  We consider the problem of portfolio optimization in a simple incomplete
market and under a general utility function. By working with the associated
Hamilton-Jacobi-Bellman partial differential equation (HJB PDE), we obtain a
closed-form formula for a trading strategy which approximates the optimal
trading strategy when the time horizon is small. This strategy is generated by
a first order approximation to the value function. The approximate value
function is obtained by constructing classical sub- and super-solutions to the
HJB PDE using a formal expansion in powers of horizon time. Martingale
inequalities are used to sandwich the true value function between the
constructed sub- and super-solutions. A rigorous proof of the accuracy of the
approximation formulas is given. We end with a heuristic scheme for extending
our small-time approximating formulas to approximating formulas in a finite
time horizon.
