Joint Sharpe-optimal optimization of execution schedule and window

Determine the jointly optimal execution fill density and execution window by maximizing the Sharpe ratio of the realized profit-and-loss functional over all admissible fill densities, rather than optimizing the window and schedule in separate stages.

Background

The paper first selects an execution window by maximizing the Sharpe ratio under a uniform execution schedule, and then selects a schedule by minimizing a separate convex envelope-cost functional. The authors explicitly note that this two-stage procedure is not a joint Sharpe-ratio optimization. A fully joint formulation would optimize the fill density directly, accounting simultaneously for expected alpha realization, concave impact, and the covariance structure of the fractional-Brownian price process. The resulting objective is a nonconvex ratio problem, and the paper leaves its solution unresolved.