标题： 连续时间内的暗池组合清算 显示英文标题

标题： Portfolio liquidation in dark pools in continuous time

摘要： 我们考虑一个流动性不足的金融市场，其中风险厌恶的投资者必须在有限的时间范围[0,T]内清算投资组合，并且可以在传统交易所（“主要场所”）和暗池中连续交易。 在主要场所，交易会产生线性价格影响。 在暗池中，不会产生价格影响成本，但订单执行是不确定的，由多维泊松过程建模。 我们通过一个线性二次泛函来描述交易成本，该泛函结合了在主要交易所交易的价格影响成本和头寸的市场风险。 清算约束意味着所得到的最小化问题的价值函数在终端时间T处存在奇点。通过HJB方程和二次假设，我们得到了一个价值函数的候选解，该解是矩阵微分方程初始值问题解序列的极限。 我们通过使用适当的矩阵不等式和Riccati矩阵方程的比较结果来证明这个极限存在。 此外，我们还得到了初始值问题解的上下界，这使我们能够证明一个验证定理。 如果要清算单一资产头寸，投资者会在主要场所缓慢地退出头寸，其余部分则在任何时间点放入暗池。 对于多资产清算，通常不是这样；例如，为了将不平衡的头寸转变为风险较小的头寸，最优策略可能是放大暗池中的订单规模。 显示英文摘要

摘要： 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.