Special Mathematics ColloquiumDate/Time/Room: Friday (3/15/2002) at 2:30pm in 304 Pickard Hall
Department of Mathematics and Statistics, SUNY at Stony Brook
``Portfolio Optimization in Mathematical Finance and PDE with Gradient Constraints''Abstract: We consider a dynamical portfolio optimization model. The portfolio consists of several risky assets (Stocks) and one risk-free asset (Bond). The rate of return on Bond is constant while the rate of return of Stocks is governed by a multidimensional logarithmic Brownian motion in the spirit of the classical approach introduced by Merton and widely used in the Black-Scholes analysis. Funds can be transferred from one asset to another, however such transaction involves penalty (brokerage fee) proportional to the size of the transaction. The objective is to find the policy which maximizes the expected rate of growth of funds.
The main probabilistic tool in the solution of this problem is the singular control theory. the analytical part of the solution of the control problem is related to the free boundary problem for the elliptic PDE with gradient constraints, similar to the ones encountered in elastic-plastic torsion problems. The existence of the classical solution cannot be proved in general but one can show an existence of a viscosity solution to these equations.
The optimal policy is to keep the vector of fractions of funds invested in different assets in an optimal (a priori) unknown boundary. This boundary is found from a solution to a variational inequalities related to a second order elliptic linear partial differential equation with gradient constraints. We show how to find these boundaries explicitly in the case of one risky and one risk-free asset when the problem becomes one dimensional. The free boundary problem can be reduced in this case to the classical Stephan problem for an ODE.