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

On regularized optimal execution problems and their singular limits