  We consider an illiquid financial market where a risk averse investor has to
liquidate a portfolio within a finite time horizon [0,T] and can trade
continuously at a traditional exchange (the "primary venue") and in a dark
pool. At the primary venue, trading yields a linear price impact. In the dark
pool, no price impact costs arise but order execution is uncertain, modeled by
a multi-dimensional Poisson process. We characterize the costs of trading by a
linear-quadratic functional which incorporates both the price impact costs of
trading at the primary exchange and the market risk of the position. The
liquidation constraint implies a singularity of the value function of the
resulting minimization problem at the terminal time T. Via the HJB equation and
a quadratic ansatz, we obtain a candidate for the value function which is the
limit of a sequence of solutions of initial value problems for a matrix
differential equation. We show that this limit exists by using an appropriate
matrix inequality and a comparison result for Riccati matrix equations.
Additionally, we obtain upper and lower bounds of the solutions of the initial
value problems, which allow us to prove a verification theorem. If a single
asset position is to be liquidated, the investor slowly trades out of her
position at the primary venue, with the remainder being placed in the dark pool
at any point in time. For multi-asset liquidations this is generally not the
case; it can, e.g., be optimal to oversize orders in the dark pool in order to
turn a poorly balanced portfolio into a portfolio bearing less risk.
