  We investigate the portfolio execution problem under a framework in which
volatility and liquidity are both uncertain. In our model, we assume that a
multidimensional Markovian stochastic factor drives both of them. Moreover, we
model indirect liquidity costs as temporary price impact, stipulating a power
law to relate it to the agent's turnover rate. We first analyze the regularized
setting, in which the admissible strategies do not ensure complete execution of
the initial inventory. We prove the existence and uniqueness of a continuous
and bounded viscosity solution of the Hamilton-Jacobi-Bellman equation, whence
we obtain a characterization of the optimal trading rate. As a byproduct of our
proof, we obtain a numerical algorithm. Then, we analyze the constrained
problem, in which admissible strategies must guarantee complete execution to
the trader. We solve it through a monotonicity argument, obtaining the optimal
strategy as a singular limit of the regularized counterparts.
