We consider an optimal execution problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modelled by functions of the market activity. We consider a random drift so to include the impact of market orders, and we describe the arrival of limit orders and of order cancellations by means of Poisson random measures. In the situation where the trades take place only in the exchange, we find that the optimal execution strategy depends significantly on the resilience of the limit order book. We assume that the trading price in the dark pool is the mid-price and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal order-size placement in both venues. Since the mid-price is taken from the exchange, the resilience of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the value function can be characterised by the unique continuous viscosity solution to the associated system of partial integro differential equations. We present a numerical example of which model parameters are analysed in detail.
