  We start with a stochastic control problem where the control process is of
finite variation (possibly with jumps) and acts as integrator both in the state
dynamics and in the target functional. Problems of such type arise in the
stream of literature on optimal trade execution pioneered by Obizhaeva and Wang
(models with finite resilience). We consider a general framework where the
price impact and the resilience are stochastic processes. Both are allowed to
have diffusive components. First we continuously extend the problem from
processes of finite variation to progressively measurable processes. Then we
reduce the extended problem to a linear quadratic (LQ) stochastic control
problem. Using the well developed theory on LQ problems we describe the
solution to the obtained LQ one and trace it back up to the solution to the
(extended) initial trade execution problem. Finally, we illustrate our results
by several examples. Among other things the examples show the Obizhaeva-Wang
model with random (terminal and moving) targets, the necessity to extend the
initial trade execution problem to a reasonably large class of progressively
measurable processes (even going beyond semimartingales) and the effects of
diffusive components in the price impact process and/or in the resilience
process.
