  We study a constrained stochastic control problem with jumps; the jump times
of the controlled process are given by a Poisson process. The cost functional
comprises quadratic components for an absolutely continuous control and the
controlled process and an absolute value component for the control of the jump
size of the process. We characterize the value function by a "polynomial" of
degree two whose coefficients depend on the state of the system; these
coefficients are given by a coupled system of ODEs. The problem hence reduces
from solving the Hamilton Jacobi Bellman (HJB) equation (i.e., a PDE) to
solving an ODE whose solution is available in closed form. The state space is
separated by a time dependent boundary into a continuation region where the
optimal jump size of the controlled process is positive and a stopping region
where it is zero. We apply the optimization problem to a problem faced by
investors in the financial market who have to liquidate a position in a risky
asset and have access to a dark pool with adverse selection.
