Legendre Transform-Dual Solution for a Class of Investment and Consumption Problems with HARA Utility

This paper provides a Legendre transform method to deal with a class of investment and consumption problems, whose objective function is to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. Assume that risk preference of the investor is described by hyperbolic absolute risk aversion (HARA)utility function,which includes power utility, exponential utility, and logarithmutility as special cases.The optimal investment and consumption strategy forHARA utility is explicitly obtained by applying dynamic programming principle and Legendre transform technique. Some special cases are also discussed.


Introduction
The investment and consumption problem was originated from the seminal work of Merton [1,2], who first used stochastic optimal control to deal with a continuous-time portfolio selection problem with consumption behavior and obtained the closed-form solution of the optimal investment and consumption strategy under power utility and logarithm utility.In the following years, many scholars began to pay attention to the investment and consumption problems and obtained many research results.One can refer to the papers of Chacko and Viceira [3], Liu [4], Noh and Kim [5], Chang and Rong [6], and Chang et al. [7].But these results were generally achieved under power utility, which was taken as a special case of hyperbolic absolute risk aversion (HARA) utility function.
HARA utility includes power utility, exponential utility, and logarithm utility as special cases.From the expression of HARA utility, we can see that HARA utility has more complicated structure than other utility functions, which leads to that there is a little work on portfolio selection problems with HARA utility in the existing literature.As a matter of fact, Grasselli [8] investigated a portfolio selection problem with HARA utility in the stochastic interest rate environment and verified that the optimal policy corresponding to HARA utility converges to the one corresponding to exponential utility and logarithm utility.Jung and Kim [9] provided a Legendre transform method to deal with an optimal portfolio model with HARA utility.
In recent years, Legendre transform is often used to deal with the complicated portfolio selection problems, for example, the optimal investment problems under the CEV model.One can refer to the work of Jonsson and Sircar [10], Xiao et al. [11], and Gao [12,13].The advantage of Legendre transform is to transform the nonlinear HJB equation into a linear dual one and the structure of the solution under HARA utility is easy to conjecture.Therefore, it is very possible for us to use Legendre transform to obtain the optimal investment strategy of more complicated portfolio selection problems.
In this paper, we consider a class of investment and consumption problems, whose objective function is to maximize the expected discount utility of intermediate consumption and terminal wealth, and wish to obtain the optimal investment and consumption strategy for HARA utility.We provide a Legendre transform method to deal with this problem and obtain the explicit expression of the optimal investment and consumption strategy under HARA utility.Some special cases are also discussed.There are two main 2 Mathematical Problems in Engineering contributions in this paper.(i) We study a class of investment and consumption problems, whose objective function is more complicated than that of the optimal investment problems.In other words, we not only hope to obtain the optimal investment strategy, but also hope to obtain the optimal strategy of consumption.(ii) We obtain the explicit expression of the optimal investment and consumption strategy under HARA utility, which has rarely been studied in the optimal portfolio selection problems.It is all well known that most of portfolio selection problems are studied under power utility or exponential utility.
The structure of this paper is as follows.Section 2 formulates a class of investment and consumption problems with HARA utility, which wish to maximize the expected discount utility of intermediate consumption and terminal wealth.Section 3 uses Legendre transform to obtain the dual one of the HJB equation.Section 4 obtains the optimal investment and consumption strategy for HARA utility and Section 5 concludes the paper.
Assume that the financial market is composed of  + 1 assets, which are continuously traded on [0, ].One is the risk-free asset (i.e., a bond), whose price at time  is denoted by  0 ().Then  0 () satisfies the following equation: where () > 0 is the risk-free interest rate.
Assume that the set of all admissible investment and consumption strategies ((), ()) is denoted by Γ = {((), ()) : 0 ⩽  ⩽ }.In this paper, we assume that the investor wishes to maximize the following objective function: where  1 () and  2 () are all utility functions and  > 0 is the subjective discount rate.The parameter  determines the relative importance of intermediate consumption and terminal wealth.When  = 0, expected utility only depends on terminal wealth and the problem ( 5) is reduced to an asset allocation problem.
In this paper, we choose hyperbolic absolute risk aversion (HARA) utility function for our analysis.HARA utility function with parameters , , and  is given by As a matter of fact, HARA utility function recovers power utility, exponential utility, and logarithm utility as special cases.
For HARA utility, it is very difficult to directly conjecture the form of the solution of (14).Therefore, we introduce the following Legendre transform technique.Definition 2. Let  : R  → R be a convex function.Legendre transform can be defined as follows: and then the function () is called the Legendre dual function of () (cf.Jonsson and Sircar [10], Xiao et al. [11], and Gao [12,13]).
If () is strictly convex, the maximum in (15) will be attained at just one point, which we denoted by  0 .We can attain at the unique solution by the first-order condition: So we have Following Jonsson and Sircar [10], Xiao et al. [11], and Gao [12,13], Legendre transform can be defined by where  > 0 denotes the dual variable to .The value of  where this optimum is attained is denoted by (, ); so we have The relationship between Ĥ(, ) and (, ) is given by Hence, we can choose either one of two functions (, ) and Ĥ(, ) as the dual function of (, ).In this paper we choose (, ).Moreover, we have Differentiating ( 21) with respect to  and , we get (24) Differentiating ( 24) with respect to  and using (20), we derive with boundary condition given by (, ) = U−1 2 (/((1 − ) − )).

The Optimal Investment and Consumption Strategy
Applying ( 22) to ( 13), we get Assume that a solution (, ) of (25) with terminal condition (26) is given by Then the partial derivatives are derived as follows: Plugging ( 27)-( 31) into (25), we obtain Further, (32) can be decomposed into the following two equations: Solving (33), we get where Applying (20), ( 22), (28), and (30), we have Meantime, (27) can be rewritten as On the other hand, according to (, ) =  and (28), we derive Considering   = , we obtain the following optimal value function: Summarizing what is mentioned above and considering () = , we can draw the following conclusions.Theorem 3.Under HARA utility (6), the optimal investment and consumption strategy of the problem (5) is given by where  1 () and  2 () are given by (34) and (35), respectively.
The proof is completed.
In particular, we can obtain the following three special cases.