2016/11/28 by Kumar, Rohini, Nasralah, Hussein
#60G99 (Secondary) #91G10 #93E20 (Primary) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Portfolio Management (q-fin.PM) #Probability (math.PR)
paper · doi:10.48550/arxiv.1611.09300
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.