  We study optimal trade execution strategies in financial markets with
discrete order flow. The agent has a finite liquidation horizon and must
minimize price impact given a random number of incoming trade counterparties.
Assuming that the order flow $N$ is given by a Poisson process, we give a full
analysis of the properties and computation of the optimal dynamic execution
strategy. Extensions, whereby (a) $N$ is a fully-observed regime-switching
Poisson process; and (b) $N$ is a Markov-modulated compound Poisson process
driven by a hidden Markov chain, are also considered.
  We derive and compare the properties of the three cases and illustrate our
results with computational examples.
