  We model the behavior of three agent classes acting dynamically in a limit
order book of a financial asset. Namely, we consider market makers (MM),
high-frequency trading (HFT) firms, and institutional brokers (IB). Given a
prior dynamic of the order book, similar to the one considered in the
Queue-Reactive models [14, 20, 21], the MM and the HFT define their trading
strategy by optimizing the expected utility of terminal wealth, while the IB
has a prescheduled task to sell or buy many shares of the considered asset. We
derive the variational partial differential equations that characterize the
value functions of the MM and HFT and explain how almost optimal control can be
deduced from them. We then provide a first illustration of the interactions
that can take place between these different market participants by simulating
the dynamic of an order book in which each of them plays his own (optimal)
strategy.
