Some computations for optimal execution with monotone strategies

We study an optimal execution problem in the infinite horizon setup. Our financial market is given by the Black-Scholes model with a linear price impact. The main novelty of the current note is that we study the constrained case where the number of shares and the selling rate are non-negative processes. For this case we give a complete characterization of the value and the optimal control via a solution of a non-linear ordinary differential equation (ODE). Furthermore, we provide an example where the non-linear ODE can be solved explicitly. Our approach is purely probabilistic.