#### 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.

#### 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 be a fixed terminal time, a complete probability space, and the transpose of the vector or matrix .

##### 2.1. The Financial Market

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

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

##### 2.2. Liability

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

*Remark 1. *When is independent of , that is, , is equal to . When , can be expressed as a linear combination of .

*Remark 2. *Since , , and are the parameters for describing the financial market and and are used to describe the exogenous liability, it is reasonable to assume that , , and are -adapted for , and , , and are -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 and a liability . We denote by the net total wealth of the investor at time and by , , the market value of the investor’s wealth in the th stock. Then, is a portfolio. The net total wealth satisfies the following equation:where .

Next, we introduce the following notations.

One has , where .

is the set of -adapted, -valued stochastic processes on such that

is the set of -adapted essentially bounded stochastic processes on with continuous sample paths.

is the set of -adapted, -valued stochastic processes on such that

is the set of -adapted, -valued stochastic processes on with -a.s. continuous sample paths such that .

is the set of -adapted, -valued stochastic processes on such that

is the set of -measurable, square-integrable random variables.

is the set of -adapted essentially bounded stochastic processes on .

*Definition 3. *A portfolio policy is said to be admissible if and there exists a unique solution of (8). In this case, we refer to as an admissible pair.

In this paper, we study the classical mean-variance asset-liability management problem where the liability is an exogenous liability . The objective of the investor is to find a portfolio 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:where 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 ,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:where is the identity matrix. Note that is the so-called nondegeneracy condition and implies that 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:where . Throughout this paper, a pair of processes is called a solution of BSDE (15) if it satisfies BSDE (15) and On the other hand, a pair is called a solution of BSDE (16) if satisfies BSDE (16) and

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 of [22]).

Lemma 5. *If Assumption 4 holds, then the following BSDE,has a solution where . Moreover, if and are solutions of (19), then .*

Here, we claim that holds. In fact, by applying Itô’s formula to , we have where , , and Once again, it follows from Itô’s formula that Thus we have From Lemma 5, we know that and so This implies that holds.

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

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

Lemma 6. *If Assumption 4 holds, thenis a standard Brownian motion under where *

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 for (13). Denote by the net wealth process associated with the optimal control . The optimal condition (see [26]) implies thatwhere is the unique solution of the following linear BSDE (called adjoint equation):By using Itô’s formula, we haveIt follows from (27) that . Further, Comparing with BSDE (16), we conclude that is a solution of (16), where Now we show the uniqueness of the solution for (16). Assume that and are two solutions of BSDE (16). It follows from Itô’s formula thatwhere , , , and .

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

This completes the proof.

The following lemma is a generalization of Lemma in [22].

Lemma 8. *Suppose that Assumption 4 holds. Let be given and fixed. If net wealth equation (8) corresponding to has a unique solution such that and , then is admissible.*

*Proof. *Assume that is given and fixed and SDE (8) corresponding to has a unique solution . It follows from Itô’s formula thatUnder Assumption 4, we have , , and , which imply that is a martingale and so Because is continuous (and bounded on , a.s.), we have and is a local martingale. Therefore, there exists a localizing sequence for the local martingale such that is a martingale.

Putting and taking expectations on both sides of (34), we have Then it can be rewritten as Since we have Rewriting the inequality above, we have From Fatou’s lemma, we obtain where the last inequality comes from Assumption 4 and .

Since , we have which implies that is admissible.

This completes the proof.

The following result concerns the admissibility of (46).

Proposition 9. *If Assumption 4 holds, then(replace (8) by (46)) has a unique solution with and . Moreover,is admissible.*

*Proof. *Consider the following SDE:where It can be shown that is the unique solution of (47), where By using Itô’s formula, we have that is the unique solution of SDE (45).

It follows from Itô’s formula thatThen, we have Taking into account, we conclude that is a local martingale under Assumption 4. Let be a localizing sequence for the local martingale above. Then, for any , It follows from Fatou’s lemma and Assumption 4 that