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We study an optimal liquidation problem with multiplicative price impact inwhich the trend of the asset's price is an unobservable Bernoulli randomvariable. The investor aims at selling over an infinite time-horizon a fixedamount of assets in order to maximize a net expected profit functional, andlump-sum as well as singularly continuous actions are allowed. Our mathematicalmodelling leads to a singular stochastic control problem featuring afinite-fuel constraint and partial observation. We provide the completeanalysis of an equivalent three-dimensional degenerate problem under fullinformation, whose state process is composed of the asset's price dynamics, theamount of available assets in the portfolio, and the investor's belief aboutthe true value of the asset's trend. The optimal execution rule and theproblem's value function are expressed in terms of the solution to a trulytwo-dimensional optimal stopping problem, whose associated belief-dependentfree boundary $b$ triggers the investor's optimal selling rule. The curve $b$is uniquely determined through a nonlinear integral equation, for which wederive a numerical solution through an application of the Monte-Carlo method.This allows us to understand the sensitivity of the problem's solution withrespect to the relevant model's parameters as well as the value of informationin our model.

Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return