Monotone optimal policies in portfolio liquidation problems

This work considers the problem of optimal liquidation of a single risky asset portfolio as a denumerable Markov Decision Processes (MDP) control problem. The model is defined over discrete time, state, and action sets, and the optimal liquidation strategy is the solution to Bellman´s equation. It is shown that the optimal strategy is monotone in the number of shares owned, the time remaining to liquidation, and the price of the underlying asset. This structural result can be exploited to estimate the optimal policy via the simultaneous perturbation stochastic approximation (SPSA) algorithm. Therefore, the optimal policy can be estimated without knowledge of the parameters of the model.