  In this paper, we study optimal liquidation problems in a randomly-terminated
horizon. We consider the liquidation of a large single-asset portfolio with the
aim of minimizing a combination of volatility risk and transaction costs
arising from permanent and temporary market impact. Three different scenarios
are analyzed under Almgren-Chriss's market impact model to explore the relation
between optimal liquidation strategies and potential inventory risk arising
from the uncertainty of the liquidation horizon. For cases where no closed-form
solutions can be obtained, we verify comparison principles for viscosity
solutions and characterize the value function as the unique viscosity solution
of the associated Hamilton-Jacobi-Bellman (HJB) equation.
