  We develop a dynamic trading strategy in the Linear Quadratic Regulator (LQR)
framework. By including a price mean-reversion signal into the optimization
program, in a trading environment where market impact is linear and stage costs
are quadratic, we obtain an optimal trading curve that reacts opportunistically
to price changes while retaining its ability to satisfy smooth or hard
completion constraints. The optimal allocation is affine in the spot price and
in the number of outstanding shares at any time, and it can be fully derived
iteratively. It is also aggressive in the money, meaning that it accelerates
whenever the price is favorable, with an intensity that can be calibrated by
the practitioner. Since the LQR may yield locally negative participation rates
(i.e round trip trades) which are often undesirable, we show that the
aforementioned optimization problem can be improved and solved under positivity
constraints following a Model Predictive Control (MPC) approach. In particular,
it is smoother and more consistent with the completion constraint than putting
a hard floor on the participation rate. We finally examine how the LQR can be
simplified in the continuous trading context, which allows us to derive a
closed formula for the trading curve under further assumptions, and we document
a two-step strategy for the case where trades can also occur in an additional
dark pool.
