In this paper, we study a free boundary problem, which arises from an optimal trading problem of a stock that is driven by a uncertain market status process. The free boundary problem is a variational inequality system of three functions with a degenerate operator. The main contribution of this paper is that we not only prove all the four switching free boundaries are no-overlapping, monotonic and $C^{ \infty}$-smooth, but also completely determine their relative localities and provide the optimal trading strategies for the stock trading problem.
Categories: math.AP q-fin.MF q-fin.RM