖(𝑡)𝜂_𝑖(𝑡)=𝑒^{∫𝑇𝑡(𝐴(𝑠)−‖𝑏𝑖(𝑠)‖2)𝑑𝑠𝑇}_𝑡𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧. (26) Substituting these expressions into (23), we obtain for 𝑖=1,2𝑐_𝑖(𝑡)=^𝑇_𝑡𝑓(𝑣)−12𝑏_𝑖(𝑣)²𝑒^{−∫𝑇𝑣𝐴(𝑠)𝑑𝑠}×^𝑇_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧⋅𝑒^{∫𝑇𝑣(𝐴(𝑠)−‖𝑏𝑖(𝑠)‖2)𝑑𝑠𝑇}_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧𝑑𝑣. (27) Because ^𝑇_𝑡𝑓(𝑣)𝑒^{∫𝑇𝑣(𝐴(𝑠)−‖𝑏𝑖(𝑠)‖2)𝑑𝑠𝑇}_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}=𝑑𝑧𝑑𝑣_𝑇𝑡𝑒^{∫𝑇𝑣−‖𝑏𝑖(𝑠)‖2𝑑𝑠}𝑓(𝑣)𝑒^{∫𝑇𝑣𝐴(𝑠)𝑑𝑠}×^𝑇_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}1𝑑𝑧𝑑𝑣=−2^𝑇_𝑡𝑒^{∫𝑇𝑣−‖𝑏𝑖(𝑠)‖2𝑑𝑠}𝑑^𝑇_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧²=12𝑒^{∫𝑇𝑡−‖𝑏𝑖(𝑠)‖2𝑑𝑠𝑇}_𝑡𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧²+^𝑇_𝑡𝑏_𝑖(𝑣)²⋅𝑒^{∫𝑇𝑣−‖𝑏𝑖(𝑠)‖2𝑑𝑠}×^𝑇_𝑣𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧²𝑑𝑣. (28)

We have for 𝑖=1,2𝑐_𝑖(𝑡)=12𝑒^{∫𝑇𝑡−‖𝑏𝑖(𝑠)‖2𝑑𝑠𝑇}_𝑡𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧². (29)

3.2. Value Function and Optimal Control

Firstly, we will construct a viscosity solution for HJB equation. Let 𝜂(𝑡)∶=𝑒^{−∫𝑇𝑡𝐴(𝑠)𝑑𝑠𝑇}_𝑡𝑓(𝑧)𝑒^{∫𝑇𝑧𝐴(𝑠)𝑑𝑠}𝑑𝑧=𝜂₁(𝑡)=𝜂₂(𝑡). (30)

Define the spaces Γ₁)∈[](𝑡),Γ∶=(𝑡,𝑥0,𝑇×𝐑∣𝑥+𝜂<0₂)∈[](𝑡),Γ∶=(𝑡,𝑥0,𝑇×𝐑∣𝑥+𝜂>0₃∶=(𝑡,𝑥)∈[0,𝑇]×𝐑∣𝑥+𝜂(𝑡)=0. (31)

We denote 𝛼(𝑡,𝑥)∶=𝑥+𝜂(𝑡)=𝑥+𝜂₁(𝑡)=𝑥+𝜂₂(𝑡).

Now let us check that 𝑉(𝑡,𝑥)=(1/2)𝑃_𝑖(𝑡)𝑥²+𝑔_𝑖(𝑡)𝑥+𝑐_𝑖(𝑡) satisfies HJB equation on Γ_𝑖, for 𝑖=1,2. Considering (20) for 𝑖=1, we have 𝑉_𝑡()=1𝑡,𝑥2𝑃̇₁(𝑡)𝑥²+𝑔̇₁(𝑡)𝑥+𝑐₁̇(𝑡),𝑉_𝑥(𝑡,𝑥)=𝑃₁(𝑡)𝑥+𝑔₁(𝑡),𝑉_𝑥𝑥(𝑡,𝑥)=𝑃₁(𝑡). (32) Substitute into (11) for (𝑡,𝑥)∈Γ₁LHS=𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥(𝑡,𝑥)𝐴(𝑡)𝑥+𝑓(𝑡)+inf_{𝑢∈𝐑𝑑}𝑉_𝑥𝑥()1𝑡,𝑥2𝑢^𝑇𝐷^𝑇₁𝐷₁𝑢+𝑉_𝑥()×𝐵𝑡,𝑥₁(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₁(𝑡)𝑥²+𝑔̇₁(𝑡)𝑥+𝑐₁̇(𝑡)+𝑃₁(𝑡)𝑥+𝑔₁(𝑡×𝐴(𝑡)(𝑡)𝑥+𝑓)+inf_{𝑢∈𝐑𝑑}12𝑃₁(𝑡)𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢+𝑃₁(𝑡)𝑥+𝑔₁(𝑡×𝐵₁)(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₁(𝑡)+𝐴(𝑡)𝑃₁(𝑡)𝑥²+𝑔̇₁(𝑡)+𝐴(𝑡)𝑔₁(𝑡)+𝑓(𝑡)𝑃₁(𝑡𝑥+𝑐̇₁)(𝑡)+𝑓(𝑡)𝑔₁(𝑡)+𝑃₁(𝑡)×inf_{𝑢∈𝐑𝑑}12𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢+𝑥+𝜂₁(𝑡×𝐵₁)(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₁(𝑡)+𝐴(𝑡)𝑃₁(𝑡)𝑥²+𝑔̇₁(𝑡)+𝐴(𝑡)𝑔₁(𝑡)+𝑓(𝑡)𝑃₁(𝑡𝑥+𝑐̇₁)(𝑡)+𝑓(𝑡)𝑔₁(𝑡)−𝑃₁(𝑡)12𝑥+𝜂₁(𝑡)²×𝑏₁(𝑡,𝑥)²=12𝑃̇₁(𝑡)+2𝐴(𝑡)𝑃₁(𝑡)−𝑏₁(𝑡,𝑥)²𝑃₁(𝑡)𝑥²+𝑔̇₁(𝑡)+𝐴(𝑡)𝑔₁(𝑡)+𝑓(𝑡)𝑃₁(𝑡)−𝑃₁(𝑡)𝜂₁(𝑡)𝑏₁(𝑡,𝑥)²𝑥+𝑐₁̇(𝑡)+𝑓(𝑡)𝑔₁(𝑡)−12𝑃₁(𝑡)𝜂₁(𝑡)²𝑏₁(𝑡,𝑥)²=0. (33) Since 𝛼(𝑡,𝑥)=𝑥+𝜂₁(𝑡)<0 in Γ₁, substituting the minimizer by Lemma 2 we can see 𝑉(𝑡,𝑥) satisfies HJB equation in Γ₁.

By the same way, we can show 𝑉(𝑡,𝑥) also satisfies HJB equation in Γ₂. For 𝑖=2, we have 𝑉_𝑡()=1𝑡,𝑥2𝑃̇₂(𝑡)𝑥²+𝑔̇₂(𝑡)𝑥+𝑐₂̇(𝑡),𝑉_𝑥(𝑡,𝑥)=𝑃₂(𝑡)𝑥+𝑔₂(𝑡),𝑉_𝑥𝑥(𝑡,𝑥)=𝑃₂(𝑡). (34) Substitute into (11) for (𝑡,𝑥)∈Γ₂LHS=𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥(𝑡,𝑥)𝐴(𝑡)𝑥+𝑓(𝑡)+inf_{𝑢∈𝐑𝑑}𝑉_𝑥𝑥()1𝑡,𝑥2𝑢^𝑇𝐷^𝑇₁𝐷₁𝑢+𝑉_𝑥()×𝐵𝑡,𝑥₁(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₂(𝑡)𝑥²+𝑔̇₂(𝑡)𝑥+𝑐₂̇(𝑡+𝑃₂)(𝑡)𝑥+𝑔₂(𝑡)𝐴(𝑡)𝑥+𝑓(𝑡)+inf_{𝑢∈𝐑𝑑}12𝑃₂(𝑡)𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢+𝑃₂(𝑡)𝑥+𝑔₂(𝑡×𝐵₁)(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₂(𝑡)+𝐴(𝑡)𝑃₂(𝑡)𝑥²+𝑔̇₂(𝑡)+𝐴(𝑡)𝑔₂(𝑡)+𝑓(𝑡)𝑃₂(𝑡𝑥+𝑐̇₂)(𝑡)+𝑓(𝑡)𝑔₂(𝑡)+𝑃₂(𝑡)×inf_{𝑢∈𝐑𝑑}12𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢+𝑥+𝜂₂(𝑡×𝐵₁)(𝑡)𝑢⁺−𝐵₂(𝑡)𝑢⁻=12𝑃̇₂(𝑡)+𝐴(𝑡)𝑃₂(𝑡)𝑥²+𝑔̇₂(𝑡)+𝐴(𝑡)𝑔₂(𝑡)+𝑓(𝑡)𝑃₂(𝑡𝑥+𝑐̇₂)(𝑡)+𝑓(𝑡)𝑔₂(𝑡)−𝑃₂(𝑡)×12𝑥+𝜂₂(𝑡)²𝑏₂(𝑡,𝑥)²=12𝑃̇₂(𝑡)+2𝐴(𝑡)𝑃₂(𝑡)−𝑏₂(𝑡,𝑥)²𝑃₂(𝑡)𝑥²+𝑔̇₂(𝑡)+𝐴(𝑡)𝑔₂(𝑡)+𝑓(𝑡)𝑃₂(𝑡)−𝑃₂(𝑡)𝜂₂(𝑡)𝑏₂(𝑡,𝑥)²𝑥+𝑐₂̇(𝑡)+𝑓(𝑡)𝑔₂(𝑡)−12𝑃₂(𝑡)𝜂₂(𝑡)²𝑏₂(𝑡,𝑥)²=0. (35) Since 𝑥+𝜂₂(𝑡)>0, substituting the minimizer we can see 𝑉(𝑡,𝑥) satisfies HJB equation in Γ₂.

On Γ₃, we have 𝑥=−𝜂(𝑡),𝑔_𝑖(𝑡)=𝑃_𝑖(𝑡)𝜂(𝑡)=−𝑥𝑃_𝑖(𝑡), then we have 𝑉()=1𝑡,𝑥2𝑃_𝑖(𝑡)𝑥²+𝑔_𝑖(𝑡)𝑥+𝑐_𝑖(𝑡)=12𝑃_𝑖(𝑡)𝑥²−𝑃_𝑖(𝑡)𝑥²+𝑐_𝑖(𝑡)=0, (36) by (29), for 𝑖=1,2. So 𝑉(𝑡,𝑥) is continuous at (𝑡,𝑥)∈Γ₃.

Therefore, 𝑉_𝑡()=1𝑡,𝑥2𝑃̇₁(𝑡)𝑥²+𝑔̇₁(𝑡)𝑥+𝑐₁̇(𝑡)=12𝑃̇₂(𝑡)𝑥²+𝑔̇₂(𝑡)𝑥+𝑐₂̇(𝑡)𝑉=0,_𝑥(𝑡,𝑥)=𝑃₁(𝑡)𝑥+𝑔₁(𝑡)=𝑃₂(𝑡)𝑥+𝑔₂(𝑡)=0, (37) that is to say, 𝑉(𝑡,𝑥) is differentiable on Γ₃. However, 𝑉 may be not smooth on Γ₃. It is clear 𝑉_𝑥𝑥 does not exist on Γ₃, since 𝑃₁(𝑡)≠𝑃₂(𝑡). This means that the classical solution of HJB equation (11) does not exist. Therefore, we have to work within the framework of viscosity solution (see the appendix of [6]).

By the definition of viscosity solutions, we have for all (𝑡,𝑥)∈Γ₃, 𝐷^1,2,+_𝑡,𝑥𝑉(𝑡,𝑥)={0}×{0}×𝑃₂(𝑡),𝐷,+∞_{1,2,−𝑡,𝑥}𝑉(𝑡,𝑥)={0}×{0}×−∞,𝑃₁(𝑡). (38)

Now let 𝐺=1𝑡,𝑥,𝑢,𝑝,𝑃2𝑃𝑢^𝑇𝐷^𝑇₁𝐷₁𝑢+𝑝𝐵₁𝑢⁺−𝐵₂𝑢⁻. (39) For all (𝑞,𝑝,𝑃)∈𝐷^1,2,+_𝑡,𝑥𝑉(𝑡,𝑥), (𝑡,𝑥)∈Γ₃, we have 𝑞+inf_{𝑢∈𝐑𝑑}𝐺𝑡,𝑥,𝑢,𝑝,𝑃=inf_{𝑢∈𝐑𝑑}12𝑃(𝑡)𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢≥inf_{𝑢∈𝐑𝑑}12𝑃₂(𝑡)𝑢^𝑇𝐷(𝑡)^𝑇𝐷(𝑡)𝑢=0. (40) Therefore, 𝑉 is the subsolution of HJB equation (11). On the other hand, for all (𝑞,𝑝,𝑃)∈𝐷^1,2,−_𝑡,𝑥𝑉(𝑡,𝑥), (𝑡,𝑥)∈Γ₃, we have 𝑞+inf_{𝑢∈𝐑𝑑}𝐺𝑡,𝑥,𝑢,𝑝,𝑃=inf_{𝑢∈𝐑𝑑}12𝑃(𝑡)𝑢^𝑇𝐷^𝑇₁(𝑡)𝐷₁(𝑡)𝑢≤inf_{𝑢∈𝐑𝑑}12𝑃₁(𝑡)𝑢^𝑇𝐷(𝑡)^𝑇𝐷(𝑡)𝑢=0. (41)

Therefore, by the definition we know 𝑉(𝑡,𝑥) is the supersolution of HJB equation (11). Finally, it is easy to see that the terminal condition 𝑉(𝑇,𝑥)=(1/2)𝑥² is satisfied. Therefore, by definition, 𝑉(𝑡,𝑥) is the viscosity solution of HJB equation (11). Moreover, for all (𝑡,𝑥)∈Γ₃, let (𝑞^∗(𝑡,𝑥),𝑝^∗(𝑡,𝑥),𝑃^∗(𝑡,𝑥),𝑢^∗(𝑡,𝑥))∶=(0,0,𝑃₂(𝑡),0)∈𝐷^1,2,+_𝑡,𝑥𝑉(𝑡,𝑥)×𝓊_𝑠 then 𝑞^∗(𝑡,𝑥)+𝐺𝑡,𝑥,𝑙^∗(𝑡,𝑥),𝑝^∗(𝑡,𝑥),𝑃^∗(𝑡,𝑥)=0. (42) So it satisfies the conditions in verification theorem [12, Theorem 3.1].

The main theorem in this section follows.

Theorem 3. Let 𝑃_𝑖(𝑡),𝑔_𝑖(𝑡),𝑐_𝑖(𝑡) for 𝑖=1,2 be the solution of (21)–(23). The regions Γ_𝑖,𝑖=1,2,3 are defined by (31). Then the function defined by 𝑉()=1𝑡,𝑥2𝑃₁(𝑡)𝑥²+𝑔₁(𝑡)𝑥+𝑐₁(𝑡),(𝑡,𝑥)∈Γ₁,12𝑃₂(𝑡)𝑥²+𝑔₂(𝑡)𝑥+𝑐₂(𝑡),(𝑡,𝑥)∈Γ₂∪Γ₃ (43) is the viscosity solution of HJB equation (11). Control function 𝜋^∗()=()𝜆𝑡,𝑥−𝛼𝑡,𝑥₁(𝑡),(𝑡,𝑥)∈Γ₁,−𝛼(𝑡,𝑥)𝜆₂(𝑡),(𝑡,𝑥)∈Γ₂∪Γ₃ (44) is the optimal feedback control of system (8) for objective functional (9) with 𝑠=0, the optimal value is 𝑉(0,𝑥). Here 𝜆₁,𝜆₂ are constructed by Lemmas 1 and 2.

4. Optimal Strategy and Efficient Frontier

4.1. Optimal Strategy

Back to the portfolio problem. From [6] problem (7) is equivalent to 𝐴(𝜌) for some 𝜌: 1min𝐸2𝑋(𝑇)−𝜇−𝜌²𝜋(⋅)s.t.∈𝐿²_ℱ0,𝑇,𝐑^𝑑,𝑋(⋅)satisfy(5). (45)

In order to make use of the result in Section 3, let 𝑥(𝑡)=𝑋(𝑡)−(𝜇−𝜌). We study optimal control problem 1min2𝐸𝑥²(𝑇(𝑡)=𝑟(𝑡)𝑥(𝑡)+(𝑏(𝑡)(𝑡)𝟏))s.t.𝑑𝑥−𝑟^𝑇𝜋(𝑡)−𝜆𝑟(𝑡)×𝟏^𝑇𝜋⁻(𝑡)+𝜇−𝜌𝑟(𝑡)𝑑𝑡+𝜋^𝑇(𝑡)𝜎(𝑡)(𝑡),𝑥(0).𝑑𝑊=𝑣−𝜇−𝜌 (46)

If we denote 𝐴(𝑡)=𝑟(𝑡), 𝐵₁(𝑡)=𝑏(𝑡)−𝑟(𝑡)𝟏^𝑇, 𝐵₂(𝑡)=𝜆𝑟(𝑡)𝟏^𝑇, 𝑓(𝑡)=(𝜇−𝜌)𝑟(𝑡), 𝐷₁(𝑡)=𝜎^𝑇(𝑡), (46) becomes 1min2𝐸𝑥²(𝑇(𝑡)=𝐴(𝑡)𝑥(𝑡))s.t.𝑑𝑥+𝐵₁(𝑡)𝜋(𝑡)−𝐵₀(𝑡)𝜋⁻(𝑡)+𝑓(𝑡)𝑑𝑡+𝜋^𝑇(𝑡)𝐷^𝑇₁(𝑡)(𝑡),𝑥(0).𝑑𝑊=𝑣−𝜇−𝜌 (47)

Before analyzing the efficient frontier of the original problem (7), we firstly give the optimal strategy of 𝐴(𝜌). The translated problem is a special case studied in the previous section, the optimal strategy is given in (44). In this problem, 𝛼()(𝑡)𝑡,𝑥=𝑥+𝜂=𝑥+𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠𝑇}_𝑡𝜇−𝜌𝑑−𝑒^{∫𝑇𝑧𝑟(𝑠)}𝑑𝑧=𝑥−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}𝜇−𝜌1−𝑒^{∫𝑇𝑡𝑟(𝑠)𝑑𝑠}=𝑥+𝜇−𝜌1−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}. (48)

The optimal strategy is 𝜋^∗()=−𝜆𝑡,𝑥¹(𝑡)𝑥+𝜇−𝜌1−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},if𝑥+𝜇−𝜌1−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}−𝜆<0,₂(𝑡)𝑥+𝜇−𝜌1−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},if𝑥+𝜇−𝜌1−𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0. (49) Substituting 𝑥 with 𝑋−(𝜇−𝜌), we get the following.

Theorem 4. For problem 𝐴(𝜌), the optimal portfolios are 𝜋^∗()=−𝜆𝑡,𝑋¹(𝑡)𝑋−𝜇−𝜌𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},𝑖𝑓𝑋−𝜇−𝜌𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}−𝜆<0,₂(𝑡)𝑋−𝜇−𝜌𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},𝑖𝑓𝑋−𝜇−𝜌𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0. (50) Here 𝜆₁,𝜆₂ are constructed by Lemmas 1 and 2.

4.2. Efficient Frontier

In this part, we will compute the efficient frontier of problem 𝐴(𝜌). Because 12𝐸𝑥(𝑇)²=12𝐸𝑋(𝑇)−𝜇−𝜌²=12𝐸𝑋(𝑇)−𝜇²𝑋(𝑇)+1+𝜌𝐸−𝜇2𝜌². (51)

For fixed 𝜌, we have min_{𝑢(⋅)∈𝓊}𝐸12𝑋(𝑇)−𝜇²(𝑇)((0−1+𝜌𝐸𝑋−𝜇=𝑉0,𝑥))2𝜌²=12𝑃(0)𝑥(0)²(0)𝑥(0)(0)−1+𝑔+𝑐2𝜌²=12𝑃(0)𝑣−𝜇−𝜌²(0(0)−1+𝑔)𝑣−𝜇−𝜌+𝑐2𝜌², (52) where 𝑃(⋅), 𝑔(⋅), 𝑐(⋅) are either 𝑃₁(⋅), 𝑔₁(⋅), 𝑐₁(⋅) or 𝑃₂(⋅), 𝑔₂(⋅), 𝑐₂(⋅), respectively, on 𝑋−(𝜇−𝜌)𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}<0 and 𝑋−(𝜇−𝜌)𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0. Now, if 𝑋−(𝜇−𝜌)𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}<0, we have a concave quadratic function in 𝜌min_{𝑢(⋅)[]∈𝓊0,𝑇}𝐸12𝑋(𝑇)−𝜇²(𝑇)=1+𝜌𝐸𝑋−𝜇2𝑃₁(0)𝑋₀−𝜇−𝜌²+𝑔₁(0)𝑋₀−𝜇−𝜌+𝑐₁(0)−12𝜌²=12𝑒^{−∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−12𝜌². (53)

If 𝑋−(𝑑−𝜌)𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0, we obtain min_{𝑢(⋅)∈𝓊[0,𝑇]}𝐸12𝑋(𝑇)−𝜇²(𝑇)=1+𝜌𝐸𝑋−𝜇2𝑃₂(0)𝑋₀−𝜇−𝜌²+𝑔₂(0)𝑋₀−𝜇−𝜌+𝑐₂(0)−12𝜌²=12𝑒^{−∫𝑇0‖𝑏2(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−12𝜌². (54)

The minimum variance (optimal value functional) is min_{𝑢(⋅)∈𝓊}𝐸[𝑋(𝑇)−𝑑]²[𝑋(𝑇)]=𝑒+2𝜌𝐸−𝑑_{−∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−𝜌²,if𝑣−𝜇−𝜌𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}𝑒<0;_{−∫𝑇0‖𝑏2(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−𝜌²,if𝑣−𝜇−𝜌𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0. (55)

Since 𝜌 is Lagrange multiplier. We must maximize it. Note that 𝜇≥𝜇₀=𝑣𝑒^{∫𝑇𝑡𝑟(𝑠)𝑑𝑠}. If 𝑣−(𝜇−𝜌)𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}≤0, then 𝜌≤𝜇−𝜇₀. Maximize 𝑒^{−∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−𝜌², (56) we get 𝜌^∗₁=𝜇−𝜇₀1−𝑒^{∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}<𝜇−𝜇₀ (57) with maximum value 𝜇−𝜇₀²𝑒^{∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}.−1 (58) If 𝑣−(𝜇−𝜌)𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}≥0, then 𝜌≥𝜇−𝜇₀. Maximize 𝑒^{−∫𝑇0‖𝑏2(𝑠)‖2𝑑𝑠}𝑣𝑒^{∫𝑇0𝑟(𝑠)𝑑𝑠}−𝜇−𝜌²−𝜌². (59) We get 𝜌^∗₂=𝜇−𝜇₀ with maximum value −(𝜌^∗₂)²<0. Therefore the maximum of (55) equals (58).

Now we have the theorem to conclude the section.

Theorem 5. The efficient strategy of mean-variance problem can be formulated as 𝜋^∗()=−𝜆𝑡,𝑥¹(𝑡)𝑥−𝜇−𝜌^∗𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},𝑖𝑓𝑥−𝜇−𝜌^∗𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}−𝜆<0,₂(𝑡)𝑥−𝜇−𝜌^∗𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠},𝑖𝑓𝑥−𝜇−𝜌^∗𝑒^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}≥0, (60) where 𝜌^∗=𝜌^∗₁ is defined in (53), 𝜆₁, 𝜆₂, come from Lemmas 1 and 2. The minimum variance is var[𝑋(𝑇)]=(𝜇−𝜇₀)²𝑒^{∫𝑇0‖𝑏1(𝑠)‖2𝑑𝑠}.−1 (61) Here 𝑏₁ is determined by Lemmas 1 and 2.

Note. When coefficients are all constant then 𝜆₁, 𝜆₂ and 𝑏₁ are all constant vectors.

5. Two Examples

Example 6. Suppose 𝑑=1, 𝑇=1, 𝑟(𝑡)=𝑟, 𝑏(𝑡)=𝑏, 𝜎(𝑡)=1. Then the wealth equation is 𝑑𝑋(𝑡)=[𝑟𝑋(𝑡)+(𝑏−𝑟)𝜋(𝑡)−𝜆𝑟𝜋⁻(𝑡(𝑡)(𝑡),𝑋(0))]𝑑𝑡+𝜋𝑑𝑊=𝑣. (62)
The problem 𝐴(𝜌) can be formulated as 1min2𝐸𝑥²(𝑇(𝑡)=[(𝑡)+()𝜋(𝑡))s.t.𝑑𝑥𝑟𝑥𝑏−𝑟−𝜆𝑟𝜋⁻(𝑡)+𝑟(𝑡)(𝑡),𝑥(0).𝜇−𝜌𝑑𝑡+𝜋𝑑𝑊=𝑣−𝜇−𝜌 (63)
The correspondence HJB equation is 𝑣_𝑡(𝑡,𝑥)+inf_𝑢∈𝐑𝑣_𝑥()𝑔()+1𝑡,𝑥𝑡,𝑥,𝑢2𝑣_𝑥𝑥(𝑡,𝑥)𝑢²𝑣()=1=0,𝑇,𝑥2𝑥². (64) Here 𝑔(𝑡,𝑥,𝑢)=𝑟𝑥(𝑡)+(𝑏−𝑟)𝑢−𝜆𝑟𝑢⁻+(𝜇−𝜌)𝑟. Letting 𝛼(𝑡,𝑥)∶=𝑣_𝑥(𝑡,𝑥)/𝑣_𝑥𝑥(𝑡,𝑥), we need to minimize ℎ(𝑢)=12𝑢²+[(𝑏−𝑟)𝑢−𝜆𝑟𝑢⁻]. (65)
We assume 𝑏>𝑟>0, then when 𝛼(𝑡,𝑥)<0, (𝑢)(𝛼(1minℎ=ℎ𝑡,𝑥)(𝑏−𝑟))=−2[𝛼(𝑡,𝑥)(𝑏−𝑟)]². (66) When 𝛼(𝑡,𝑥)≥0, (𝑢)((1minℎ=ℎ−𝛼𝑡,𝑥)(𝑏−𝑟+𝑟𝜆))=−2[𝛼(𝑡,𝑥)(𝑏−𝑟+𝑟𝜆)]². (67)
By (53) we have the optimal 𝜌^∗=(𝜇−𝜇₀)/(1−𝑒^{(𝑏−𝑟)2}) (note 𝑏²₁=(𝑏−𝑟)² and 𝜇₀=𝑒^𝑟) and the optimal value (𝜇−𝜇₀)²/(𝑒^{(𝑏−𝑟)2}−1); they are all independent of margin rate 𝜆.
Because 𝑓(𝑡)=(𝜇−𝜌^∗)𝑟, we have Γ₃={(𝑡,𝑥)∶𝑥−(𝜇−𝜌^∗)𝑒^{𝑟(1−𝑡)}=0}. By (64) and (65) we know 𝜆₁(𝑡)=𝑏−𝑟,𝜆₂(𝑡)=−(𝑏−𝑟+𝜆𝑟). By Theorem 5, the efficient strategy of mean-variance problem can be formulated as 𝜋^∗()=−(𝑡,𝑥𝑏−𝑟)𝑥−𝜇−𝜌^∗𝑒^{−𝑟(1−𝑡)},if𝑥−𝜇−𝜌^∗𝑒^{−𝑟(1−𝑡)}(<0,𝑏−𝑟+𝜆𝑟)𝑥−𝜇−𝜌^∗𝑒^{−𝑟(1−𝑡)},if𝑥−𝜇−𝜌^∗𝑒^{−𝑟(1−𝑡)}≥0. (68)
It is interesting to see that although the optimal strategies are different when 𝜆 changes, Γ₃ (which is the boundary of two regions Γ₁ and Γ₂) and the optimal values remain the same and therefore two regions Γ₁,Γ₂ and the efficient frontiers keep unchanged. But it is not always the same case, next example we will show that the optimal values and the efficient frontiers vary with margin rate 𝜆.

Example 7. Let 𝑑=2, 𝑇=1, 𝑟(𝑡)=0.01, 𝑏(𝑡)=(^0.050.02), 𝜎(𝑡)=²⁰₁₂. In order to solve the HJB equation, we must minimize ℎ(𝑢)=12𝑢^𝑇𝜎𝜎^𝑇𝑢+𝛼[(𝑏−𝑟𝟏)𝑢−𝑟𝜆𝑢⁻]=124𝑢²₁+4𝑢₁𝑢₂+5𝑢²₂+𝛼0.04𝑢₁−0.01𝜆𝑢⁻₁+0.01𝑢₂−0.01𝜆𝑢⁻₂. (69)
When 𝜆=0, we have 𝑢^∗=−𝛼^0.01125_−0.0025,‖𝑏₁‖²=0.000425.
When 𝜆=+∞, short-selling is not allowed. We have 𝑢^∗()=()0()00()𝑡,𝑥−𝛼𝑡,𝑥0.01,if𝛼𝑡,𝑥<0,,if𝛼𝑡,𝑥≥0. (70)‖𝑏₁‖²=0.0004.
When 𝜆=0.2, we consider four cases: 𝑢₁, 𝑢₂≥0; 𝑢₁,𝑢₂≤0; 𝑢₁≥0, 𝑢₂≤0; 𝑢₁≤0, 𝑢₂≥0, then take the minimum. We have 𝑢^∗()=()()()()𝑡,𝑥−𝛼𝑡,𝑥0.011−0.002,if𝛼𝑡,𝑥<0,−𝛼𝑡,𝑥0.0118750.00275,if𝛼𝑡,𝑥≥0. (71)‖𝑏₁‖²=0.000416.
From the example we know 𝑏₁ depends on 𝜆. By Theorem 5, we know the optimal parameter 𝜌^∗, the boundary of two regions Γ₁ and Γ₂, the optimal value are determined by 𝑏₁ therefore they all vary with 𝜆. For the same return 𝜇, the risk is lowest when 𝜆=0 (i.e., no margin required) and the risk is highest when short-selling is prohibited 𝜆=+∞. The results of our paper cover these two extreme cases.

6. Conclusion

In this paper, we study the mean-variance portfolio selection problems for short selling with margin required. The problem is more complicated than free short-selling (𝜆=0) and short-selling prohibiting (𝜆=+∞). We found the optimal strategies in the feedback form and the optimal value function which is the viscosity solution of the HJB equation. We have also given examples to show how the margin rate affects the optimal portfolio, optimal variance, and efficient frontier.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10971127).