We consider a portfolio execution problem where a possibly risk-averse agent needs to trade a fixed number of shares in multiple stocks over a short time horizon. Our price dynamics can capture linear but stochastic temporary and permanent price impacts as well as stochastic volatility. In general it is not possible to solve even numerically for the optimal policy in this model, however, and so we must instead search for good suboptimal policies. Our principal policy is a variant of an open-loop feedback control (OLFC) policy, and we show how the corresponding OLFC value function may be used to construct good primal and dual bounds on the optimal value function. The dual bound is constructed using the recently developed duality methods based on information relaxations. One of the contributions of this paper is the identification of sufficient conditions to guarantee convexity, and hence tractability, of the associated dual problem instances. That said, we do not claim that the only plausible models are th...