The Optimal Mean–Variance Selling Problem With Finite Horizon

The optimal mean–variance selling problem seeks to determine a dynamically optimal stopping time in the nonlinear problem , where is a geometric Brownian motion with strictly positive drift, the supremum is taken over stopping times of , and is a given and fixed constant. The solution to the problem is known when the horizon is infinite, however, the method of proof developed to solve the problem in that case is not applicable in the case when the horizon is finite. In this paper, we develop a new method of proof, which solves the problem when the horizon is finite. In this way, we find that the dynamically optimal stopping time is given by , where the function can be characterized as a unique solution to a nonlinear Volterra integral equation. We also prove that the dynamically optimal stopping time satisfies the smooth fit principle. To our knowledge, this is the first time that such a nonlinear phenomenon of “dynamic smooth fit” has been derived in the literature. MSC2020 Classification : 60G40, 35R35, 60J65, 90C30, 45G10, 91B06