  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.
