An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

Abstract

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.

1. Introduction

The first treatment of the investment and consumption problem in a continuous-time framework originated in the seminal papers of Merton [1, 2]. Merton used dynamic programming principle to construct an explicit optimal portfolio for power and logarithm utility. Later on, this problem attracted more and more attention and inspired literally hundreds of extensions and applications. For example, Fleming and Zariphopoulou [3] and Vila and Zariphopoulou [4] studied optimal consumption and portfolio decisions with borrowing constraints to maximize the total expected discounted utility coming from consumption; Duffie et al. [5] investigated the optimal investment and consumption strategy in incomplete markets; Dumas and Luciano [6], Shreve and Soner [7], Liu and Loewenstein [8], and Dai et al. [9] focused on the investment and consumption problem with transaction costs under different market assumptions. But these models were investigated under the assumption that risk-free interest rate and the volatility of the stock were supposed to be constants.

However, two aspects are worthy to be further explored based on the above-mentioned literature. On the one hand, risk-free interest rate in most of the above-mentioned literature is assumed to be constant, which is contrary to the fact. Interest rate is not always fixed in our real life. There is much literature documenting the term structure of interest rate, such as Ho and Lee [10], Vasicek [11], and Cox et al. [12]. Therefore, many investors find that the optimal portfolios with stochastic interest rates are more practical. In recent years, many scholars have studied the portfolio selection problem under stochastic interest rate models. Korn and Kraft [13] and Grasselli [14] considered the optimal portfolios under Ho-Lee model and Vasicek model and CIR interest rate dynamics, respectively, and obtained the explicit solutions under the expected utility criterion. Deelstra et al. [15], Josa-Fombellida and Rincón-Zapatero [16], and Gao [17] investigated the investment strategies for pension funds with stochastic interest rate in different situations. Deelstra et al. [18] analyzed the optimal investment problem in a CIR framework and derived the closed-form solution of the optimal investment and consumption strategy by assuming the completeness of financial markets. Fleming and Pang [19] applied stochastic optimal control theory to an investment and consumption problem with stochastic interest rate and wished to maximize the total expected discounted utility of consumption but have no ways to obtain the explicit expressions and only prove the existence of solution by using subsupersolution method. Munk and Sørensen [20] characterized the solution to the investment and consumption problem with stochastic changes in the opportunity set and suggested that the hedge portfolio is more sensitive to the form of the term structure than to the dynamics of interest rate.

On the other hand, there is more and more attention paid to the portfolio selection problem with stochastic volatility (SV). Stochastic volatility has been recognized as an important factor of stock dynamics and can demonstrate many well-known empirical features, such as volatility smile and volatility clustering. Heston's SV model [21] is the simplest but important one. Kraft [22] studied the optimal portfolio with stochastic volatility and obtained the closed-form solution. Fleming and Hernández-Hernández [23] and Chacko and Viceira [24] analyzed an investment and consumption problem with SV in different market models, respectively. Li et al. [25] considered SV model in the insurance market and obtained the closed-form solution to the optimal investment and insurance strategies in a mean-variance framework. Sasha and Zariphopoulou [26], Castañeda-leyva and Hernández-Hernández [27], and Liu [28] applied different methods to investigate an investment and consumption model in different stochastic environments, where interest rate and return rate and volatility are supposed to be stochastic. Michelbrink and Le [29] proposed a martingale approach to tackle a consumption model with jump diffusions and obtained the optimal investment and consumption strategies. Li and Wu [30] assumed that risk-free interest rate is driven by the CIR model and the volatility of the stock is described by Heston's SV model and obtained the optimal investment strategy to maximize the power utility of terminal wealth. Noh and Kim [31] investigated an investment and consumption model with stochastic interest rate and stochastic volatility, where interest rate model and the volatility model are correlated with the stock price. Noh and Kim analyzed the existence of the optimal portfolio, but did not obtain the closed-form solution. This paper considers one optimal consumption problem based on Li and Wu's model and uses the same approach as Liu [28] to solve corresponding nonlinear partial different equation. In addition, we obtain the explicit expressions of the optimal investment and consumption strategy in the power and logarithm utility cases.

As far as we know, there is little work in the literature on the investment and consumption problem with stochastic interest rate and stochastic volatility, and only Noh and Kim [31] studied this problem so far and choose the total expected discounted utility coming from consumption as the objective function. Therefore, in this paper, we assume that risk-free interest rate is driven by the CIR model and the volatility of stock is described by Heston's stochastic volatility model. Our objective is to maximize the expected discounted utility of consumption and terminal wealth. Firstly, we use dynamic programming principle to obtain the HJB equation for the value function and choose power utility and logarithm utility as our analysis. Secondly, by applying separate variable approach and variable change technique, we obtain the closed-form solutions to the optimal investment and consumption strategy. A numerical example is given to illustrate the results obtained and analyze the impact of market parameters on the optimal investment and consumption strategy. This paper has three main contributions: (i) the optimal investment and consumption strategies under stochastic interest rate model and stochastic volatility model are studied, which is the more important extension of the work of Li and Wu [30]; (ii) the expected discounted utility of consumption and terminal wealth is chosen as the objective function, which is different from Noh and Kim [31], and our objective function is more practical; (iii) we use the same approach as Liu [28] to solve the nonlinear second-order partial different equation such that the closed-form solutions are obtained, which is the most important innovation of this paper.

The remainder of this paper is organized as follows. The financial market and wealth process and optimization criterion are presented in Section 2. In Section 3, we use dynamic programming principle to derive the HJB equation for the value function and obtain the closed-form solutions to the optimal investment and consumption strategy in the power and logarithm cases. In Section 4, we propose numerical illustration and sensitivity analysis. Section 5 concludes the paper.

2. Problem Formulation

Assume that there are no transaction costs or taxes in the financial market and all assets can be traded continuously. (𝑊₁(𝑡),𝑊₂(𝑡)) is a two-dimensional well-defined and independent adapted Brownian motion in a given filtered complete probability space (Ω,ℱ,{ℱ_𝑡}_{𝑡∈[0,𝑇]},𝑃), where 𝑇 is a positive constant representing fixed and finite investment horizon; {ℱ_𝑡}_{𝑡∈[0,𝑇]} is an information filtration generated by (𝑊₁(𝑡),𝑊₂(𝑡)).

2.1. The Financial Market

Assume that the financial market is composed of two assets: one risk-free asset (e.g., a bank account) and one risky asset (e.g., a stock). The price process of the risk-free asset at time 𝑡 is denoted by 𝐵(𝑡), which satisfies 𝑑𝐵(𝑡)=𝑟(𝑡)𝐵(𝑡)𝑑𝑡, (1) where 𝑟(𝑡) is the stochastic process and is supposed to be driven by the CIR model 𝑑𝑟(𝑡)=(𝜃−𝑐𝑟(𝑡))𝑑𝑡+𝜎₀𝑟(𝑡)𝑑𝑊₁(𝑡),𝑟(0)=𝑟₀>0, (2) where 𝜃, 𝑐, and 𝜎₀ are positive constants satisfying 2𝜃>𝜎²₀. It is well known that 𝑟(𝑡)>0 for all 𝑡≥0 (see Cox et al. [12]).

The price process 𝑆(𝑡) of the risky asset is as follows: 𝑑𝑆(𝑡)=𝑆(𝑡)𝑟(𝑡)+𝑘𝜂(𝑡)𝑑𝑡+𝜎₁𝜂(𝑡)𝑑𝑊₂(𝑡,𝑆(0))=𝑠₀>0, (3) where 𝑘 and 𝜎₁ are positive constants, and 𝜂(𝑡) is driven by another CIR model 𝑑𝜂(𝑡)=𝑏−𝑎𝜂(𝑡)𝑑𝑡+𝜎₂𝜂(𝑡)𝑑𝑊₂(𝑡),𝜂(0)=𝜂₀>0, (4) where 𝑏, 𝑎, and 𝜎₂ are positive constants satisfying 2𝑏>𝜎²₂. We have also 𝜂(𝑡)>0 for all 𝑡≥0.

2.2. The Wealth Process

During the time horizon [0,𝑇], we suppose that the initial wealth of an investor is denoted by 𝑥₀>0, and the wealth process at time 𝑡 is denoted by 𝑋(𝑡). A trading strategy is a pair of stochastic processes denoted by (𝜋(𝑡),𝐶(𝑡)), where 𝜋(𝑡) is the amount invested in the stock, and 𝐶(𝑡) is the consumption rate. The amount invested in the risk-free asset is 𝑋(𝑡)−𝜋(𝑡). Therefore, wealth process evolves according to the following stochastic differential equation: (𝑡)=(𝑋(𝑡)(𝑡(𝑡)𝑑𝑋−𝜋))𝑑𝐵𝐵(𝑡)+𝜋_𝑡𝑑𝑆(𝑡)𝑆(𝑡)(𝑡)𝑋(0)−𝐶𝑑𝑡,=𝑥₀>0. (5) Considering (1) and (3), we get (𝑡)=𝑟(𝑡)𝑋(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)𝜎𝑑𝑋+𝜋𝑘𝜂−𝐶𝑑𝑡+𝜋₁𝜂(𝑡)𝑑𝑊₂(𝑡),𝑋(0)=𝑥₀>0. (6)

2.3. The Optimization Criterion

Definition 1 (admissible strategy). An investment and consumption strategy (𝜋(𝑡),𝐶(𝑡)) is said to be admissible if the following conditions are satisfied.(i)(𝜋(𝑡),𝐶(𝑡)) is progressively measurable.(ii)𝐸[∫^𝑇₀(𝜋(𝑡)𝜎₁𝜂(𝑡))²𝑑𝑡]<∞.(iii)For all initial conditions (𝑡₀,𝑟₀,𝜂₀,𝑥₀)∈[0,𝑇]×(0,∞)³, the wealth process 𝑋(𝑡) with 𝑋(0)=𝑥₀>0 has a pathwise unique solution.

Assume that the set of all admissible strategies is denoted by Γ={(𝜋(𝑡),𝐶(𝑡))∶𝑡∈[0,𝑇]}. The investor wishes to maximize the expected discounted utility of the intermediate consumption and terminal wealth. Mathematically, the objective function is expressed as Maximize_{(𝜋(𝑡),𝐶(𝑡))∈Γ}𝐸𝛼^𝑇₀𝑒^−𝛽𝑡𝑈₁(𝐶(𝑡+()𝑒))𝑑𝑡1−𝛼^−𝛽𝑇𝑈₂(𝑋(𝑇)), (7) where 𝑈₁(⋅) and 𝑈₂(⋅) are all utility functions, which satisfy the following conditions: the first-order derivative is greater than zero and the second-order derivative is less than zero. The parameter 𝛽 is the subjective discount rate and 𝛼 determines the relative importance of the intermediate consumption and the bequest. When 𝛼=0, the expected utility only depends on the terminal wealth and the problem (7) is reduced to an asset allocation problem.

3. The Optimal Portfolios

In this section, we apply dynamic programming principle to obtain the HJB equation of the value function and study the optimal investment and consumption strategy under power utility and logarithm utility function. By using variable change technique, we obtain the closed-form solutions of the optimal portfolios.

The value function is defined as 𝐻𝑡,𝑟,𝜂,𝑥=Sup_{(𝜋𝑡,𝐶𝑡)∈Γ}𝐸𝛼^𝑇₀𝑒^−𝛽𝑡𝑈₁(𝐶(𝑡+()𝑒))𝑑𝑡1−𝛼^−𝛽𝑇𝑈₂(𝑋(𝑇∣𝑋(𝑡)(𝑡)(𝑡)))=𝑥,𝑟=𝑟,𝜂=𝜂 (8) with boundary condition 𝐻(𝑇,𝑟,𝜂,𝑥)=(1−𝛼)𝑒^−𝛽𝑇𝑈₂(𝑥).

Dynamic programming principle leads to the following HJB equation for 𝐻(𝑡,𝑟,𝜂,𝑥): Sup_{(𝜋𝑡,𝐶𝑡)∈Γ}𝐻_𝑡+𝑟𝑥+𝜋(𝑡)𝑘𝜂−𝐶(𝑡)𝐻_𝑥+12𝜋²(𝑡)𝜎²₁𝜂𝐻_𝑥𝑥+(𝜃−𝑐𝑟)𝐻_𝑟+12𝜎²₀𝑟𝐻_𝑟𝑟+𝑏−𝑎𝜂𝐻_𝜂+12𝜎²₂𝜂𝐻_𝜂𝜂+𝜋(𝑡)𝜎₁𝜎₂𝐻_𝑥𝜂+𝛼𝑒^−𝛽𝑡𝑈₁(𝐶(𝑡))=0, (9) where 𝐻_𝑡, 𝐻_𝑥, 𝐻_𝑥𝑥, 𝐻_𝑟, 𝐻_𝑟𝑟, 𝐻_𝜂, 𝐻_𝜂𝜂, and 𝐻_𝑥𝜂 denote partial derivatives of first-order and second-order with respect to the time 𝑡, wealth process 𝑋(𝑡), interest rate 𝑟(𝑡), and volatility 𝜂(𝑡). We also use similar notation for higher derivatives and the derivatives of other functions.

The first-order maximizing conditions for the optimal investment and consumption strategy are 𝜋^∗(𝑡)𝑘=−𝜎²₁⋅𝐻_𝑥𝐻_𝑥𝑥−𝜎₂𝜎₁⋅𝐻_𝑥𝜂𝐻_𝑥𝑥,𝑈^′₁𝐶^∗(𝑡)=𝐻_𝑥𝛼𝑒^−𝛽𝑡. (10) Putting (10) in (9), we obtain the HJB equation as follows: 𝐻_𝑡+𝑟𝑥𝐻_𝑥+(𝜃−𝑐𝑟)𝐻_𝑟+12𝜎²₀𝑟𝐻_𝑟𝑟+𝑏−𝑎𝜂𝐻_𝜂+12𝜎²₂𝜂𝐻_𝜂𝜂−𝑘²𝜂2𝜎²₁⋅𝐻²_𝑥𝐻_𝑥𝑥−𝑘𝜂𝜎₂𝜎₁⋅𝐻_𝑥𝐻_𝑥𝜂𝐻_𝑥𝑥−12𝜎²₂𝜂𝐻²_𝑥𝜂𝐻_𝑥𝑥−𝐶^∗(𝑡)𝐻_𝑥+𝛼𝑒^−𝛽𝑡𝑈₁𝐶^∗(𝑡)=0. (11)

In the following subsection, we choose power utility and logarithm utility function as our analysis and apply variable change technique to solve (11).

3.1. Power Utility

Power utility is given by 𝑈₁(𝑥)=𝑈₂(𝑥)=𝑥^𝛿𝛿,𝛿<1,𝛿≠0, (12) where 𝛿 is the risk aversion factor.

We conjecture a solution to (11) with the following form: 𝐻𝑡,𝑟,𝜂,𝑥=𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓𝑡,𝑟,𝜂,𝑓𝑇,𝑟,𝜂=1−𝛼. (13) Then the partial derivatives are as follows: 𝐻_𝑡=−𝛽𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓+𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝑡,𝐻_𝑥𝑥=(𝛿−1)𝑒^−𝛽𝑡𝑥^𝛿−2𝐻𝑓,_𝑥=𝑒^−𝛽𝑡𝑥^𝛿−1𝐻𝑓,_𝑟=𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝑟,𝐻_𝑟𝑟=𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝑟𝑟,𝐻_𝜂=𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝜂,𝐻_𝜂𝜂=𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝜂𝜂,𝐻_𝑥𝜂=𝑒^−𝛽𝑡𝑥^𝛿−1𝑓_𝜂. (14) Therefore, the optimal consumption strategy is given by 𝐶^∗(𝑡)=𝛼^{−1/(𝛿−1)}𝑓^1/(𝛿−1)𝑥. (15)

Introducing (14) and (15) into (11), we obtain 𝑒^−𝛽𝑡𝑥^𝛿𝛿𝑓_𝑡+𝑟𝛿−𝛽−𝛿𝑘²𝜂2(𝛿−1)𝜎²₁𝑓+(𝜃−𝑐𝑟)𝑓_𝑟+12𝜎²₀𝑟𝑓_𝑟𝑟+𝑏−𝑎𝜂−𝛿𝑘𝜂𝜎₂(𝛿−1)𝜎₁𝑓_𝜂+12𝜎²₂𝜂𝑓_𝜂𝜂−𝛿𝜎²₂𝜂2()𝑓𝛿−1𝑓^2𝜂+(1−𝛿)𝛼^1/(𝛿−1)𝑓^{𝛿/(𝛿−1)}=0. (16) Eliminating the dependence on 𝑥, we get 𝑓_𝑡+𝑟𝛿−𝛽−𝛿𝑘²𝜂2(𝛿−1)𝜎²₁𝑓+(𝜃−𝑐𝑟)𝑓_𝑟+𝑏−𝑎𝜂−𝛿𝑘𝜂𝜎₂(𝛿−1)𝜎₁𝑓_𝜂+12𝜎²₀𝑟𝑓_𝑟𝑟+12𝜎²₂𝜂𝑓_𝜂𝜂−𝛿𝜎²₂𝜂2()𝑓𝛿−1𝑓^2𝜂+(1−𝛿)𝛼^1/(𝛿−1)𝑓^{𝛿/(𝛿−1)}=0. (17)

Suppose that the solution structure of (17) is of the form 𝑓𝑡,𝑟,𝜂=𝑔𝑡,𝑟,𝜂^1−𝛿,𝑔𝑇,𝑟,𝜂=(1−𝛼)^1/(𝛿−1). (18) Then, we have 𝑓_𝑡=(1−𝛿)𝑔^−𝛿𝑔_𝑡,𝑓_𝑟=(1−𝛿)𝑔^−𝛿𝑔_𝑟,𝑓_𝜂=(1−𝛿)𝑔^−𝛿𝑔_𝜂,𝑓_𝑟𝑟=(1−𝛿)(−𝛿)𝑔^−𝛿−1𝑔²_𝑟+(1−𝛿)𝑔^−𝛿𝑔_𝑟𝑟,𝑓_𝜂𝜂=(1−𝛿)(−𝛿)𝑔^−𝛿−1𝑔²_𝜂+(1−𝛿)𝑔^−𝛿𝑔_𝜂𝜂. (19)

Substituting (19) into (17), after some simplification, we have (1−𝛿)𝑔^−𝛿𝑔_𝑡+𝛿𝑟−𝛽+𝛿1−𝛿(𝛿−1)²⋅𝑘²𝜂2𝜎²₁𝑔+(𝜃−𝑐𝑟)𝑔_𝑟+12𝜎²₀𝑟𝑔_𝑟𝑟−12𝜎²₀𝑔𝑟𝛿^2𝑟𝑔+𝛿𝑏−𝑎𝜂−⋅𝛿−1𝑘𝜂𝜎₂𝜎₁𝑔_𝜂+12𝜎²₂𝜂𝑔_𝜂𝜂+𝛼^1/(1−𝛿)=0. (20) This leads to another equation for 𝑔 as 𝑔_𝑡+𝛿𝑟−𝛽+𝛿1−𝛿(𝛿−1)²⋅𝑘²𝜂2𝜎²₁𝑔+(𝜃−𝑐𝑟)𝑔_𝑟+12𝜎²₀𝑟𝑔_𝑟𝑟−12𝜎²₀𝑔𝑟𝛿^2𝑟𝑔+𝛿𝑏−𝑎𝜂−⋅𝛿−1𝑘𝜂𝜎₂𝜎₁𝑔_𝜂+12𝜎²₂𝜂𝑔_𝜂𝜂+𝛼^1/(1−𝛿)=0. (21)

Equation (21) is a nonlinear second-order partial differential equation which is difficult to solve directly, for there exists the term 𝛼^1/(1−𝛿). Inspired by the paper of Liu [28], we assume that 𝑔 is of the following form: 𝑔𝑡,𝑟,𝜂=𝛼^{1/(1−𝛿)𝑇}_𝑡𝑔+()𝑢,𝑟,𝜂𝑑𝑢1−𝛼^1/(1−𝛿)𝑔𝑡,𝑟,𝜂. (22) One can prove that (21) is reduced to 𝑔_𝑡+𝛿𝑟−𝛽+𝛿1−𝛿(𝛿−1)²⋅𝑘²𝜂2𝜎²₁𝑔+(𝜃−𝑐𝑟)𝑔_𝑟+12𝜎²₀𝑟𝑔_𝑟𝑟−12𝜎²₀𝑔𝑟𝛿^2𝑟𝑔+𝛿𝑏−𝑎𝜂−⋅𝛿−1𝑘𝜂𝜎₂𝜎₁𝑔_𝜂+12𝜎²₂𝜂𝑔_𝜂𝜂=0,𝑔𝑇,𝑟,𝜂=1. (23)

In fact, if we define a differential operator ∇ on any function 𝑔 by ∇𝑔=𝛿𝑟−𝛽+𝛿1−𝛿(𝛿−1)²⋅𝑘²𝜂2𝜎²₁𝑔+(𝜃−𝑐𝑟)𝑔_𝑟+12𝜎²₀𝑟𝑔_𝑟𝑟−12𝜎²₀𝑔𝑟𝛿^2𝑟𝑔+𝛿𝑏−𝑎𝜂−⋅𝛿−1𝑘𝜂𝜎₂𝜎₁𝑔_𝜂+12𝜎²₂𝜂𝑔_𝜂𝜂. (24) Then (23) can be rewritten as 𝜕𝑔𝑔𝜕𝑡+∇𝑔=0,𝑇,𝑟,𝜂=1. (25)

Furthermore, we find 𝜕𝑔𝜕𝜕𝑡+∇𝑔=𝛼𝜕𝑡^{1/(1−𝛿)𝑇}_𝑡𝑔𝛼𝑢,𝑟,𝜂𝑑𝑢+∇^{1/(1−𝛿)𝑇}_𝑡𝑔+()𝑢,𝑟,𝜂𝑑𝑢1−𝛼^1/(1−𝛿)𝜕𝑔𝑔.𝜕𝑡𝑡,𝑟,𝜂+∇𝑡,𝑟,𝜂 (26) Taking (25) into consideration, we have 𝜕𝑔𝜕𝑡+∇𝑔=−𝛼^1/(1−𝛿)𝑔𝑡,𝑟,𝜂+𝛼^{1/(1−𝛿)𝑇}_𝑡∇𝑔𝑢,𝑟,𝜂𝑑𝑢=−𝛼^1/(1−𝛿)𝑔𝑡,𝑟,𝜂−𝛼^{1/(1−𝛿)𝑇}_𝑡𝜕𝑔𝑢,𝑟,𝜂𝜕𝑢𝑑𝑢=−𝛼^1/(1−𝛿)𝑔𝑡,𝑟,𝜂−𝛼^1/(1−𝛿)𝑔𝑇,𝑟,𝜂−𝑔𝑡,𝑟,𝜂=−𝛼^1/(1−𝛿). (27)

Therefore, we obtain 𝜕𝑔𝜕𝑡+∇𝑔+𝛼^1/(1−𝛿)=0,𝑔𝑇,𝑟,𝜂=(1−𝛼)^1/(1−𝛿). (28) This is to say that (21) holds.

For (23), we can fit a solution of the following form: 𝑔𝑡,𝑟,𝜂=exp𝜙(𝑡)𝜂+ℎ(𝑡)𝑟+Ψ(𝑡) (29) with boundary condition 𝜙(𝑇)=ℎ(𝑇)=Ψ(𝑇)=0.

Introducing (29) into (23) yields 𝑔𝜂𝜙^′(𝑡)+𝛿(𝛿−1)²⋅𝑘²2𝜎²₁−𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁𝜙(𝑡)+12𝜎²₂𝜙²(𝑡ℎ)+𝑟^′(𝑡)+𝛿(𝑡)+11−𝛿−𝑐ℎ2(1−𝛿)𝜎²₀ℎ²(𝑡)+Ψ^′(𝑡)(𝑡)(𝑡)−𝛽+𝑏𝜙+𝜃ℎ1−𝛿=0. (30) We can decompose (30) into the following three equations in order to eliminate the dependence on 𝑟 and 𝜂: 𝜙^′(𝑡)+𝛿(𝛿−1)²⋅𝑘²2𝜎²₁−𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁𝜙(𝑡)+12𝜎²₂𝜙²(𝑡)(𝑇)ℎ=0,𝜙=0,_′(𝑡)+𝛿(𝑡)+11−𝛿−𝑐ℎ2(1−𝛿)𝜎²₀ℎ²(𝑡)(𝑇)Ψ=0,ℎ=0,_′(𝑡)(𝑡)(𝑡)−𝛽+𝑏𝜙+𝜃ℎ(𝑇)1−𝛿=0,Ψ=0. (31)(32)(33)

Rewritting (31), we have 𝜙^′(𝑡)1=−2𝜎²₂𝜙²(𝑡)−2𝜎²₂𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁𝜙(𝑡)+2𝜎²₂⋅𝛿(𝛿−1)²⋅𝑘²2𝜎²₁. (34) Let Δ_𝜙 denote the discriminant of the quadratic equation 𝜙²(𝑡)−2𝜎²₂𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁𝜙(𝑡)+2𝜎²₂⋅𝛿(𝛿−1)²⋅𝑘²2𝜎²₁=0 (35) and then we have Δ_𝜙=4𝜎⁴₂−𝑎²+𝛿𝛿−1𝛿−1𝑎+𝑘𝜎₂𝜎₁². (36) Suppose that Δ_𝜙>0, that is, 𝑎𝛿<²𝑎+𝑘𝜎₂/𝜎₁²<1. (37)

Under the condition (37), integrating (34) on both sides with respect to 𝑡, we obtain 1𝜆₁−𝜆₂^𝑇_𝑡1𝜙(𝑡)−𝜆₁−1𝜙(𝑡)−𝜆₂(𝑡)1𝑑𝜙=−2𝜎²₂(𝑇−𝑡), (38) where 𝜆₁ and 𝜆₂ are two real roots of (35), namely, 𝜆_1,2=1𝜎²₂𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁±1𝜎²₂−𝑎²+𝛿𝛿−1𝛿−1𝑎+𝑘𝜎₂𝜎₁². (39) Solving (38) with boundary condition 𝜙(𝑇)=0, we derive 𝜙(𝑡)=𝜆₁𝜆₂1−exp−(1/2)𝜎²₂𝜆₁−𝜆₂(𝑇−𝑡)𝜆₁−𝜆₂exp−(1/2)𝜎²₂𝜆₁−𝜆₂(.𝑇−𝑡) (40)

For (32), using the same analysis as 𝜙(𝑡), we obtain ℎ(𝑡)=𝜆₃𝜆₄1−exp−((1−𝛿)/2)𝜎²₀𝜆₃−𝜆₄(𝑇−𝑡)𝜆₃−𝜆₄exp−((1−𝛿)/2)𝜎²₀𝜆₃−𝜆₄(𝑇−𝑡) (41) under the condition 𝑐𝛿<²2𝜎²₀, (42) where 𝜆₃ and 𝜆₄ are two real roots of the equation ℎ²(𝑡)−2𝑐(1−𝛿)𝜎²₀ℎ(𝑡)+𝛿⋅21−𝛿(1−𝛿)𝜎²₀=0. (43) Easy calculation leads to 𝜆_3,4=1(1−𝛿)𝜎²₀±𝑐²−2𝛿𝜎²₀(1−𝛿)𝜎²₀. (44)

For (33), we have Ψ(𝑡)=𝑏^𝑇_𝑡𝜙(𝑠)𝑑𝑠+𝜃^𝑇_𝑡ℎ(𝑠)𝛽𝑑𝑠−().1−𝛿𝑇−𝑡 (45) Combining (37) with (42), we find that the risk aversion factor 𝛿 should satisfy 𝑎𝛿<Min²𝜎²₁𝑎𝜎₁+𝑘𝜎₂²,𝑐²2𝜎²₀,𝛿≠0. (46)

According to (14) and (19), we have 𝐻_𝑥𝐻_𝑥𝑥=1𝐻𝛿−1𝑥,^𝑥𝜂𝐻_𝑥𝑥𝑔=−^𝜂𝑔𝑥. (47)

Finally, taking (10) and (15) into consideration, we obtain the optimal investment and consumption strategy under the power utility function.

Proposition 2. If utility function 𝑈₁(𝑥)=𝑈₂(𝑥)=𝑥^𝛿/𝛿, with the conditions 𝛿<Min{𝑎²𝜎²₁/(𝑎𝜎₁+𝑘𝜎₂)²,𝑐²/2𝜎²₀} and 𝛿≠0, the optimal investment and consumption strategy of the problem (7) is given by 𝜋^∗_𝑡=𝑘(1−𝛿)𝜎²₁𝑋_𝑡+𝜎₂𝜎₁⋅𝑔_𝜂𝑔𝑋_𝑡,𝐶^∗(𝑡)=𝛼^1/(1−𝛿)𝑔⁻¹𝑋_𝑡, (48) where 𝑔=𝑔𝑡,𝑟,𝜂=𝛼^{1/(1−𝛿)𝑇}_𝑡𝜙(𝑠)(𝑠)(𝑠+()exp𝜂+ℎ𝑟+Ψ)𝑑𝑠1−𝛼^1/(1−𝛿)exp𝜙(𝑡)𝜂+ℎ(𝑡)𝑟+Ψ(𝑡). (49) In addition, 𝜙(𝑡), ℎ(𝑡), and Ψ(𝑡) are determined by (40), (41) and (45), respectively.

In order to compare the results obtained by Proposition 2 with those in the existing literatures, we consider the following five special cases in the power utility case.

Special Case 1. If 𝜃=𝑐=𝜎₀=0, this is to say that risk-free interest rate 𝑟(𝑡) will become a constant. Then the problem (7) is degenerated into an investment and consumption problem with stochastic volatility. After easy calculation, we obtain 𝜋^∗_𝑡=𝑘(1−𝛿)𝜎²₁𝑋_𝑡+𝜎₂𝜎₁⋅𝑔_𝜂𝑔𝑋_𝑡,𝐶^∗(𝑡)=𝛼^1/(1−𝛿)𝑔⁻¹𝑋_𝑡 (50) under the conditions 𝛿<𝑎²𝜎²₁/(𝑎𝜎₁+𝑘𝜎₂)² and 𝛿≠0, where 𝜆_1,2=1𝜎²₂𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁±1𝜎²₂−𝑎²+𝛿𝛿−1𝛿−1𝑎+𝑘𝜎₂𝜎₁²,𝜙(𝑡)=𝜆₁𝜆₂1−exp−(1/2)𝜎²₂𝜆₁−𝜆₂(𝑇−𝑡)𝜆₁−𝜆₂exp−(1/2)𝜎²₂𝜆₁−𝜆₂(,ℎ(𝑡)=𝛿𝑇−𝑡)(),Ψ(𝑡)1−𝛿𝑇−𝑡=𝑏^𝑇_𝑡𝜙(𝑠)𝛽𝑑𝑠−(),1−𝛿𝑇−𝑡𝑔=𝑔𝑡,𝑟,𝜂=𝛼^{1/(1−𝛿)𝑇}_𝑡𝜙(𝑠)(𝑠)(𝑠+()exp𝜂+ℎ𝑟+Ψ)𝑑𝑠1−𝛼^1/(1−𝛿)exp𝜙(𝑡)𝜂+ℎ(𝑡)𝑟+Ψ(𝑡). (51)

Special Case 2. If 𝑎=𝑏=𝜎₂=0, namely, volatility term 𝜂(𝑡) reduces to a constant, then the problem (7) becomes an investment and consumption problem with stochastic interest rate. Therefore, we have 𝜋^∗_𝑡=𝑘(1−𝛿)𝜎²₁𝑋_𝑡,𝐶^∗(𝑡)=𝛼^1/(1−𝛿)𝑔⁻¹𝑋_𝑡 (52) under the conditions 𝛿<Min{𝑐²/2𝜎²₀,1} and 𝛿≠0, where 𝑔=𝑔𝑡,𝑟,𝜂=𝛼^{1/(1−𝛿)𝑇}_𝑡𝜙(𝑠)(𝑠)(𝑠+()exp𝜂+ℎ𝑟+Ψ)𝑑𝑠1−𝛼^1/(1−𝛿)exp𝜙(𝑡)𝜂+ℎ(𝑡)𝑟+Ψ(𝑡) (53) with 𝜙(𝑡)=𝛿(𝛿−1)²⋅𝑘²2𝜎²₁(),Ψ(𝑡)𝑇−𝑡=𝜃^𝑇_𝑡ℎ(𝑠)𝛽𝑑𝑠−(),𝜆1−𝛿𝑇−𝑡_3,4=1(1−𝛿)𝜎²₀±𝑐²−2𝛿𝜎²₀(1−𝛿)𝜎²₀,ℎ(𝑡)=𝜆₃𝜆₄1−exp−((1−𝛿)/2)𝜎²₀𝜆₃−𝜆₄(𝑇−𝑡)𝜆₃−𝜆₄exp−((1−𝛿)/2)𝜎²₀𝜆₃−𝜆₄(.𝑇−𝑡) (54)

Special Case 3. If 𝜃=𝑐=𝜎₀=0 and 𝑎=𝑏=𝜎₂=0, that is, risk-free interest rate 𝑟(𝑡) and volatility term 𝜂(𝑡) of the stock become the constants, then the problem (7) is reduced to an investment and consumption problem with constant interest rate and constant volatility rate. In addition, we get 𝜋^∗_𝑡=𝑘(1−𝛿)𝜎²₁𝑋_𝑡,𝐶^∗(𝑡)=𝛼^1/(1−𝛿)𝑔⁻¹𝑋_𝑡 (55) under the conditions 𝛿<1 and 𝛿≠0, where 𝑔=𝑔𝑡,𝑟,𝜂=𝛼^1/(1−𝛿)𝑒^{𝜔(𝑇−𝑡)}−1𝜔+(1−𝛼)^1/(1−𝛿)𝑒^{𝜔(𝑇−𝑡)},𝜔=𝛿𝜂(𝛿−1)²⋅𝑘²2𝜎²₁+𝛿𝑟−𝛽1−𝛿.1−𝛿 (56)

Special Case 4. If 𝛼=0 and 𝛽=0, the problem (7) is degenerated into a dynamic asset allocation problem with stochastic interest rate and stochastic volatility. Therefore, the optimal investment strategy is given by 𝜋^∗_𝑡=𝑘(1−𝛿)𝜎²₁𝑋_𝑡+𝜎₂𝜎₁⋅𝜙(𝑡)𝑋_𝑡 (57) with the conditions 𝛿<Min{𝑎²𝜎²₁/(𝑎𝜎₁+𝑘𝜎₂)²,𝑐²/2𝜎²₀} and 𝛿≠0, where 𝜆_1,2=1𝜎²₂𝛿𝑎+⋅𝛿−1𝑘𝜎₂𝜎₁±1𝜎²₂−𝑎²+𝛿𝛿−1𝛿−1𝑎+𝑘𝜎₂𝜎₁²,𝜙(𝑡)=𝜆₁𝜆₂1−exp−(1/2)𝜎²₂𝜆₁−𝜆₂(𝑇−𝑡)𝜆₁−𝜆₂exp−(1/2)𝜎²₂𝜆₁−𝜆₂(.𝑇−𝑡) (58)

Special Case 5. If 𝛿=0, power utility will degenerate into a logarithmic utility. After easy calculation, we get 𝜙(𝑡)(𝑡)(𝑡)(),=𝛼=0,ℎ=0,Ψ=−𝛽𝑇−𝑡𝑔=𝑔𝑡,𝑟,𝜂𝛽1−𝑒^{−𝛽(𝑇−𝑡)}+(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}. (59)

As a result, the optimal investment and consumption strategy of the problem (7) in the logarithm utility case is given by 𝜋^∗_𝑡=𝑘𝜎²₁𝑋_𝑡,𝐶^∗(𝑡)=𝛼𝛼/𝛽1−𝑒^{−𝛽(𝑇−𝑡)}+(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}𝑋_𝑡. (60) The above results are in accordance with those under the logarithm utility maximizing criterion in the following subsection.

3.2. Logarithm Utility

Assume that the logarithm utility is given by 𝑈₁(𝑥)=𝑈₂(𝑥)=ln𝑥. (61) Then we conjecture a solution to (11) with the following form; 𝐻𝑡,𝑟,𝜂,𝑥=𝑢(𝑡)𝑒^−𝛽𝑡ln𝑥+𝑣𝑡,𝑟,𝜂, (62) where boundary conditions are given by 𝑢(𝑇)=1−𝛼 and 𝑣(𝑇,𝑟,𝜂)=0.

Putting (62) in (11), we get 𝑒^−𝛽𝑥ln𝑥𝑢^′(𝑡)−𝛽𝑢(𝑡)+𝛼+𝑣_𝑡+𝑟𝑢(𝑡)𝑒^−𝛽𝑡+(𝜃−𝑐𝑟)𝑣_𝑟+12𝜎²₀𝑟𝑣_𝑟𝑟+𝑏−𝑎𝜂𝑣_𝜂+12𝜎²₂𝜂𝑣_𝜂𝜂+𝑘²𝜂2𝜎²₁𝑢(𝑡)𝑒^−𝛽𝑡+(ln𝛼−1)𝛼𝑒^−𝛽𝑡−𝛼𝑒^−𝛽𝑡ln𝑢(𝑡)=0. (63) Eliminating the dependence on 𝑥, we have 𝑢^′(𝑡)(𝑡)(𝑇)𝑣−𝛽𝑢+𝛼=0,𝑢=1−𝛼,_𝑡+𝑟𝑢(𝑡)𝑒^−𝛽𝑡+(𝜃−𝑐𝑟)𝑣_𝑟+12𝜎²₀𝑟𝑣_𝑟𝑟+𝑏−𝑎𝜂𝑣_𝜂+12𝜎²₂𝜂𝑣_𝜂𝜂+𝑘²𝜂2𝜎²₁𝑢(𝑡)𝑒^−𝛽𝑡+(ln𝛼−1)𝛼𝑒^−𝛽𝑡−𝛼𝑒^−𝛽𝑡ln𝑢(𝑡)=0. (64)(65)

Solving (64), we get 𝑢(𝑡)=(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}−𝛼𝛽𝑒^{−𝛽(𝑇−𝑡)}−1. (66)

Suppose that the solution to (65) is of the structure 𝑣(𝑡)(𝑡)(𝑡)𝐷(𝑇)(𝑇)(𝑇)𝑡,𝑟,𝜂=𝐷+𝐸𝑟+𝐹𝜂,=𝐸=𝐹=0. (67) Putting (67) in (65), we derive 𝑟𝐸^′(𝑡)−𝑐𝐸(𝑡)+𝑢(𝑡)𝑒^−𝛽𝑡+𝜂𝐹^′(𝑡)(𝑡)+𝑘−𝑎𝐹²2𝜎²₁𝑢(𝑡)𝑒^−𝛽𝑡+𝐷^′(𝑡)+𝜃𝐸(𝑡)+𝑏𝐹(𝑡)+(ln𝛼−1)𝛼𝑒^−𝛽𝑡−𝛼𝑒^−𝛽𝑡ln𝑢(𝑡)=0. (68) By matching coefficients, we obtain the following three equations: 𝐸^′(𝑡)−𝑐𝐸(𝑡)+𝑢(𝑡)𝑒^−𝛽𝑡(𝑇)𝐹=0,𝐸=0,_′(𝑡)(𝑡)+𝑘−𝑎𝐹²2𝜎²₁𝑢(𝑡)𝑒^−𝛽𝑡(𝑇)𝐷=0,𝐹=0,_′(𝑡)+𝜃𝐸(𝑡)+𝑏𝐹(𝑡)+(ln𝛼−1)𝛼𝑒^−𝛽𝑡−𝛼𝑒^−𝛽𝑡ln𝑢(𝑡)=0,𝐷(𝑇)=0. (69)(70)(71)

Solving the above three equations (69)–(71), we obtain 𝐸(𝑡)=𝛼+𝛼𝛽−𝛽𝑒𝛽𝑐^−𝛽𝑇𝑒^{−𝑐(𝑇−𝑡)}−𝛼−1𝛽𝑒𝛽+𝑐^−𝛽𝑇𝑒^{−𝑐(𝑇−𝑡)}−𝑒^{𝛽(𝑇−𝑡)},𝐹(𝑡)=𝑘²2𝜎²₁⋅𝛼𝛽+𝛼−𝛽𝑒𝑎𝛽^−𝛽𝑇𝑒^{−𝑎(𝑇−𝑡)}−𝑘−1₂2𝜎²₁⋅𝛼𝛽𝑒𝑎+𝛽^−𝛽𝑇𝑒^{−𝑎(𝑇−𝑡)}−𝑒^{𝛽(𝑇−𝑡)},𝐷(𝑡)=𝜃^𝑇_𝑡𝐸(𝑠)𝑑𝑠+𝑏^𝑇_𝑡𝐹(𝑠)+()𝛼𝑑𝑠ln𝛼−1^𝑇_𝑡𝑒^−𝛽𝑠𝑑𝑠−𝛼^𝑇_𝑡𝑒^−𝛽𝑠ln𝑢(𝑠)𝑑𝑠. (72) Taking (62) into considering, we obtain 𝐻_𝑥𝐻_𝑥𝑥𝐻=−𝑥,^𝑥𝜂𝐻_𝑥𝑥=0,𝐶^∗(𝑡)=𝛼𝑢(𝑡)𝑥. (73)

Therefore, we can summarize the optimal investment and consumption strategy in the logarithm utility case as the following proposition.

Proposition 3. If utility function is given by 𝑈₁(𝑥)=𝑈₂(𝑥)=ln𝑥, the optimal investment and consumption strategy of the problem (7) is 𝜋^∗(𝑡)=𝑘𝜎²₁𝑋_𝑡,𝐶^∗(𝑡)=𝛼(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}−𝛼/𝛽𝑒^{−𝛽(𝑇−𝑡)}𝑋−1^𝑡. (74)

Remark 4. According to Proposition 3, we find the following. (i) The parameters 𝜃, 𝑐, and 𝜎₁ in the stochastic interest rate model have no influence on 𝜋^∗(𝑡) and 𝐶^∗(𝑡), (ii) The parameters 𝑏, 𝑎, and 𝜎₂ in the stochastic volatility model have no impact on 𝜋^∗(𝑡) and 𝐶^∗(𝑡) as well, (iii) The optimal amount 𝜋^∗(𝑡) invested in the stock depends only on the parameters 𝑘 and 𝜎₁. In addition, 𝜋^∗(𝑡) increases with respect to 𝑘 and decreases with respect to 𝜎₁, (iv) The optimal consumption rate 𝐶^∗(𝑡) increases with respect to 𝛼, for we can obtain 𝑑𝐶^∗(𝑡)=𝑒𝑑𝛼(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}−𝛼/𝛽^{−𝛽(𝑇−𝑡)}𝑒^{−𝛽(𝑇−𝑡)}−1²𝑋_𝑡>0. (75)

4. Numerical Analysis

In this section, we will provide a numerical example to illustrate how market parameters affect the dynamic behavior of the optimal investment and consumption strategy in the power utility case. Throughout the numerical analysis, unless otherwise stated, the basic parameters are given by 𝑐=0.15, 𝜎₀=0.2, 𝜃=0.25, 𝑟(0)=0.05, 𝜎₁=1.2, 𝑘=0.6, 𝜂(0)=0.36, 𝜎₂=0.8, 𝑏=1.0, 𝑎=0.6, 𝑡=0, 𝑇=1, 𝛼=0.4, 𝛽=0.1, 𝛿=−1, and 𝑥₀=100.

4.1. The Impact on the Optimal Investment Strategy

It can be seen from Figures 1 and 2 that we can summarize the effect of market parameters on the optimal investment strategy and provide some economic implications.(a1) The optimal investment strategy 𝜋^∗(𝑡) increases with respect to the parameter 𝑘. Recalling (3), 𝑘𝜂(𝑡) can be taken as the appreciation rate of the stock. Hence, as the parameter 𝑘 increases, then the expected income of the stock increases, which leads to the fact that the investor wishes to invest more amount of his wealth in the stock.  (a2)𝜋^∗(𝑡) increases with respect to the parameter 𝑎. The larger the value of 𝑎, the smaller the expected value of 𝜂(𝑡). This leads to the fact that the expected appreciation rate of the stock becomes less and the volatility of the stock will become much smaller. Due to the much less risk of the investment, the investor would invest more money in the stock.  (a3)𝜋^∗(𝑡) is decreasing in 𝜎₂. In fact, the larger the value of 𝜎₂, the higher the volatility term of 𝜂(𝑡), which leads to that the volatility of the stock is increasing. Therefore, the risk of investment is increasing accordingly and the investor invests the less amount in the stock.  (a4)𝜋^∗(𝑡) is increasing in 𝛿. According to the condition of Proposition 2, the value of 𝛿 should be less than 0.28. In the power utility case, the coefficient of risk aversion is denoted by 1−𝛿. It indicates that the investor can take less risk as the parameter 𝛿 is increasing. Therefore, the larger the value of 𝛿, the more amount the investor wishes to invest in the stock. On the other hand, when 𝛿=0, power utility is degenerated to logarithm utility. Therefore, we can summarize that the optimal investment strategy in the power utility case will be less than that in the logarithm utility case when 𝛿<0, while the optimal investment policy for power utility will be more than that for logarithm utility when 0<𝛿<0.28.

(a)

(b)

Figure 1 

The impact of 𝑘 and 𝑎 on the optimal investment strategy.

(a)

(b)

Figure 2 

The impact of 𝜎2 and 𝛿 on the optimal investment strategy.

4.2. The Impact on the Optimal Consumption Strategy

From Figures 3 and 4, we can summarize the following conclusions. (b1) The optimal consumption strategy 𝐶^∗(𝑡) increases with respect to the parameter 𝑘. In fact, when the value of 𝑘 becomes larger, the expected appreciation of the stock becomes larger as well. Therefore, the wealth of the investor will become much larger, which leads to the fact that the investor can consume the more amount of the wealth.  (b2)𝐶^∗(𝑡) is slowly decreasing in 𝑎. As the value of 𝑎 is increasing, the expected value of 𝜂(𝑡) is decreasing. It leads to the fact that the expected value of the wealth is decreasing. Therefore, the investor should consume less money.  (b3)𝐶^∗(𝑡) will slowly decrease as the parameter 𝜎₂ increases. As a matter of fact, the bigger the value of 𝜎₂, the less the amount invested in the stock. It means that the wealth of the investor is decreasing. Therefore, the investor should keep less consumption.  (b4)𝐶^∗(𝑡) will decrease with the parameter 𝛿. That is to say, the bigger the value of 𝛿, the less the risk aversion the investor needs to take and the less the investor can consume. Moreover, we conclude that the amount the investor can consume in the power utility case is more than that in the logarithm utility case when 𝛿<0, while the investor with power utility preference can consume less amount than the one with logarithm utility preference when 0<𝛿<0.28.

(a)

(b)

Figure 3 

The impact of 𝑘 and 𝑎 on the optimal consumption strategy.

(a)

(b)

Figure 4 

The impact of 𝜎2 and 𝛿 on the optimal consumption strategy.

5. Conclusions

In this paper, we investigate an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is assumed to be driven by the CIR model and the volatility of stock price is described by Heston's stochastic volatility model. The financial market consists of one risk-free asset and one risky asset. Our objective is to maximize the expected discounted utility of consumption and terminal wealth in the fixed finite investment horizon. By applying dynamic programming principle, we obtain the explicit expressions of the optimal investment and consumption strategies in the power and logarithm utility cases. We have also given a numerical example to illustrate the results obtained and analyzed the effect of market parameters on the optimal trading strategies.

In recent years, there has been much attention to the portfolio selection problems in stochastic environments. This is due to the facts that risk-free interest rate and volatility of the stock is not always fixed in most of real-world situations and the optimal portfolios in stochastic environments will make it more practical. Under some special assumptions, we obtain the closed-form solutions of the optimal trading strategies in stochastic interest rate and stochastic volatility situations. These assumptions include the following: (i) interest rate has no correlation with risky asset dynamics; (ii) Heston's stochastic volatility and risky asset dynamics are driven by the same Brownian motion; (iii) stock price is assumed to follow geometric Brownian motion, which is a special case of the constant elasticity of variance (CEV) model. If we relax these assumptions and consider some more general cases, it will lead to more sophisticated HJB equations and will have no ways to get the closed-form solutions of the optimal trading strategy. In future research, we will explore new approaches to tackle these problems.

Acknowledgments

The author is very grateful to many anonymous reviewers for their insightful comments and suggestions. This research is supported by the Higher School Science and Technology Development Foundation of Tianjin (no. 20100821) and the Humanities and Social Science Research Youth Foundation of Ministry of Education (no. 11YJC790006).