A Random Parameter Model for Continuous-Time Mean-Variance Asset-Liability Management

Abstract

We consider a continuous-time mean-variance asset-liability management problem in a market with random market parameters; that is, interest rate, appreciation rates, and volatility rates are considered to be stochastic processes. By using the theories of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), we tackle this problem and derive optimal investment strategies as well as the mean-variance efficient frontier analytically in terms of the solution of BSDEs. We find that the efficient frontier is still a parabola in a market with random parameters. Comparing with the existing results, we also find that the liability does not affect the feasibility of the mean-variance portfolio selection problem. However, in an incomplete market with random parameters, the liability can not be fully hedged.

Hui-qiang Ma 1,2 and Meng Wu 3 and Nan-jing Huang 4

Academic Editor:Leonid Shaikhet

1, School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041, China
2, School of Finance and Statistics, Hunan University, Changsha, Hunan 410079, China
3, Business School, Sichuan University, Chengdu, Sichuan 610064, China
4, Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 15 May 2015; Accepted 6 September 2015; 7 October 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Mean-variance portfolio selection model was pioneered by Markowitz [1] in the single-period setting. In his seminal paper, Markowitz proposed the variance as the measure of the risk. The advantage of using variance for measuring the risk of a portfolio is due to the simplicity of computation. Thus, the mean-variance approach has inspired literally hundreds of extensions and applications and also has been commonly used in practical financial decisions. For example, Wang and Xia [2] gave an excellent review on portfolio selection problem. Li and Ng [3] employed the framework of multiobjective optimization and an embedding technique to obtain the exact mean-variance efficient frontier for multiperiod investment. Wu and Li [4] investigated a multiperiod mean-variance portfolio selection with regime switching and uncertain exit time. Zhou and Li [5] studied a continuous-time mean-variance portfolio selection problem under a stochastic LQ framework. Furthermore, Li et al. [6] considered a continuous-time mean-variance portfolio selection problem with no-shorting constraints. Under partial information, Xiong and Zhou [7] and Wang and Wu [8] considered a continuous-time mean-variance portfolio selection problem and a problem of hedging contingent claims by portfolios, respectively.

Among several extensions of the classic mean-variance portfolio selection model, asset and liability management problem is an important subject in both academic literatures and the real world situations. In the real world, liability is so important that almost all financial institutions and individual investors should manage their debt. Thus, incorporating liability into the portfolio selection model can make investment strategies more practical. The research on mean-variance asset-liability management also evokes recent concern. Sharpe and Tint [9] first investigated mean-variance asset-liability management in a single-period setting. Leippold et al. [10] considered a multiperiod asset-liability management problem and derived both the analytical optimal policy and the efficient frontier. Chiu and Li [11] studied a mean-variance asset-liability management problem in the continuous-time case where the liability was governed by a geometric Brownian motion (GBM). Xie et al. [12] also considered a continuous-time asset-liability management problem under the mean-variance criterion where the dynamic of liability is a Brownian motion with drift. Further, Xie [13] studied a mean-variance portfolio selection model with stochastic liability in a Markovian regime switching financial market. Zeng and Li [14] investigated an asset-liability management problem in a jump diffusion market. Yao et al. [15] studied continuous-time mean-variance asset-liability management with endogenous liabilities. By using the time-consistent approach, Wei et al. [16] considered a mean-variance asset-liability management problem with regime switching.

Among these studies, we note that all market parameters are assumed to be deterministic. However, in the real world, market parameters observed in many situations are always uncertain (see, e.g., [17-20]). In order to capture the features of optimal investment strategies with random parameters, random parameter models have drawn more attention over last few years. For example, Lim and Zhou [21] investigated a mean-variance portfolio selection problem with random parameters in a complete market and derived efficient investment strategies as well as the efficient frontier analytically in terms of the solution of BSDEs. Further, Lim [22] extended Lim and Zhou's [21] results to the case where the market is incomplete.

Up to now, the studies on the asset-liability management problem are under a common assumption that all parameters are assumed to be known with certainty. An interesting and unexplored question is what happens in a more realistic situation with random parameters. This is the main focus of our research. In view of this, we study a mean-variance asset-liability management problem with random parameters and derive both the mean-variance optimal portfolio strategies and the efficient frontier. Referring to Lim [22], we consider a market where the related market parameters are random, such as interest rate, the appreciation rates, and the volatility rates of stocks' price. Further, we routinely assume that the liability is dynamically exogenous and evolves according to a Brownian motion with drift. Note that this description of liability has been widely used (see, e.g., [12, 23, 24]). Under the above assumptions, we introduce an unconstrained stochastic control problem with random parameters and derive the optimal control strategies in terms of the solutions of BSDEs. Then, by using the Lagrange multiplier technique, we derive both the mean-variance optimal investment strategies and the efficient frontier.

Our model is most closely related to the model of Lim [22]. The main differences between our model and Lim's model are in two dimensions. Firstly, we consider a portfolio selection problem with liability. Since the liability is dynamically exogenous, the driving factors of the wealth in our model include that of stocks' price and liability, which is an essential difficulty in our model but not encountered in [22]. Secondly, due to the introduction of random liability, the wealth process derived from our model is no longer homogenous with respect to the control variables, whereas the wealth process in the model without liability (see, e.g., [21, 22]) is homogenous.

This paper proceeds as follows. In Section 2, we give some preliminaries and formulate a continuous-time mean-variance portfolio selection model with liability and random parameters. In Section 3, we introduce an unconstrained stochastic LQ control problem and derive the optimal policies and value function in closed forms in terms of the solution of BSDEs. Further, Section 4 presents the optimal investment strategies and the efficient frontier for the mean-variance asset-liability management problem with random parameters. Section 5 concludes the paper.

2. Model Formulation

In this section, we describe the financial market, the liability, and the mean-variance asset-liability management problem, respectively. Throughout this paper, let [figure omitted; refer to PDF] be a fixed terminal time, [figure omitted; refer to PDF] a complete probability space, and [figure omitted; refer to PDF] the transpose of the vector or matrix [figure omitted; refer to PDF] .

2.1. The Financial Market

Let [figure omitted; refer to PDF] be a filtered complete probability space on which a standard [figure omitted; refer to PDF] -adapted [figure omitted; refer to PDF] -dimensional Brownian motion [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is defined. It is assumed that [figure omitted; refer to PDF] . In this paper, we use [figure omitted; refer to PDF] to model the financial market incompleteness as Lim [22] did. When [figure omitted; refer to PDF] , the financial market corresponds to a complete market.

Consider a financial market with [figure omitted; refer to PDF] securities which consists of a bond and [figure omitted; refer to PDF] stocks. The price of bond [figure omitted; refer to PDF] satisfies the following differential equation: [figure omitted; refer to PDF] where the interest rate is as follows: [figure omitted; refer to PDF] . The price of the [figure omitted; refer to PDF] th stock, [figure omitted; refer to PDF] , is described by the following stochastic differential equation (SDE): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are appreciation rate and volatility rate of the [figure omitted; refer to PDF] th stock, respectively. The [figure omitted; refer to PDF] -valued process of volatility coefficients [figure omitted; refer to PDF] is known as the volatility. In addition, we assume that the market parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -adapted stochastic processes.

2.2. Liability

We assume that an exogenous accumulative liability [figure omitted; refer to PDF] is governed by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a one-dimensional standard Brownian motion. We assume that the diffusion term of the liability, [figure omitted; refer to PDF] , is correlated with [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the correlation coefficient. Then, [figure omitted; refer to PDF] can be further expressed as follows (see [figure omitted; refer to PDF] of [25] for more details): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a standard Brownian motion which is independent of [figure omitted; refer to PDF] . It follows from Itô's formula that [figure omitted; refer to PDF] Thus, the liability [figure omitted; refer to PDF] can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Further, we assume that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -adapted stochastic processes, where [figure omitted; refer to PDF] .

Remark 1.

When [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is equal to [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] can be expressed as a linear combination of [figure omitted; refer to PDF] .

Remark 2.

Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the parameters for describing the financial market and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are used to describe the exogenous liability, it is reasonable to assume that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -adapted for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -adapted.

2.3. The Mean-Variance Asset-Liability Management Model

Suppose that the trading of shares takes place continuously in a self-financing fashion and there are no transaction costs. We assume that an investor has an initial endowment [figure omitted; refer to PDF] and a liability [figure omitted; refer to PDF] . We denote by [figure omitted; refer to PDF] the net total wealth of the investor at time [figure omitted; refer to PDF] and by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the market value of the investor's wealth in the [figure omitted; refer to PDF] th stock. Then, [figure omitted; refer to PDF] is a portfolio. The net total wealth satisfies the following equation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Next, we introduce the following notations.

One has [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted, [figure omitted; refer to PDF] -valued stochastic processes on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted essentially bounded stochastic processes on [figure omitted; refer to PDF] with continuous sample paths.

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted, [figure omitted; refer to PDF] -valued stochastic processes on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted, [figure omitted; refer to PDF] -valued stochastic processes on [figure omitted; refer to PDF] with [figure omitted; refer to PDF] -a.s. continuous sample paths such that [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted, [figure omitted; refer to PDF] -valued stochastic processes on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -measurable, square-integrable random variables.

[figure omitted; refer to PDF] is the set of [figure omitted; refer to PDF] -adapted essentially bounded stochastic processes on [figure omitted; refer to PDF] .

Definition 3.

A portfolio policy [figure omitted; refer to PDF] is said to be admissible if [figure omitted; refer to PDF] and there exists a unique solution of (8). In this case, we refer to [figure omitted; refer to PDF] as an admissible pair.

In this paper, we study the classical mean-variance asset-liability management problem where the liability is an exogenous liability [figure omitted; refer to PDF] . The objective of the investor is to find a portfolio [figure omitted; refer to PDF] to minimize his/her risk which is measured by the variance of the net terminal wealth subject to archiving a prescribed expected terminal wealth. Then, the mean-variance asset-liability management problem can be formulated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the prescribed expected terminal wealth. It is clear that (12) is a linearly constrained convex program problem. Thus, it can be reduced to an unconstrained problem by introducing a Lagrange multiplier. Therefore, in Section 3, we first consider the following unconstrained problem parameterized by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and approach it from the perspective of stochastic LQ optimal control and BSDEs. Further, in Section 4, based on the results in Section 3, we employ the Lagrange multiplier method to derive the mean-variance efficient portfolio and the efficient frontier.

In addition, we assume that the following assumptions are satisfied throughout this paper.

Assumption 4.

Consider the following: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] identity matrix. Note that [figure omitted; refer to PDF] is the so-called nondegeneracy condition and implies that [figure omitted; refer to PDF] is invertible.

3. The Unconstrained Asset-Liability Management Problem

The aim of this section is to derive the optimal solution for the unconstrained problem (13).

Consider the following BSDEs: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Throughout this paper, a pair of processes [figure omitted; refer to PDF] is called a solution of BSDE (15) if it satisfies BSDE (15) and [figure omitted; refer to PDF] On the other hand, a pair [figure omitted; refer to PDF] is called a solution of BSDE (16) if [figure omitted; refer to PDF] satisfies BSDE (16) and [figure omitted; refer to PDF]

Before deriving the optimal solution for problem (13), we will prove the existence and uniqueness of solutions of BSDEs (15) and (16), respectively. The following result can be found in [22] (see Theorem [figure omitted; refer to PDF] of [22]).

Lemma 5.

If Assumption 4 holds, then the following BSDE, [figure omitted; refer to PDF] has a solution [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Moreover, if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are solutions of (19), then [figure omitted; refer to PDF] .

Here, we claim that [figure omitted; refer to PDF] holds. In fact, by applying Itô's formula to [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] Once again, it follows from Itô's formula that [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] From Lemma 5, we know that [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] holds.

Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -adapted, BSDE (19) can reduce to (15). This implies that BSDE (15) has a unique solution [figure omitted; refer to PDF] under Assumption 4. Moreover, [figure omitted; refer to PDF] is the unique solution of BSDE (19), where [figure omitted; refer to PDF] .

From the discussion of Section 4 in [22], we have the following lemma.

Lemma 6.

If Assumption 4 holds, then [figure omitted; refer to PDF] is a standard Brownian motion under [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

For the existence and uniqueness of solution of BSDE (16), we have the following result.

Proposition 7.

If Assumption 4 holds, then BSDE (16) has a unique solution.

Proof.

The assumption guarantees that there is a unique optimal control [figure omitted; refer to PDF] for (13). Denote by [figure omitted; refer to PDF] the net wealth process associated with the optimal control [figure omitted; refer to PDF] . The optimal condition (see [26]) implies that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the unique solution of the following linear BSDE (called adjoint equation): [figure omitted; refer to PDF] By using Itô's formula, we have [figure omitted; refer to PDF] It follows from (27) that [figure omitted; refer to PDF] . Further, [figure omitted; refer to PDF] Comparing with BSDE (16), we conclude that [figure omitted; refer to PDF] is a solution of (16), where [figure omitted; refer to PDF]

Now we show the uniqueness of the solution for (16). Assume that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two solutions of BSDE (16). It follows from Itô's formula that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

By using the transformation defined by (25) to (32), we have [figure omitted; refer to PDF] which is a linear BSDE and has a unique solution [figure omitted; refer to PDF] under Assumption 4. In consequence, we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

This completes the proof.

The following lemma is a generalization of Lemma [figure omitted; refer to PDF] in [22].

Lemma 8.

Suppose that Assumption 4 holds. Let [figure omitted; refer to PDF] be given and fixed. If net wealth equation (8) corresponding to [figure omitted; refer to PDF] has a unique solution [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is admissible.

Proof.

Assume that [figure omitted; refer to PDF] is given and fixed and SDE (8) corresponding to [figure omitted; refer to PDF] has a unique solution [figure omitted; refer to PDF] . It follows from Itô's formula that [figure omitted; refer to PDF] Under Assumption 4, we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , which imply that [figure omitted; refer to PDF] is a martingale and so [figure omitted; refer to PDF]

Because [figure omitted; refer to PDF] is continuous (and bounded on [figure omitted; refer to PDF] , a.s.), we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a local martingale. Therefore, there exists a localizing sequence [figure omitted; refer to PDF] for the local martingale such that [figure omitted; refer to PDF] is a martingale.

Putting [figure omitted; refer to PDF] and taking expectations on both sides of (34), we have [figure omitted; refer to PDF]

Then it can be rewritten as [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Rewriting the inequality above, we have [figure omitted; refer to PDF]

From Fatou's lemma, we obtain [figure omitted; refer to PDF] where the last inequality comes from Assumption 4 and [figure omitted; refer to PDF] .

Since [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] is admissible.

This completes the proof.

The following result concerns the admissibility of (46).

Proposition 9.

If Assumption 4 holds, then [figure omitted; refer to PDF] (replace (8) by (46)) has a unique solution with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Moreover, [figure omitted; refer to PDF] is admissible.

Proof.

Consider the following SDE: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

It can be shown that [figure omitted; refer to PDF] is the unique solution of (47), where [figure omitted; refer to PDF] By using Itô's formula, we have that [figure omitted; refer to PDF] is the unique solution of SDE (45).

It follows from Itô's formula that [figure omitted; refer to PDF] Then, we have [figure omitted; refer to PDF] Taking [figure omitted; refer to PDF] into account, we conclude that [figure omitted; refer to PDF] is a local martingale under Assumption 4. Let [figure omitted; refer to PDF] be a localizing sequence for the local martingale above. Then, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] It follows from Fatou's lemma and Assumption 4 that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , there exists a constant [figure omitted; refer to PDF] such that, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Thus, we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Further, we have [figure omitted; refer to PDF] which means that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it follows from (46) that [figure omitted; refer to PDF] . Further, we conclude from Lemma 8 that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is admissible.

This completes the proof.

Next, we formulate the optimal control policy and the cost for the unconstrained control problem (13).

Theorem 10.

Let [figure omitted; refer to PDF] be given by (46). If Assumption 4 holds, then [figure omitted; refer to PDF] is the unique optimal control policy for problem (13) and [figure omitted; refer to PDF] is the optimal cost.

Proof.

From Itô's formula, (16) and (8) give [figure omitted; refer to PDF]

By using Itô's formula again, we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is given by (46).

Then, by integrating from [figure omitted; refer to PDF] and taking expectations, we have [figure omitted; refer to PDF]

Due to the fact that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where the equality holds only when [figure omitted; refer to PDF] . This completes the proof.

Remark 11.

If there is no liability, that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , and (16) boils down to [figure omitted; refer to PDF] and the unique optimal control policy for problem (13) is [figure omitted; refer to PDF] which is the same as [figure omitted; refer to PDF] in Lim [22].

4. The Mean-Variance Asset-Liability Management Problem

An admissible portfolio [figure omitted; refer to PDF] is said to be a feasible portfolio for (12) if it satisfies the constraint in (12). Then, problem (12) is said to be feasible if it has a feasible portfolio. Following the methodology of Lim [22], we get a necessary and sufficient condition for feasibility of problem (12) as follows.

Proposition 12.

Let [figure omitted; refer to PDF] be a unique solution of the following BSDE: [figure omitted; refer to PDF] If Assumption 4 holds, then mean-variance problem (12) is feasible for any [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

Proof.

Let [figure omitted; refer to PDF] be admissible and [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is the solution of (8) corresponding to [figure omitted; refer to PDF] . It follows from Itô's formula that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] which has been shown in Yong and Zhou [27] (see pp. 353 of [27]). If (69) holds, then we can choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Hence, for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is well defined and [figure omitted; refer to PDF] This implies that (12) is feasible for any [figure omitted; refer to PDF] .

Conversely, if (12) is feasible for any [figure omitted; refer to PDF] , then, for any [figure omitted; refer to PDF] , there exists an admissible portfolio [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is independent of [figure omitted; refer to PDF] , we conclude that [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] . From (71), we know that (69) is true.

This completes the proof.

Remark 13.

The necessary and sufficient condition (69) is the same as that in [22] in which Lim studied the mean-variance portfolio problem without liability. This implies that the liability does not affect the feasibility of mean-variance problem.

Remark 14.

As claimed in [22], necessary and sufficient condition (69) is very mild.

In the case of mean-variance asset-liability management problem, we can replace the unique solution [figure omitted; refer to PDF] of BSDE (16) by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the unique solutions of the following BSDEs: [figure omitted; refer to PDF] respectively.

By employing the results in Section 3 and Lagrange multiplier technique (or duality theory), we give our main result as follows.

Theorem 15.

If Assumption 4 holds and (69) is satisfied, then mean-variance asset-liability management problem (12) is feasible for every [figure omitted; refer to PDF] , and the inequality [figure omitted; refer to PDF] holds and the following constants, [figure omitted; refer to PDF] are well defined. The efficient frontier of problem (12) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the optimal portfolio associated with the expected net terminal wealth [figure omitted; refer to PDF] is given as follows: [figure omitted; refer to PDF]

Proof.

It is easy to verify that problem (12) has a convex constrained set and a convex cost which is bounded below. These imply that (12) is a linearly constrained convex problem. Because problem (12) is feasible, it follows from Lagrange multiplier technique (see [28] for more details) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

For each fixed [figure omitted; refer to PDF] , the unconstrained problem [figure omitted; refer to PDF] has the same form as (13). Then, it follows from Theorem 10 that [figure omitted; refer to PDF] and the optimal investment strategy is [figure omitted; refer to PDF] Rewriting [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is quadratic in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is finite, we have [figure omitted; refer to PDF] In fact, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] can only be finite when [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] , which is a contradiction. So it must be the case that [figure omitted; refer to PDF] .

Rewriting (86), we have [figure omitted; refer to PDF] Then we have the optimal [figure omitted; refer to PDF] for (81). Taking [figure omitted; refer to PDF] in (88) and (85), [figure omitted; refer to PDF] This completes the proof.

We claim that [figure omitted; refer to PDF] . In fact, since [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Further, because [figure omitted; refer to PDF] is a Hilbert space, it follows from Cauchy-Schwarz's inequality that [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Remark 16.

Theorem 15 shows that efficient frontier (79) is a parabola. Further, for a given mean target, the risk that the investor has to bear is given by (79). In particular, if the investor wants to take the global minimal risk, he/she can obtain the expected terminal wealth [figure omitted; refer to PDF] by choosing the optimal strategy.

Remark 17.

Theorem 15 also shows that the global minimal risk is [figure omitted; refer to PDF] which is nonnegative. This implies that when the market parameters are random and the financial market is incomplete, the liability can not be completely hedged.

Remark 18.

Now we consider a financial market without liability; that is, [figure omitted; refer to PDF] . Then, we have that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the unique solution of BSDE (76) and the constants in Theorem 15 are given by [figure omitted; refer to PDF] It follows from Theorem 15 that the efficient frontier in this case is given by [figure omitted; refer to PDF] which is the same as that of Lim [22]. This implies that Lim's result is a special case of our results. Therefore, our results generalize and improve Lim's results.

5. Conclusions

This paper studies the mean-variance asset-liability management problem with random market parameters. Since market parameters observed in the real world are always uncertain, it is more realistic to consider how to manage both assets and liabilities in a market with random market parameters. By using the theories of stochastic LQ control and BSDE, we derive both optimal investment strategies and the mean-variance efficient frontier. Compared with the existing results, the efficient frontier is still a parabola and liability does not affect the feasibility of the mean-variance portfolio selection problem in a complete market with random parameters. However, the liability can not be fully hedged in an incomplete market with random parameters.

Future studies can go one step further by considering this problem in a more complex market, whose prices are governed by SDEs with Levy noise or Markovian switching. By using the methods and techniques proposed by Zhu [29, 30], it would be more interesting to discuss the optimal investment strategies and the efficient frontiers in the market mentioned above.

Acknowledgments

The authors are very grateful to the referee and the editor for patiently giving critical and insightful comments and suggestions, which helped to significantly improve the models, results, and presentation. The research was supported in part by National Natural Sciences Foundation of China under Grant 71571125, the Fundamental Research Funds for the Central Universities under Grant 2014SZYTD01, and Sichuan University under Grant 2014SCU04A06.

Conflict of Interests

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

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