  Assuming geometric Brownian motion as unaffected price process $S^0$,
Gatheral & Schied (2011) derived a strategy for optimal order execution that
reacts in a sensible manner on market changes but can still be computed in
closed form. Here we will investigate the robustness of this strategy with
respect to misspecification of the law of $S^0$. We prove the surprising result
that the strategy remains optimal whenever $S^0$ is a square-integrable
martingale. We then analyze the optimization criterion of Gatheral & Schied
(2011) in the case in which $S^0$ is any square-integrable semimartingale and
we give a closed-form solution to this problem. As a corollary, we find an
explicit solution to the problem of minimizing the expected liquidation costs
when the unaffected price process is a square-integrable semimartingale. The
solutions to our problems are found by stochastically solving a finite-fuel
control problem without assumptions of Markovianity.
