Continuous-Time Mean-Variance Portfolio Selection with Inflation in an Incomplete Market

This paper concerns 
a continuous-time portfolio selection problem with inflation in an incomplete 
market. By using the approach of more 
general stochastic linear quadratic control technique (SLQ), we obtain 
the optimal strategy and efficient frontier to this problem. Furthermore, a 
numerical example is also provided.


Introduction
Portfolio selection problem is a key topic in the modern finance. The seminal work of Markowitz (1952Markowitz ( , 1959 addressed the issue of allocation of wealth in order to obtain the optimal return-risk trade-off. Since then, the mean-variance model has been extended in many aspects. Merton (1969Merton ( , 1971) introduced a continuous-time model for maximizing the expected utility from investor's consumption and terminal wealth. Zhou & Li (2000) investigated a continuous-time mean-variance portfolio problem and obtained the optimal strategy and efficient frontier by using the stochastic LQ technique, which opened up possible approach to solve the problem for more constraints. Following Zhou & Li (2000), many scholars extended this model to the more complicated market situations, such as liability, bankruptcy prohibition and incomplete market. See more details in Bielecki, Jin, Pliska, & Zhou (2005), Xie, Li, & Wang (2008) and Ji (2010).
In a real world, investors must deal realistically with the problems of inflation with the growth of economy when adopting a long-term but finite horizon investment strategy. Therefore, the consideration of inflation risk in a portfolio selection model will make it more practical. However, to our knowledge, the research on mean-variance portfolio selection under inflation is limited. The existing literature on this topic is not much as can be seen Brennan & Xia (2002) and Bensoussan, Keppo, & Sethi (2009).
The main goal of this paper is to investigate a continuous-time portfolio selection problemunder inflation in an incomplete market. It is clear that this model is more suitable and practical in most of the real-world situations, especially for long-term investors. Therefore, our focus will be on two cases. On the one hand, we investigate the incomplete market with inflation, in which there are m risky assets and one risk-free asset. The price processes of risky assets are driven by an m-dimensional Brownian motion. We also assume that the inflation factors affected by the market are random, which can be described by m + 1 Brownian motion. In general, the changes in the nominal price index are not just correlated with the risky assets' nominal prices, but also with other uncertainties. It is reasonable that the other uncertainties can be represented by one Brownian motion, which is our (m + 1)-th Brownian motion. The original idea can be seen in Brennan & Xia (2002). On the other hand, we employ a stochastic linear quadratic (LQ) technique introduced by Zhou & Li (2000) to solve this problem. It should be pointed out that the introduction of inflation is by no means routine and does give rise to difficulties which are not encountered in Zhou & Li (2000). However, by using the more general stochastic LQ control technique in Yong & Zhou (1999), we can also obtain the optimal strategy and efficient frontier in closed forms.
The paper proceeds as follows. In Section 2, the model is formulated. Section 3 provides a closed-form solution of our model by using the more general stochastic LQ approach. Section 4 presents a numerical example. Finally, concluding remarks and suggestions for future work are given in Section 5.

Problem Formulation
We consider a market in which m + 1 assets are traded continuously within the time horizon [ ] 0,T . One of the assets is the risk-free whose nominal price process 0 t S is subject to the following ordinary differential equation: where 0 t r > is the nominal interest rate of the risk-free asset. The remaining m assets are risky and their nominal price processes 1 , , m t t S S  satisfy the following stochastic differential equations:  is a m-dimensional standard Brownian motion, which represents the random factors that affect risky assets' nominal prices. i t b is the appreciation rate of the ith ( ) is the volatility associated with the ith risky asset. Thus, the covariance matrix of risky assets is as follows: where the superscript " ′ " represents the transpose of a vector or a matrix. As widely adopted in the literature, we assume the non-degeneracy condition of The nominal price of real consumption goods in the economy at time t is denoted by t Ψ , which follows a diffusion process: where ( ) We denote by t X the nominal wealth of the investor at time a portfolio or a strategy of the investor. We assume that the trading of shares takes place continuously in a self-financing fashion and there are no transaction costs or taxes. We also assume that short-selling is allowable. Then we have is the risk premium. With the consideration of the inflation, the real value of any asset in the economy at time t is determined by deflating by the price index t Ψ . The real value of the investor's wealth is given by t t X Ψ . Let : Remark 2. In order to facilitate the following mathematical treatment, we give , , In fact, we can also think that: it is assumed that there are m + 1 risky assets in the market, and their nominal prices are driving by m + 1 Brownian motions. The first m risky assets are the same ones we assumed before, and the (m + 1)-th risky asset is a fictitious risky asset. The ith ( ) risky asset's volatility is given by the ith rank of t σ . Moreover, we assume the shares of the (m + 1)-th risky asset held by the investor remains 0. Therefore, we can get t π by deriving t π . The admissible strategy set under inflation with initial wealth x is defined as The objective of the investor is to maximize the expected terminal real wealth, T Eh , and at the same time to minimize the variance of the terminal real wealth, This is the mean-variance model which can be expressed by the bi-objective optimization problem: It is known from Li & Ng (2000) that Equation (9) is equivalent to the following single objective optimization problem: where the parameter 0 µ > represents the weight imposed by the investor on the objective is an optimal strategy of P P µ π π µ ∏ = . (11)

Solution to the Problem
In this section, we will apply the more general stochastic linear quadratic (LQ) control technique in Yong & Zhou (1999) to our model. Firstly, we will introduce a stochastic LQ auxiliary control problem and derive its optimal feedback control. Eventually the optimal portfolio strategy and the efficient frontier for the original mean-variance portfolio optimization problem under inflation are obtained in closed form.

Auxiliary Problem
Similar to Zhou & Li (2000), we introduce an auxiliary problem as follows: is an optimal control of , Recall Theorem 3.1 in Zhou & Li (2000) which shows the relationship between problems ( ) P µ and ( ) , A µ λ . (8) becomes the following stochastic differential equation:

Theorem 1. For any
, and the objective function of the auxiliary problem ( ) Thus the auxiliary problem ( ) , A µ λ is equivalent to the following stochastic LQ control problem:

Solution to the Auxiliary Problem
A solution of the stochastic LQ control problem ( ) A γ will involve, in an essential way, the following Riccati equation: along with the following adjoint ordinary differential equation:  (16) and (17), respectively, such that

Then Problem ( )
A γ is solvable with the optimal control being in a state feedback form, Moreover, the optimal cost value is is an optimal feedback control of state variable t Y . For any ( ) t y π ∈ Λ , let t Y be the state variable associated with the control vector t π . By applying Itô formula to 2 2 t t PY and t t g Y , and integrating them from 0 to T, taking expectations, add them together, we get and the equality holds if and only if ( ) . It shows that the feedback control given by Equation (18) is an optimal control and the optimal cost function can be obtained by Equation (19). The proof is completed.  Noting that the third constraint in Equation (16) Let t t t H g P = .
Then noting Equation (16) and Equation (17) Since the equivalence of problem ( ) A γ and ( ) , A µ λ , the optimal feedback control of the auxiliary problem ( ) , A µ λ is also given by Theorem 2: Substituting Equation (23) into Equation (24), we have

Solution to the Original Problem
Let * t h be the wealth process under the optimal feedback control * π of the auxiliary problem ( ) Applying Itô's formula to 2 t h * yields Taking expectation on both side of Equations (25) and (26), which leads respectively to Similarly, by solving Equation (28) we have Substituting Equation (23) Thus, the optimal feedback control of the problem ( ) P µ can be expressed by Correspondingly, the variance of the terminal wealth is and Equation (23) into Equation (33), we finally obtain the efficient frontier as follows: Obviously, the result of Zhou & Li (2000) is a special case in our paper.

Numerical Example
In this section, we discuss a numerical example. Suppose that the market has four assets and a risk-free asset. Let's assign the following parameters which are needed in our model: the risk-free asset 0.0350 After some transformations and calculations by using the above parameters, the efficient frontier of our model is obtained by the following.  Next, we compare our model with that of Zhou & Li (2000). The biggest difference is that our model is considered the factor of inflation in the decision-making process. Figure 1 shows the efficient frontier to continuous-time mean-variance model with and without inflation in a market. It can be seen that the frontier with inflation lies below the one without inflation. This means that the inflation plays as a penalty factor for portfolio revision. Furthermore, it tells us that the impact of it cannot be ignored in the real world when portfolio managers choose investment strategy.

Conclusion
This paper extends the work of Zhou & Li (2000) to an incomplete market with inflation. In our model, the inflation process is assumed to be a geometric Brownian motion, which is correlated with those risky assets. The driving factors of inflation are not the same ones which affect risky assets' prices. This means that the random factors affecting inflation include but do not equal to the ones of risky assets' prices. By using the more general stochastic LQ approach, we have provided a closed-form optimal strategy and efficient frontier. Comparing to Zhou & Li (2000), our results in this paper are more general. In addition, a numerical example is also provided. The search on the liability and bankruptcy prohibition in this problem is left for future work.