Optimal stock selling based on the global maximum

We aim to determine an optimal stock selling time to minimize the expectation of the square error between the selling price and the global maximum price over a given period. Assuming that stock price follows the geometric Brownian motion, we formulate the problem as an optimal stopping time problem or, equivalently, a variational inequality problem. We provide a partial differential equation (PDE) approach to characterize the resulting free boundary that corresponds to the optimal selling strategy. The monotonicity and smoothness of the free boundary are addressed as well.