  We consider an optimal trading 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 L\'evy
processes. Effects on the best bid price arising from the arrival of limit buy
orders at more favourable prices, the incoming market sell orders potentially
walking the book, and deriving from the cancellations of limit sell orders at
the best ask price are incorporated in the proposed price dynamics. A permanent
impact that occurs when 'lit' pool trades cannot be avoided is built in, and an
instantaneous impact that models the slippage, to which all 'lit' exchange
trades are subject, is also considered. 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
trading strategy in both venues. Since the mid-price is taken from the
exchange, the dynamics 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 partial integro differential equation. We present two explicit
examples of the price and the spread models, and derive the associated optimal
trading strategy numerically. We discuss the various degrees of the agent's
risk aversion and further show that roundtrips, i.e. posting the remaining
inventory in the dark pool at every point in time, are not necessarily
beneficial.
