In this review article, we present recent work on the regularity of dynamical market impact models and their associated optimal order execution strategies. In particular, we address the question of the stability and existence of optimal strategies, showing that in a large class of models, there is price manipulation and no well-behaved optimal order execution strategy. We also address issues arising from the use of dark pools and predatory trading.
Gatheral, Jim and Schied, Alexander, Dynamical Models of Market Impact and Algorithms for Order Execution (March 21, 2012). Available at SSRN: http://ssrn.com/abstract=2034178 or http://dx.doi.org/10.2139/ssrn.2034178