General Closed-Form Solutions to the Dynamic Optimization Problem in Incomplete Markets

In this paper, we provide general closed-form solutions to the incomplete-market random-coefficient dynamic optimization problem without the restrictive assumption of exponential or HARA utility function. Moreover, we explicitly express the optimal portfolio as a function of the optimal consumption and show the impact of optimal consumption on the optimal portfolio.


Introduction
Dynamic optimization has been used extensively in the economic and financial literature.Examples include incomplete markets, stochastic volatility and random coefficients models.The contemporary literature usually adopts random coefficient models (the parameters of the model are dependent on a random external economic factor) or non-tradable assets models.Examples include Bayraktar and Young [1], Bayraktar and Ludkovsky [2], Alghalith [3], Focardi and Fabozzi [4], and Fleming [5].
In order to derive explicit solution to the optimization problem, the previous studies relied exclusively on exponential or HARA utility functions.This assumption is restrictive and sometimes unrealistic, since other common and more appropriate functional forms exist.
In this paper, we relax the exponential or HARA utility assumption.In doing so, we derive general closed form solutions to the random-coefficient incompletemarket dynamic optimization problem without imposing restrictions on the functional form of utility.Furthermore, we explicitly derive a functional relationship between the optimal portfolio and optimal consumption and show the impact of consumption on the optimal portfolio.
In Section 2, we present the theoretical model and the results.In Section 3, we draw conclusions.

The Model
We consider a standard investment-consumption model, which includes a risky asset, a risk-free asset and a random external economic factor (see, for example, Fleming [5] for background information on the investment-consumption model).This implies a two-dimensional stan- , where    , is the augmentation of filtration.The risk-free asset price process is is the rate of return and s Y is the economic factor.The risky asset price process is given by are the rate of return and the volatility, respectively.The economic factor process is given by where 1   is the correlation factor between the two Brownian motions and     The wealth process is given by The investor's objective is to maximize the expected utility of wealth and consumption is a differentiable, bounded and strictly concave utility function.
The value function satisfies the Hamiltonian-Jacobi-Bellman PDE . Differentiating the above equation with respect π t and c t , respectively and rearranging, we obtain the optimal solutions Lemma.We can obtain an exact fixed-coeffiecient Taylor expansion of The solution yields and thus Now since ˆt c depends on the value of a , choose a specific value of a a  such that ˆt The above equation can be rewritten as where i b is a constant.Using the same procedure we obtain the following exact expansion of This is a general explicit formula that holds for any utility function.Moreover, this formula allows us to determine the impact of consumption on the portfolio.To show this This approach is empirically very convenient since the parameters in (4) b 1 , b 2 , b 4 and b 5 can be easily estimated using a nonlinear regression based on historical data.

Conclusion
This approach is superior to the existing approaches in several ways.First, we can obtain closed-form solutions without the assumption of exponential or HARA utility function.Secondly, we can easily obtain comparative statics results.For example, there is a trade-off between consumption and investment.Moreover, our approach can be easily utilized by future empirical studies.