Optimal Mean-Variance Selling Strategies

where t runs from 0 onwards, the supremum is taken over stopping times τ of X , and c > 0 is a given and fixed constant. Using direct martingale arguments we first show that when μ ≤ 0 it is optimal to stop at once and when μ ≥ σ/2 it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for 0 < μ < σ/2 can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given by