Optimal execution with multiplicative price impact and incomplete information on the return

We study an optimal liquidation problem with multiplicative price impact in which the trend of the asset's price is an unobservable Bernoulli random variable. The investor aims at selling over an infinite time-horizon a fixed amount of assets in order to maximize a net expected profit functional, and lump-sum as well as singularly continuous actions are allowed. Our mathematical modelling leads to a singular stochastic control problem featuring a finite-fuel constraint and partial observation. We provide the complete analysis of an equivalent three-dimensional degenerate problem under full information, whose state process is composed of the asset's price dynamics, the amount of available assets in the portfolio, and the investor's belief about the true value of the asset's trend. The optimal execution rule and the problem's value function are expressed in terms of the solution to a truly two-dimensional optimal stopping problem, whose associated belief-dependent free boundary b triggers the investor's optimal selling rule. The curve b is uniquely determined through a nonlinear integral equation, for which we derive a numerical solution allowing to understand the sensitivity of the problem's solution with respect to the relevant model's parameters.