Global Closed-form Approximation of Free Boundary for Optimal Investment\n Stopping Problems

In this paper we study a utility maximization problem with both optimal\ncontrol and optimal stopping in a finite time horizon. The value function can\nbe characterized by a variational equation that involves a free boundary\nproblem of a fully nonlinear partial differential equation. Using the dual\ncontrol method, we derive the asymptotic properties of the dual value function\nand the associated dual free boundary for a class of utility functions,\nincluding power and non-HARA utilities. We construct a global closed-form\napproximation to the dual free boundary, which greatly reduces the\ncomputational cost. Using the duality relation, we find the approximate\nformulas for the optimal value function, trading strategy, and exercise\nboundary for the optimal investment stopping problem. Numerical examples show\nthe approximation is robust, accurate and fast.\n