  We consider a finite-horizon market-making problem faced by a dark pool that
executes incoming buy and sell orders. The arrival flow of such orders is
assumed to be random and, for each transaction, the dark pool earns a per-share
commission no greater than the half bid-ask spread. Throughout the entire
period, the main concern is inventory risk, which increases as the number of
held positions becomes critically small or large. The dark pool can control its
inventory by choosing the size of the commission for each transaction, so to
encourage, e.g., buy orders instead of sell orders. Furthermore, it can submit
lit-pool limit orders, of which execution is uncertain, and market orders,
which are expensive. In either case, the dark pool risks an information
leakage, which we model via a fixed penalty for trading in the lit pool. We
solve a double-obstacle impulse-control problem associated with the optimal
management of the inventory, and we show that the value function is the unique
viscosity solution of the associated system of quasi variational inequalities.
We explore various numerical examples of the proposed model, including one that
admits a semi-explicit solution.
