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This paper studies a continuous-time market under a stochastic environment where an agent, having specified an investment horizon and a target terminal mean return, seeks to minimize the variance of the return with multiple stocks and a bond. In the model considered here, the mean returns of individual assets are explicitly affected by underlying Gaussian economic factors. Using past and present information of the asset prices, a partial-information stochastic optimal control problem with random coefficients is formulated. Here, the partial information is due to the fact that the economic factors can not be directly observed. Using dynamic programming theory, we show that the optimal portfolio strategy can be constructed by solving a deterministic forward Riccati-type ordinary differential equation and two linear deterministic backward ordinary differential equations.

Mean-variance is an important investment decision rule in financial portfolio selection, which is first proposed and solved in the single-period setting by Markowitz in his Nobel-Prize-winning works [

The dynamic extension of the Markowitz model has been established in subsequent years by employing the martingale theory, convex duality and stochastic control. The pioneer work for continuous time portfolio management is [

In [

This paper attempts to deal with the mean-variance portfolio selection under partial information based on the model of [

The rest of the paper is organized as follows. In Section 2, we formulate the mean-variance portfolio selection model under partial information, and an auxiliary problem is introduced. Section 3 gives the optimal strategy of the auxiliary problem by the dynamic programming method. Section 4 studies the original problem, while Section 5 gives some concluding remarks.

Throughout this paper

There is a capital market containing

where

where

and the factors are Gaussian processes. To be precise, denoting

where the constant matrices

Consider an agent with an initial endowment

where

Let

As pointed out by [

Definition 2.1. A portfolio

The agent’s objective is to find an admissible portfolio

is minimized. The problem of finding such a portfolio

Definition 2.2. The mean-variance portfolio selection problem, with respect to the initial wealth

mulated as a constrained stochastic optimization problem parameterized by

The problem is called feasible (with respect to

We impose the basic assumption:

Assumption (PD). For any

Let

with

By the definition of

By Itô’s formula we have

Define

then

By (2.7), a simple calculation shows that

Substituting (2.9), we have an equivalent representation of the wealth process

where

This is the separation principle developed by [

By general convex optimization theory, the constrained optimal problem (12) with

which is equivalent to the following (denoting

in the sense that two problems have exactly the same optimal strategy. In the following, we will call problem (2.14) the auxiliary problem of the original problem (2.12).

The problem (2.14) can be viewed as an unconstrained special stochastic optimal control problem with random coefficients in system equation and zero integral term in the performance index. Different from existing results using BSDEs methodology, in this section, we derive the optimal portfolio strategy from dynamic programming directly. This enables us to derive the optimal policy by solving just two linear deterministic backward ODEs and a Riccati-type forward deterministic ODE.

Let

where

To evaluate

By Itô’s formula, it follows that

where

which makes

In this and the following PDEs and ODEs, the arguments

Noticing that the terminal condition of

Simple calculation shows

Substituting

By the special structure of (3.5), the following separation form of

which will be proved in Theorem 3.1. Therefore, the optimal control (3.2) has the following structure

which is linear in

Clearly, if

then

Notice that the left hand side of the first equation in (3.8) is linear in

with

Theorem 3.1. For problem (2.14), the optimal strategy is given by

where

Proof. Bearing the form (3.9) of

Therefore, (3.8) is equivalent to

which is equivalent to

(3.14)

The left hand of above PDE can be decomposed into three terms:

1) the term that is irrespective of

2) the term that is linear in

3) the term that is quadratic in

So, if the

we can determine the function

Thus,

by innovation process

completes the proof.

We will give a brief discussion about the solvability in theory of (2.8) (3.11) (3.12). Clearly,

with

with

where

Therefore,

Clearly, (3.12) is a Lyapunov differential equation, which is solved by introducing the following operator

where

where

Let

Therefore,

where

In this section, we proceed to derive the efficient frontier for the original portfolio selection problem under partial information. To begin with, we prove a lemma which shows the feasibility of the original problem.

Lemma 4.1. Problem (5) is feasible, and the minimal mean-variance of the terminal wealth process is finite.

Proof. The proof follows directly from results of Section 5 in [

where

Clearly,

Now, we state our main theorem.

Theorem 4.1. The efficient strategy of Problem (2.5) with the terminal expected wealth constraint

Here,

where

and

Proof. By Lemma 4.1, we know that the constraint Problem (2.5) is feasible, and its minimal terminal mean- variance

where the equality is true by general convex constraint optimization theory (see, for example, [

In terms of

Clearly,

Thus

where

For any fixed

To obtain the optimal mean-variance value and the optimal portfolio strategy of Problem (2.5), we should maximize (4.6) over

And we can assert that

If this is not true, the optimal cost will be infinite, which contradicts (4.5).

In this paper, we have studied the continuous-time mean-variance portfolio selection problem with stochastic drifts. In particular, drifts are assumed to be linear functions of economic factor processes. Because the factor processes cannot be observed directly, partial information is assumed together with a filter process. Conse- quently, by dynamic programming technique and the method of separation of variables, we have derived the explicit optimal strategy via the solution of a system of ODEs. As a future extension, it would be of interest to study the solutions with real financial data and carry out appropriate economic analysis. Also, regime-switching model [

This work is supported by the PolyU grant G-YL05 and A-PL62, and the JRI of the Department of Applied Mathematics, The Hong Kong Polytechnic University.