This paper studies how to sell a portfolio of several assets in dark pools when the people who might trade with you could have better information. Dark pools are trading venues that hide order details until a trade happens. They reduce the price impact of big trades but raise the risk of adverse selection — the risk that a counterparty trades because they know something you do not. The authors build a model that combines these features for multiple, possibly correlated assets and study the optimal way to liquidate a position.

The authors set up a continuous-time liquidation problem where a trader can post active orders in the public exchange and passive orders in dark pools. They model dark-pool executions as jumps of independent Poisson processes. The cost function includes three parts: the usual temporary price impact from trading in the public market (leading to quadratic costs), a penalty for holding inventory, and a quadratic, reduced-form penalty for adverse selection in the dark pool. To keep the model tractable, they assume dark-pool orders can be updated continuously, executions are all-or-nothing (no partial fills), and adverse selection is captured by a quadratic cost matrix.

Mathematically, the control problem leads to a matrix-valued backward stochastic differential equation (BSDE) with jumps and a singular terminal condition. “Backward” here means the equation is solved backward from the end of the trading horizon; the singular terminal condition arises from the requirement that holdings must be zero at the final time. The paper’s main technical achievements are proving that this BSDE has a solution (existence) and that the solution is unique. To get these results the authors use a linearization idea to build a comparison principle, approximate the singular terminal condition by truncated problems, and derive sharp a-priori estimates to pass to the limit. They emphasize that uniqueness results of this type are rare in the multidimensional, singular setting even without jumps.