  We study the optimal portfolio liquidation problem over a finite horizon in a
limit order book with bid-ask spread and temporary market price impact
penalizing speedy execution trades. We use a continuous-time modeling
framework, but in contrast with previous related papers (see e.g. [24] and
[25]), we do not assume continuous-time trading strategies. We consider instead
real trading that occur in discrete-time, and this is formulated as an impulse
control problem under a solvency constraint, including the lag variable
tracking the time interval between trades. A first important result of our
paper is to show that nearly optimal execution strategies in this context lead
actually to a finite number of trading times, and this holds true without
assuming ad hoc any fixed transaction fee. Next, we derive the dynamic
programming quasi-variational inequality satisfied by the value function in the
sense of constrained viscosity solutions. We also introduce a family of value
functions converging to our value function, and which is characterized as the
unique constrained viscosity solutions of an approximation of our dynamic
programming equation. This convergence result is useful for numerical purpose,
postponed in a further study.
