Facilitation and Internalization Optimal Strategy in a Multilateral   Trading Context
  This paper studies four trading algorithms of a professional trader at a multilateral trading facility, observing a realistic two-sided limit order book whose dynamics are driven by the order book events. The identity of the trader can be either internalizing or regular, either a hedge fund or a brokery agency. The speed and cost of trading can be balanced by properly choosing active strategies on the displayed orders in the book and passive strategies on the hidden orders within the spread. We shall show that the price switching algorithms provide lower and upper bounds of the mixed trading algorithms. Especially, when the internalization premium is zero, an internalizing trader's optimal mixed trading strategy can be achieved among the set of price switching strategies. For both an internalizing trader and a regular trader, the optimal price switching strategy exists and is expressed in terms of the value function. A parallelizable algorithm to numerically compute the value function and optimal price switching strategy for the discretized state process is provided. 
Qinghua Li
q-fin.TR math.OC