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Abstract: 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 N is a Markov-modulated compound Poisson process are also considered. We derive and compare the properties of the various cases and illustrate our results with computational examples.