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

Authors: Hui-qiang Ma; Meng Wu; Nan-jing Huang
Journal: Mathematical Problems in Engineering
 (2015)

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## 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.


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## Body

## 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 Section2, 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, letT be a fixed terminal time, (Ω,F,P) a complete probability space, and M′ the transpose of the vector or matrix M.


### 2.1. The Financial Market

Let(Ω,F,P,{Ft}t≥0) be a filtered complete probability space on which a standard {Ft}t≥0-adapted m+d-dimensional Brownian motion W¯(t)≔(W(t)′,B(t)′)′≔(W1(t),…,Wm(t),B1(t),…,Bd(t))′ for m≥1 and d≥0 is defined. It is assumed that Ft=σ{W¯(s):s≤t}. In this paper, we use B(t) to model the financial market incompleteness as Lim [22] did. When d=0, the financial market corresponds to a complete market.Consider a financial market withm+1 securities which consists of a bond and m stocks. The price of bond A0(t) satisfies the following differential equation: (1)dA0t=rtA0tdt,t∈0,T,A00=A00>0,where the interest rate is as follows: r(t)>0. The price of the ith stock, Ai(t), is described by the following stochastic differential equation (SDE): (2)dAit=Aitμitdt+∑j=1mσijtdWjt,Ai0=A0i>0,where μi(t)>0 and σi(t)≔(σi1(t),…,σim(t)) are appreciation rate and volatility rate of the ith stock, respectively. The Rm×m-valued process of volatility coefficients (3)σt≔σ1t′,…,σmt′′ is known as the volatility. In addition, we assume that the market parameters r(·), μi(·), and σij(·) are {Ft}t≥0-adapted stochastic processes.


### 2.2. Liability

We assume that an exogenous accumulative liabilityL(t) is governed by (4)dLt=utdt+vtdBLt,L0=L0>0,where BL(t) is a one-dimensional standard Brownian motion. We assume that the diffusion term of the liability, BL, is correlated with W¯(t), and ρ=(ρ1,…,ρm+d)′ is the correlation coefficient. Then, BL(t) can be further expressed as follows (see (2.6) of [25] for more details): (5)BLt=ρ′W¯t+1-ρ′ρW0t,where W0(t) is a standard Brownian motion which is independent of W¯(t). It follows from Itô’s formula that (6)dBLt=ρ′dW¯t+1-ρ′ρdW0t. Thus, the liability L(t) can be rewritten as (7)dLt=utdt-δ1t′dWt-δ2t′dBt-δ0tdW0t,where δ1(t)≔-v(t)(ρ1,…,ρm)′, δ2(t)≔-v(t)(ρm+1,…,ρm+d)′, and δ0(t)≔-v(t)1-ρ′ρ. Further, we assume that u(·) and δ(·)≔(δ1(·)′,δ2(·)′,δ0(·))′ are {F^t}t≥0-adapted stochastic processes, where F^t≔σW¯(s)′,W0(s)′:s≤t.Remark 1.
WhenBL(t) is independent of W¯(t), that is, ρ=0, BL(t) is equal to W0(t). When ρ′ρ=1, BL(t) can be expressed as a linear combination of W1(t),…,Wm(t),B1(t),…,Bd(t).Remark 2.
Sincer(·), μi(·), and σij(·) are the parameters for describing the financial market and u(·) and v(·) are used to describe the exogenous liability, it is reasonable to assume that r(·), μi(·), and σij(·) are {Ft}t≥0-adapted for i,j=1,…,m, and u(·), v(·), and δ(·) are {F^t}t≥0-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 endowmentw and a liability L(t). We denote by X(t) the net total wealth of the investor at time t∈[0,T] and by πi(t), i=1,…,m, the market value of the investor’s wealth in the ith stock. Then, π(t)≔(π1(t),…,πm(t))′ is a portfolio. The net total wealth satisfies the following equation:(8)dXt=rtXt+btπt-utdt+πt′σt+δ1t′dWt+δ2t′dBt+δ0tdW0t,X0=X0=w-L0,where b(t)=(μ1(t)-r(t),…,μm(t)-r(t)).Next, we introduce the following notations.One hasx≔∑i=1nxi2, where x=(x1,…,xn)′∈Rn.L
F
2
(
0
,
T
;
R
n
) is the set of {Ft}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (9)E∫0Tft2dt<∞.L
F
∞
(
Ω
;
C
(
0
,
T
;
R
)
) is the set of {Ft}t≥0-adapted essentially bounded stochastic processes on [0,T] with continuous sample paths.L
F
^
2
(
0
,
T
;
R
n
) is the set of {F^t}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (10)E∫0Tft2dt<∞.L
F
^
2
(
Ω
;
C
(
0
,
T
;
R
)
) is the set of {F^t}t≥0-adapted, R-valued stochastic processes on [0,T] with P-a.s. continuous sample paths such that Esupt∈0,T⁡ft2<∞.L
F
^
2
,
l
o
c
(
0
,
T
;
R
n
) is the set of {F^t}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (11)∫0Tft2dt<∞,P-a.s.L
F
^
T
2
(
Ω
;
R
) is the set of F^T-measurable, square-integrable random variables.L
F
^
∞
(
0
,
T
;
R
m
) is the set of {F^t}t≥0-adapted essentially bounded stochastic processes on [0,T].Definition 3.
A portfolio policyπ(·) is said to be admissible if π(·)∈LF^2(0,T;Rm) and there exists a unique solution of (8). In this case, we refer to (X(·),π(·)) as an admissible pair.In this paper, we study the classical mean-variance asset-liability management problem where the liability is an exogenous liabilityL(t). The objective of the investor is to find a portfolio π(·) 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:(12)J∗≔minVar⁡XT=EXT-c2,subject  to:EXT=c,X·,π·is  admissible  for8,where c∈R 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 l∈R,(13)minEXT-l2,subject  to:X·,π·is  admissible  for8,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:(14)r·,μi·,σij·∈LF∞0,T;R,i,j=1,…,m;u·,v·∈LF^∞0,T;R;δ·∈LF^∞0,T;Rm+d+1;σtσt′≥ϵIm,∀t∈0,T,for  someϵ>0,where Im is the m×m identity matrix. Note that σ(t)σ(t)′≥ϵIm is the so-called nondegeneracy condition and implies that σ(t) is invertible.


## 2.1. The Financial Market

Let(Ω,F,P,{Ft}t≥0) be a filtered complete probability space on which a standard {Ft}t≥0-adapted m+d-dimensional Brownian motion W¯(t)≔(W(t)′,B(t)′)′≔(W1(t),…,Wm(t),B1(t),…,Bd(t))′ for m≥1 and d≥0 is defined. It is assumed that Ft=σ{W¯(s):s≤t}. In this paper, we use B(t) to model the financial market incompleteness as Lim [22] did. When d=0, the financial market corresponds to a complete market.Consider a financial market withm+1 securities which consists of a bond and m stocks. The price of bond A0(t) satisfies the following differential equation: (1)dA0t=rtA0tdt,t∈0,T,A00=A00>0,where the interest rate is as follows: r(t)>0. The price of the ith stock, Ai(t), is described by the following stochastic differential equation (SDE): (2)dAit=Aitμitdt+∑j=1mσijtdWjt,Ai0=A0i>0,where μi(t)>0 and σi(t)≔(σi1(t),…,σim(t)) are appreciation rate and volatility rate of the ith stock, respectively. The Rm×m-valued process of volatility coefficients (3)σt≔σ1t′,…,σmt′′ is known as the volatility. In addition, we assume that the market parameters r(·), μi(·), and σij(·) are {Ft}t≥0-adapted stochastic processes.


## 2.2. Liability

We assume that an exogenous accumulative liabilityL(t) is governed by (4)dLt=utdt+vtdBLt,L0=L0>0,where BL(t) is a one-dimensional standard Brownian motion. We assume that the diffusion term of the liability, BL, is correlated with W¯(t), and ρ=(ρ1,…,ρm+d)′ is the correlation coefficient. Then, BL(t) can be further expressed as follows (see (2.6) of [25] for more details): (5)BLt=ρ′W¯t+1-ρ′ρW0t,where W0(t) is a standard Brownian motion which is independent of W¯(t). It follows from Itô’s formula that (6)dBLt=ρ′dW¯t+1-ρ′ρdW0t. Thus, the liability L(t) can be rewritten as (7)dLt=utdt-δ1t′dWt-δ2t′dBt-δ0tdW0t,where δ1(t)≔-v(t)(ρ1,…,ρm)′, δ2(t)≔-v(t)(ρm+1,…,ρm+d)′, and δ0(t)≔-v(t)1-ρ′ρ. Further, we assume that u(·) and δ(·)≔(δ1(·)′,δ2(·)′,δ0(·))′ are {F^t}t≥0-adapted stochastic processes, where F^t≔σW¯(s)′,W0(s)′:s≤t.Remark 1.
WhenBL(t) is independent of W¯(t), that is, ρ=0, BL(t) is equal to W0(t). When ρ′ρ=1, BL(t) can be expressed as a linear combination of W1(t),…,Wm(t),B1(t),…,Bd(t).Remark 2.
Sincer(·), μi(·), and σij(·) are the parameters for describing the financial market and u(·) and v(·) are used to describe the exogenous liability, it is reasonable to assume that r(·), μi(·), and σij(·) are {Ft}t≥0-adapted for i,j=1,…,m, and u(·), v(·), and δ(·) are {F^t}t≥0-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 endowmentw and a liability L(t). We denote by X(t) the net total wealth of the investor at time t∈[0,T] and by πi(t), i=1,…,m, the market value of the investor’s wealth in the ith stock. Then, π(t)≔(π1(t),…,πm(t))′ is a portfolio. The net total wealth satisfies the following equation:(8)dXt=rtXt+btπt-utdt+πt′σt+δ1t′dWt+δ2t′dBt+δ0tdW0t,X0=X0=w-L0,where b(t)=(μ1(t)-r(t),…,μm(t)-r(t)).Next, we introduce the following notations.One hasx≔∑i=1nxi2, where x=(x1,…,xn)′∈Rn.L
F
2
(
0
,
T
;
R
n
) is the set of {Ft}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (9)E∫0Tft2dt<∞.L
F
∞
(
Ω
;
C
(
0
,
T
;
R
)
) is the set of {Ft}t≥0-adapted essentially bounded stochastic processes on [0,T] with continuous sample paths.L
F
^
2
(
0
,
T
;
R
n
) is the set of {F^t}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (10)E∫0Tft2dt<∞.L
F
^
2
(
Ω
;
C
(
0
,
T
;
R
)
) is the set of {F^t}t≥0-adapted, R-valued stochastic processes on [0,T] with P-a.s. continuous sample paths such that Esupt∈0,T⁡ft2<∞.L
F
^
2
,
l
o
c
(
0
,
T
;
R
n
) is the set of {F^t}t≥0-adapted, Rn-valued stochastic processes on [0,T] such that (11)∫0Tft2dt<∞,P-a.s.L
F
^
T
2
(
Ω
;
R
) is the set of F^T-measurable, square-integrable random variables.L
F
^
∞
(
0
,
T
;
R
m
) is the set of {F^t}t≥0-adapted essentially bounded stochastic processes on [0,T].Definition 3.
A portfolio policyπ(·) is said to be admissible if π(·)∈LF^2(0,T;Rm) and there exists a unique solution of (8). In this case, we refer to (X(·),π(·)) as an admissible pair.In this paper, we study the classical mean-variance asset-liability management problem where the liability is an exogenous liabilityL(t). The objective of the investor is to find a portfolio π(·) 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:(12)J∗≔minVar⁡XT=EXT-c2,subject  to:EXT=c,X·,π·is  admissible  for8,where c∈R 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 l∈R,(13)minEXT-l2,subject  to:X·,π·is  admissible  for8,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:(14)r·,μi·,σij·∈LF∞0,T;R,i,j=1,…,m;u·,v·∈LF^∞0,T;R;δ·∈LF^∞0,T;Rm+d+1;σtσt′≥ϵIm,∀t∈0,T,for  someϵ>0,where Im is the m×m identity matrix. Note that σ(t)σ(t)′≥ϵIm is the so-called nondegeneracy condition and implies that σ(t) 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:(15)dpt=-2rt+θt2pt+2θt′Λ1t+1ptΛ1t2dt+Λ1t′dWt+Λ2t′dBt,pT=1,pt>0,∀t∈0,T,(16)dht=rtht+θt′η1t-Λ2t′ptη2t-ut-θt′δ1t+δ2t′Λ2tptdt+η1t′dWt+η2t′dBt+η0tdW0t,hT=l,where θ(t)=σ(t)-1b(t)′. Throughout this paper, a pair of processes (p(·),Λ(·)) is called a solution of BSDE (15) if it satisfies BSDE (15) and (17)Λ·=Λ1·′,Λ2·′′,p·,Λ·∈LF∞Ω;C0,T;R·LF20,T;Rm+d,1p·∈LF∞Ω;C0,T;R.On the other hand, a pair (h(·),η(·)) is called a solution of BSDE (16) if (h(·),η(·)) satisfies BSDE (16) and (18)η·=η1·′,η2·′,η0·′,h·,η·∈LF^2Ω;C0,T;R·LF^20,T;Rm+d+1.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 6.1 of [22]).Lemma 5.
If Assumption4 holds, then the following BSDE,(19)dpt=-2rt+θt2pt+2θt′Λ1t+Λ1t2ptdt+Λ1t′dWt+Λ~2t′dB~t,pT=1,pt>0,∀t∈0,T,has a solution (p(·),Λ~(·)) where B~(·)≔(B(·)′,W0(·))′. Moreover, if (p¯(·),Λ¯(·)) and (p(·),Λ~(·)) are solutions of (19), then p¯(·)≡p(·).Here, we claim thatΛ¯(·)≡Λ~(·) holds. In fact, by applying Itô’s formula to Δp(t)≔p(t)-p¯(t), we have (20)dΔpt=ΔFtdt+ΔΛ1t′dWt+ΔΛ2t′dB~t,ΔpT=0, where ΔΛ1(·)≔Λ~1(·)-Λ¯1(·), ΔΛ2(·)≔Λ~2(·)-Λ¯2(·), and (21)ΔF·≔-2r·+θ·2Δp·+2θ·′ΔΛ1·+1p·Λ~1·2-Λ¯1·2. Once again, it follows from Itô’s formula that(22)dΔpt2=2ΔptΔFt+ΔΛ1t2+ΔΛ2t2dt+2ΔptΔΛ1t′dWt+2ΔptΔΛ2t′dB~t,ΔpT2=0. Thus we have (23)-EΔpt2=E∫tT2ΔpsΔFs+ΔΛ1s2+ΔΛ2s2ds. From Lemma 5, we know that Δp(·)≡0 and so (24)0=E∫tTΔΛ1s2+ΔΛ2s2ds. This implies that Λ¯(·)≡Λ~(·) holds.Sincer(·) and θ(·) are {Ft}t≥0-adapted, BSDE (19) can reduce to (15). This implies that BSDE (15) has a unique solution (p(·),Λ(·)) under Assumption 4. Moreover, (p(·),Λ~(·)) is the unique solution of BSDE (19), where Λ~(·)=(Λ(·)′,0)′.From the discussion of Section 4 in [22], we have the following lemma.Lemma 6.
If Assumption4 holds, then(25)X~tY~t=WtB~t+∫0tθs-Λ~2spsdsis a standard Brownian motion under P¯ where (26)dP¯dP≔exp⁡-∫0Tθs′dWs+∫0TΛ~2s′psdB~s-12∫0Tθs2+Λ~2sps2ds.For the existence and uniqueness of solution of BSDE (16), we have the following result.Proposition 7.
If Assumption4 holds, then BSDE (16) has a unique solution.Proof.
The assumption guarantees that there is a unique optimal controlπ^(·)∈LF^2(0,T;Rm) for (13). Denote by X^(·) the net wealth process associated with the optimal control π^(·). The optimal condition (see [26]) implies that(27)bt′yt+σtz1t=0,where (y(·),z(·)) is the unique solution of the following linear BSDE (called adjoint equation):(28)dyt=-rtytdt+z1t′dWt+z2t′dBt+z0tdW0t,yT=X^T-l.By using Itô’s formula, we have(29)d1pt=2rt-θt2pt-2θt′Λ1tp2t+Λ2t2p3tdt-Λ1t′p2tdWt-Λ2t′p2tdBt,dytpt=rt-θt2ytpt-2θt′Λ1tyt+Λ1t′z1t+Λ2t′z2tp2t+ytΛ2t2p3tdt+z1t′pt-ytΛ1t′p2tdWt+z2t′pt-ytΛ2t′p2tdBt+z0tptdW0t,dX^t-ytpt=rtX^t-ytpt+θt′σt′π^t+δ1t-z1tpt+ytΛ1tp2tdt-Λ2t′ptδ2t-z2tpt+ytΛ2tp2t+ut+θ′tδ1t-δ2t′Λ2tptdt+ytθt2pt+θt′z1tpt+ytθt′Λ1t+Λ1t′z1tp2tdt+σt′π^t+δ1t-z1tpt+ytΛ1tp2t′dWt+δ2t-z2tpt+ytΛ2tp2t′dBt+δ0t-z0tptdW0t.It follows from (27) that θ(t)y(t)+z1(t)=0. Further, (30)dX^t-ytpt=rtX^t-ytpt+θt′σt′π^t+δ1t-z1tpt+ytΛ1tp2tdt-Λ2t′ptδ2t-z2tpt+ytΛ2tp2t+ut+θt′δ1t-δ2t′Λ2tptdt+σt′π^t+δ1t-z1tpt+ytΛ1tp2t′dWt+δ2t-z2tpt+ytΛ2tp2t′dBt+δ0t-z0tptdW0t.Comparing with BSDE (16), we conclude that (h~(·),η~(·)) is a solution of (16), where (31)h~t≔X^t-ytpt,η~1t≔σt′π^t+δ1t-z1tpt+ytΛ1tp2t,η~2t≔δ2t-z2tpt+ytΛ2tp2t,η~0t≔δ0t-z0tpt.
Now we show the uniqueness of the solution for (16). Assume that (h(·),η(·)) and (h~(·),η~(·)) are two solutions of BSDE (16). It follows from Itô’s formula that(32)dΔht=rtΔht+θt′Δη1t-Λ2t′ptΔη2tdt+Δη1t′dWt+Δη2t′dBt+Δη0tdW0t,ΔhT=0,where Δh(·)≔h(·)-h~(·), Δη1(·)≔η1(·)-η~1(·), Δη2(·)≔η2(·)-η~2(·), and Δη0(·)≔η0(·)-η~0(·).
By using the transformation defined by (25) to (32), we have (33)dΔht=rtΔhtdt+Δη1t′dX~t+Δη2t′,Δη0tdY~t,ΔhT=0 which is a linear BSDE and has a unique solution (0,0) under Assumption 4. In consequence, we have h(·)=h~(·) and η(·)=η~(·).
This completes the proof.The following lemma is a generalization of Lemma3.1 in [22].Lemma 8.
Suppose that Assumption4 holds. Let π¯(·)∈LF^2,loc(0,T;Rm) be given and fixed. If net wealth equation (8) corresponding to π¯(·) has a unique solution X(·) such that X(·)∈LF^2(0,T;R) and X(T)∈LF^T2(Ω;R), then π¯(·)∈LF^2(0,T;Rm) is admissible.Proof.
Assume thatπ¯(·)∈LF^2,loc(0,T;Rm) is given and fixed and SDE (8) corresponding to π¯(·) has a unique solution X(·)∈LF^2(0,T;R). It follows from Itô’s formula that(34)Xt2=X02+∫0t2XsrsXs+bsπ¯s-usds+∫0tσs′π¯s+δ1s2+δ2s2+δ0s2ds+∫0t2Xsδ1s′dWs+∫0t2Xsδ2s′dBs+∫0t2Xsδ0sdW0s+∫0t2Xsσs′π¯s′dWs.Under Assumption 4, we have 2X(·)δ1(·)∈LF^2(0,T;Rm), 2X(·)δ2(·)∈LF^2(0,T;Rd), and 2X(·)δ0(·)∈LF^2(0,T;R), which imply that (35)∫0t2Xsδ1s′dWs+∫0t2Xsδ2s′dBs+∫0t2Xsδ0sdW0s is a martingale and so (36)E∫0t2Xsδ1s′dWs+∫0t2Xsδ2s′dBs+∫0t2Xsδ0sdW0s=0.
BecauseX(·) is continuous (and bounded on [0,T], a.s.), we have 2X(·)σ(·)′π¯(·)∈LF^2,loc(0,T;Rm) and ∫0t2X(s)(σ(s)′π¯(s))′dW(s) is a local martingale. Therefore, there exists a localizing sequence {τi} for the local martingale such that (37)∫0t∧τi2Xsσs′π¯s′dWs is a martingale.
Puttingt=T∧τi and taking expectations on both sides of (34), we have (38)EXT∧τi2=X02+E∫0T∧τi2XsrsXs+bsπ¯s-usds+E∫0T∧τiσs′π¯s+δ1s2+δ2s2+δ0s2ds.
Then it can be rewritten as(39)X02+E∫0T∧τiπ¯s′σsσs′π¯sds+E∫0T∧τiδ1s2+δ2s2+δ0s2ds=EXT∧τi2-E∫0T∧τi2rsXs2ds+E∫0T∧τi2Xsusds-E∫0T∧τi2δ1s′σs′π¯sds-E∫0T∧τi2Xsbsπ¯sds. Since (40)-2Xsbsπ¯s=-24ϵXsbsϵ4π¯s≤4ϵXs2bs2+ϵ4π¯s2,-2δ1s′σs′π¯s=-24ϵδ1s′σs′ϵ4π¯s≤4ϵσsδ1s2+ϵ4π¯s2, we have (41)X02+E∫0T∧τiπ¯s′σsσs′π¯sds+E∫0T∧τiδ1s2+δ2s2+δ0s2ds≤EXT∧τi2-E∫0T∧τi2rsXs2ds+E∫0T∧τi2Xsusds+E∫0T∧τi4ϵσsδ1s2+ϵ4π¯s2ds+E∫0T∧τi4ϵXs2bs2+ϵ4π¯s2ds. Rewriting the inequality above, we have (42)X02+E∫0T∧τiδ1s2+δ2s2+δ0s2ds+E∫0T∧τiπ¯s′σsσs′π¯sds-ϵ2E∫0T∧τiπ¯s2ds≤EXT∧τi2+E∫0T∧τiXs2-2rs+4ϵbs2ds+E∫0T∧τi2Xsusds+4ϵE∫0T∧τiσsδ1s2ds.
From Fatou’s lemma, we obtain(43)X02+E∫0Tδ1s2+δ2s2+δ0s2ds+E∫0Tπ¯s′σsσs′π¯sds-ϵ2E∫0Tπ¯s2ds≤EXT2+E∫0TXs2-2rs+4ϵbs2ds+E∫0T2Xsusds+4ϵE∫0Tσsδ1s2ds<+∞, where the last inequality comes from Assumption 4 and X(·)∈LF^2(0,T;R).
Sinceσ(t)σ(t)′≥ϵIm, we have (44)ϵ2E∫0Tπ¯s2ds≤X02+E∫0Tδ1s2+δ2s2+δ0s2ds+E∫0Tπ¯s′σsσs′π¯sds-ϵ2E∫0Tπ¯s2ds<+∞which implies that π¯(·)∈LF^2(0,T;Rm) is admissible.
This completes the proof.The following result concerns the admissibility of (46).Proposition 9.
If Assumption4 holds, then(45)dXt=rtXt-θt′δ1t-η1t+Xt-htθt+Λ1tpt-utdt+η1t-Xt-htθt+Λ1tpt′dWt+δ2t′dBt+δ0tdW0t,X0=X0(replace (8) by (46)) has a unique solution with X(·)∈LF^2(0,T;R) and X(T)∈LF^T2(Ω;R). Moreover,(46)π¯t≔-σt-1′δ1t-η1t+Xt-htθt+Λ1tptis admissible.Proof.
Consider the following SDE:(47)dYt=-rtYtdt-Ytθt′dWt+Ytα1t+α2t′dBt+β2tdW0t,Y0=p0X0-h0,where (48)α1t=Λ2tpt,α2t=ptδ2t-η2t,β2t=ptδ0t-η0t.
It can be shown that(49)Yt=Φtp0X0-h0-∫0tα1s′α2sΦ-1sds+Φt∫0tα2s′Φ-1sdBs+∫0tβ2sΦ-1sdW0sis the unique solution of (47), where (50)Φt=exp⁡-12∫0tθs2+α1s2+2rsds-∫0tθs′dWs+∫0tα1s′dBs.By using Itô’s formula, we have that (51)Xt=ht+Ytptis the unique solution of SDE (45).
It follows from Itô’s formula that(52)dXt-ht=rtXt-ht-Xt-htθt′θt+Λ1tpt+Λ2t′ptη2t-δ2tdt-Xt-htθt+Λ1tpt′dWt+δ2t-η2t′dBt+δ0t-η0tdW0t,dXt-ht2=Xt-ht22rt-θt2+Λ1tpt2+2Xt-htη2t-δ2t′Λ2tptdt+δ2t-η2t2+δ0t-η0t2dt-2Xt-ht2θt+Λ1tpt′dWt+2Xt-htδ2t-η2t′dBt+2Xt-htδ0t-η0tdW0t,dptXt-ht2=ptδ2t-η2t2+δ0t-η0t2dt-ptXt-ht22θt+Λ1tpt′dWt+2ptXt-htδ2t-η2t′+Xt-ht2Λ2t′dBt+2ptXt-htδ0t-η0tdW0t.Then, we have (53)ptXt-ht2=p0X0-h02+∫0tpsδ2s-η2s2+δ0s-η0s2ds-∫0tpsXs-hs22θs+Λ1sps′dWs+∫0t2psXs-hsδ2s-η2s′+Xs-hs2Λ2s′dBs+∫0t2psXs-hsδ0s-η0sdW0s. Taking X(t)=h(t)+Y(t)/p(t) into account, we conclude that (54)-∫0tpsXs-hs22θs+Λ1sps′dWs+∫0t2psXs-hsδ2s-η2s′dBs+∫0tXs-hs2Λ2s′dBs+∫0t2psXs-hsδ0s-η0sdW0s=-∫0tYs2ps2θs+Λ1sps′dWs+∫0t2Ysδ2s-η2s′dBs+∫0tYs2ps2Λ2s′dBs+∫0t2Ysδ0s-η0sdW0sis a local martingale under Assumption 4. Let {τi} be a localizing sequence for the local martingale above. Then, for any t∈[0,T], (55)Ept∧τiXt∧τi-ht∧τi2=p0X0-h02+E∫0t∧τipsδ2s-η2s2+δ0s-η0s2ds. It follows from Fatou’s lemma and Assumption 4 that (56)EptXt-ht2≤p0X0-h02+E∫0tpsδ2s-η2s2+δ0s-η0s2ds≤p0X0-h02+E∫0Tpsδ2s-η2s2+δ0s-η0s2ds.Since p(·)>0 and 1/p(·)∈LF∞(Ω;C(0,T;R)), there exists a constant ε>0 such that, for any t∈[0,T], p(t)≥ε. Thus, we have (57)εEXt-ht2≤EptXt-ht2≤H, where H≔p(0)(X(0)-h(0))2+E∫0Tp(s)(δ2s-η2s2+δ0s-η0s2)ds<+∞.
Further, we have(58)EXT-hT2≤1εEpTXT-hT2≤Hε<+∞,E∫0TXt-ht2dt=∫0TEXt-ht2dt≤HεT<+∞which means that (X(T)-h(T))∈LF^T2(Ω;R) and (X(·)-h(·))∈LF^2(0,T;R).
Becauseh(T)∈LF^T2(Ω;R) and h(·)∈LF^2(0,T;R), we have (59)X·=X·-h·+h·∈LF^20,T;R,XT=XT-hT+hT∈LF^T2Ω;R.
SinceΛ1(·)∈LF2(0,T;Rm) and h(·)∈LF^2(Ω;C(0,T;R)), it follows from (46) that π¯(·)∈LF^2,loc(0,T;Rm). Further, we conclude from Lemma 8 that π¯(·)∈LF^2(0,T;Rm) and π¯(·) is admissible.
This completes the proof.Next, we formulate the optimal control policy and the cost for the unconstrained control problem (13).Theorem 10.
Letπ¯(·) be given by (46). If Assumption 4 holds, then π¯(·) is the unique optimal control policy for problem (13) and (60)J∗=p0X0-h02+E∫0Tptδ2t-η2t2+δ0t-η0t2dtis the optimal cost.Proof.
From Itô’s formula, (16) and (8) give(61)dXt-ht=rtXt-ht+btπt+θt′δ1t-η1t+Λ2t′ptη2t-δ2tdt+σt′πt+δ1t-η1t′dWt+δ2t-η2t′dBt+δ0t-η0tdW0t,dXt-ht2=2Xt-htrtXt-ht+btπt+θt′δ1t-η1t+Λ2t′ptη2t-δ2tdt+2Xt-htσt′πt+δ1t-η1t′dWt+2Xt-htδ2t-η2t′dBt+2Xt-htδ0t-η0tdW0t+σt′πt+δ1t-η1t2+δ2t-η2t2+δ0t-η0t2dt.
By using Itô’s formula again, we have(62)dptXt-ht2=ptπt′σtσt′πt+2πt′σtδ1t-η1t+Xt-htbt′+σtΛ1tptdt+ptδ0t-η0t2+δ2t-η2t2+δ1t-η1t+Xt-htθt+Λ1tpt2dt+Xt-htXt-htΛ1t+2ptσt′πt+δ1t-η1t′dWt+Xt-htXt-htΛ2t+2ptδ2t-η2t′dBt+2ptXt-htδ0t-η0tdW0t,or(63)dptXt-ht2=ptπt-π¯t′σtσt′πt-π¯tdt+ptδ2t-η2t2+δ0t-η0t2dt+Xt-htXt-htΛ1t+2ptσt′πt+δ1t-η1t′dWt+Xt-htXt-htΛ2t+2ptδ2t-η2t′dBt+2ptXt-htδ0t-η0tdW0t,where π¯(t) is given by (46).
Then, by integrating from[0,T] and taking expectations, we have (64)EpTXT-hT2=p0X0-h02+E∫0Tptπt-π¯t′σtσt′πt-π¯tdt+E∫0Tptδ2t-η2t2+δ0t-η0t2dt+E∫0TXt-htXt-htΛ1t+2ptσt′πt+δ1t-η1t′dWt+E∫0TXt-htXt-htΛ2t+2ptδ2t-η2t′dBt+E∫0T2ptXt-htδ0t-η0tdW0t=p0X0-h02+E∫0Tptπt-π¯t′σtσt′πt-π¯tdt+E∫0Tptδ2t-η2t2+δ0t-η0t2dt.
Due to the fact thatp(t)>0 and σ(t)σ(t)′>ϵIm, we have (65)EXT-l2=EPTXT-hT2≥p0X0-h02+E∫0Tptδ2t-η2t2+δ0t-η0t2dt, where the equality holds only when π(·)=π¯(·). This completes the proof.Remark 11.
If there is no liability, that is,u(·)=v(·)=0 and Ft=F^t, then δ(·)=0, and (16) boils down to (66)dht=rtht+θt′η1t-Λ2t′ptη2tdt+η1t′dWt+η2t′dBt+η0tdW0t,hT=l, and the unique optimal control policy for problem (13) is (67)π¯t=-σt-1′-η1t+Xt-htθt+Λ1tpt,which is the same as (12) in Lim [22].


## 4. The Mean-Variance Asset-Liability Management Problem

An admissible portfolioπ 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(Ψ(·),ξ(·)) be a unique solution of the following BSDE: (68)dΨt=-rtΨtdt+ξ1t′dWt+ξ2t′dBt,ΨT=1. If Assumption 4 holds, then mean-variance problem (12) is feasible for any c∈R if and only if(69)E∫0TΨtbt′+σtξ1t2dt>0.Proof.
Letπ(·) be admissible and π~(·)=λπ(·) for some λ∈R. Assume that X~(·) is the solution of (8) corresponding to π~(·). It follows from Itô’s formula that X~(t)=Z1(t)+λZ2(t), where (70)dZ1t=rtZ1t-utdt+δ1t′dWt+δ2t′dBt+δ0tdW0t,Z10=X0,dZ2t=rtZ2t+btπtdt+πt′σtdWt,Z20=0.Then we have EX~(T)=EZ1(T)+λEZ2(T), where(71)EZ2T=E∫0TΨtbt′+σtξ1t′πtdt,which has been shown in Yong and Zhou [27] (see pp. 353 of [27]). If (69) holds, then we can choose π(t)=Ψ(t)b(t)′+σ(t)ξ1(t) such that (72)EZ2T=E∫0TΨtbt′+σtξ1t2dt>0.Hence, for any c∈R, λc=(EZ2(T))-1(c-EZ1(T)) is well defined and (73)EX~T=EZ1T+λcEZ2T=c.This implies that (12) is feasible for any c∈R.
Conversely, if (12) is feasible for any c∈R, then, for any c∈R, there exists an admissible portfolio π(·) such that EX(T)=EZ1(T)+EZ2(T)=c. Since EZ1(T) is independent of π(·), we conclude that EZ2(T)≠0 for some π(·). 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(h(·),η(·)) of BSDE (16) by (74)ht=hTg1t+g2t,η1t=hTq1t+q¯1t,η2t=hTq2t+q¯2t,η0t=q¯0t, where (g1(·),q(·)) and (g2(·),q¯(·)) are the unique solutions of the following BSDEs: (75)dg1t=rtg1t+θt′q1t-Λ2t′ptq2tdt+q1t′dWt+q2t′dBt,g1T=1,(76)dg2t=rtg2t+θt′q¯1t-Λ2t′ptq¯2t-ut-θt′δ1t+δ2t′Λ2tptdt+q¯1t′dWt+q¯2t′dBt+q¯0tdW0t,g2T=0,respectively.By employing the results in Section3 and Lagrange multiplier technique (or duality theory), we give our main result as follows.Theorem 15.
If Assumption4 holds and (69) is satisfied, then mean-variance asset-liability management problem (12) is feasible for every c∈R, and the inequality (77)1-M-p0g120>0holds and the following constants, (78)M≔E∫0Tptq2t2dt,k1≔E∫0Tptq2t′δ2t-q¯2tdt,k≔-c+p0g10X0-g20+k11-M-p0g120,D≔E∫0Tptδ2t-q¯2t2+δ0t-q¯0t2dt, are well defined. The efficient frontier of problem (12) is given by(79)Var⁡X∗T=M+p0g1201-M-p0g120c-k1+p0g10X0-g20M+p0g1202+D+p0X0-g202-k1+p0g10X0-g202M+p0g120,where c=EX∗(T) and the optimal portfolio associated with the expected net terminal wealth c is given as follows: (80)π∗t=-σt-1′δ1t-q¯1t+kq1t+Xt-g2t+kg1tθt+Λ1tpt.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(81)J∗=maxλ∈RinfX·,π·is  admissibleJπ·,λ<+∞,where (82)Jπ·,λ≔EXT-c2+2λEXT-c=EXT-c+λ2-λ2.
For each fixedλ, the unconstrained problem (83)Jλ≔infX·,π·is  admissibleJπ·,λ has the same form as (13). Then, it follows from Theorem 10 that(84)Jλ=-λ2+p0X0-h02+E∫0Tptδ2t-η2t2+δ0t-η0t2dt=E∫0Tptδ2t-c-λq2t-q¯2t2+δ0t-q¯0t2dt-λ2+p0X0-c-λg10-g202, and the optimal investment strategy is(85)πt=-σt-1′δ1t-η1t+Xt-htθt+Λ1tpt=-σt-1′δ1t-c-λq1t-q¯1t-σt-1′Xt-c-λg1t-g2tθt+Λ1tpt.Rewriting J(λ), we have(86)Jλ=λ-c2-1+p0g120+E∫0Tptq2t2dt+2λ-c-c+g10p0X0-g20+E∫0Tptq2t′δ2t-q¯2tdt-c2+p0X0-g202+D=λ-c2-1+p0g120+M+2λ-c-c+p0g10X0-g20+k1-c2+p0X0-g202+D.Since J(λ) is quadratic in λ and J∗ is finite, we have (87)1-M-p0g120>0. In fact, if 1-M-p(0)g12(0)=0, then J∗ can only be finite when -c+g1(0)p(0)(X0-g2(0))+k1=0 for any c, which is a contradiction. So it must be the case that 1-M-p(0)g12(0)>0.
Rewriting (86), we have(88)Jλ=-1-M-p0g120λ-c-k2+1-M-p0g120k2-c2+p0X0-g202+D.Then we have the optimal λ∗=c+k for (81). Taking λ∗ in (88) and (85), (89)J∗=1-M-p0g120k2-c2+p0X0-g202+D=-c+k1+p0g10X0-g2021-M-p0g120-c2+p0X0-g202+D=M+p0g1201-M-p0g120c-k1+p0g10X0-g20M+p0g1202+D+p0X0-g202-k1+p0g10X0-g202M+p0g120,π∗t=-σt-1′δ1t+kq1t-q¯1t+Xt+kg1t-g2tθt+Λ1tpt.This completes the proof.We claim thatD+p(0)(X0-g2(0))2-[k1+p(0)g1(0)(X0-g2(0))]2/M+p(0)g12(0)≥0. In fact, since (90)p·∈LF∞Ω;C0,T;R,q2·,q¯2·∈LF^20,T;Rd,δ2·∈LF^∞0,T;Rd, we have p(·)δ2·-q¯2·, p(·)q2·∈LF^2(0,T;R). Further, because LF^2(0,T;R) is a Hilbert space, it follows from Cauchy-Schwarz’s inequality that (91)
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, and so (92)M+p0g120D+p0X0-g202-k1+p0g10X0-g202M+p0g120=p0g120E∫0Tptδ2t-q¯2t2+δ0t-q¯0t2dt+p0g10X0-g202+p0X0-g202E∫0Tptq2t2dt+E∫0Tptδ2t-q¯2t2dtE∫0Tptq2t2dt+E∫0Tptδ0t-q¯0t2dtE∫0Tptq2t2dt-k1+p0g10X0-g202≥p0g120E∫0Tptδ0t-q¯0t2dt+E∫0Tptδ0t-q¯0t2dtE∫0Tptq2t2dt≥0. Since M+p(0)g12(0)>0, we have (93)D+p0X0-g202-k1+p0g10X0-g202M+p0g120≥0.Remark 16.
Theorem15 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 k1+p(0)g1(0)(X0-g2(0))/M+p(0)g12(0) by choosing the optimal strategy.Remark 17.
Theorem15 also shows that the global minimal risk is (94)D+p0X0-g202-k1+p0g10X0-g202M+p0g120 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,u(·)=v(·)=0. Then, we have that δ(·)=0, (0,0) is the unique solution of BSDE (76) and the constants in Theorem 15 are given by (95)M=E∫0Tptq2t2dt,k1=0,k=-c+p0g10X01-M-p0g120,D=0. It follows from Theorem 15 that the efficient frontier in this case is given by (96)Var⁡X∗T=M+p0g1201-M-p0g120c-p0g10X0M+p0g1202+Mp0X02M+p0g120, 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 Lévy 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.



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*Source: 687428-2015-10-07.xml*